The University of
Warwick
THTCRY OF
COMPUTATON
REPORT
No. I
DECISlON PROCEDURES FOR FAMILIES
DETERMINI
.}
Department
STIC
OF
PUSHDO|'IN AUTOMATA
LESLIE G. VALIANT
of
University of
Computer
Warwick
COVENTRY CV4 7AL
ENGLAN D.
Science
August
1973
DECISION PROCEDI'RES FOR FA},TILIES OF
DETERMINISTIC PUSHDOIIN AUTOI'{ATA
Leslie G. Valiant
Department
of
Computer Science,
l.Ian"rick,
Coventryl England.
Itniversity of
A dissertation submitted for the degree of Doctor
of Philosophy. July
1973.
PREFACE
I ehould like to thank my superviaor, Michael Patergon for
hie continued intereat
discuseions
and encouragement and
for the numerous
nith hirn that provided the etinulation for this
research. The contente of Chapter 5 is the reeult of work
done
jointly with hin.
I
am
aleo grateful to Albert Meyer for bringing the regularity
problen to uy attentionr to the Science Research Council for
financial support,
this theeie.
and
to Jill Pladdys and Roeeuary llent for typing
ABSTRACT
The existence and corrpLexity
fanilies of deterministic
of decision
pushdown automata
procedures for
are investigated, with
special eurphasis on positive decidability results for those questions,
such as equivalence, which are known
deterninistic restriction is
The equivalence problem
to
become undecidable when the
removed.
is
proved decidable
for the following
three determini.stic families, all of which are already extensive
enough
to have undecidable inclusion
(a) nonsingular
automata
problems:
- a realtirne subfamily, which
the largest corresponding classes with previously
extends
known
equival-ence tests,
(b) finite-turn
aut,omata
- characterised by having a bound
the nrmrber of times the direction of the stack
on
rnovement can
change, and
(c) one-counter
aLphabet
tlhe prubLem
automata
to just
- defined by restricting the stack
one symbol.
of whether a )-anguage defined by a rnachine in
one
fani.ly1 can be recognised by one in another, is a convenienE
formulation of numerous decidabLe or potentially decidable queseions,
l,Ie ehow
that
such questions as whether a detennisusc:Le conte:rt-free
language can be recognised by a noachine
named
in
any one
of the above
classes, muet be, if decidable at all, at least as difficult
to decide as wheeher
such a l-anguage
regularity test of Stearns,
is regular.
We re-exasnine the
and obtain an improved
algorithm.
We
do
this by reducing by an exponential order the upper
possibLe state conplexity
pushdoltn aut.omata
bound on the
of regular sets recognised by deterministic
of a given size, to a level close to
one knovn
to be achievable.
I{e pursue an application
problem.
We
of this analysis to a schema theoretic
coneider the succinctness with which certain functional
schemas can be used
to
e)rpress equivalent large fLowchart schemas, and
obtain closely matching upper and lower bounds for a measure of this.
CONTENTS
Abstract
Introduction
I Prelininaries
1:
Chapter
Determinis tic Pushdown Autonat'a
Elementary ProPerties
Subf aniLi.es
Some Famil.y ProPerties
1.1
T,2
1.3
1.4
1.5
1.5
L.7
1.8
Derivations
complexity Measures
Tradeoffs in DescriPtion
Famil-y RelationshiPs
Chapter
2,L
2,2
2.3
2.4
2.5
II
DEFINITIONS
2:
Introduction
Emptiness
Finiteness
A NotmaL Form
Totality
chapter 3 : A
Nonsingutar Automata
Al"ternate Stacking
Main Results
UndecidabilitY of Inclueion
A Ccnjeeture about Rg
4.1
4.2
4.3
4.4
4.5
4,6
5:
2L
23
27
29
30
33
34
37
39
42
44
46
47
54
57
FINITE-TURNAIJT0UATA
lntroduction
Definitions
Proof Strategy
ParalLel Stacking
Existence of Verifying Machine
The Decision Procedure
Chapter
5.1
5.2
5,3
5.4
4:
15
18
REALTIME FAMILY
Introduction
C:hepter
t5
EASILY DECIDABLE PROBLEMS
The EquivaLence ProbLeg
3.1.
3,2
3.3
3.4
3.5
3.6
9
L2
59
60
63
64
67
72
ONE COUNTER AIEOI'{AIA
Introducrion
PrelioinarY Results
ProprietY
Decision Procedure
75
77
82
87
III
Containment Probleme
Chapter
l6,2
6.3
6.
IV
Asl
Introduction
93
94
101
Conments
7:
TIIE REGULARITY
PROBLEM
Introduction
NuLl--transparency and {,-invisibiLity
Main Theorem
Bounds for SubfamiLies
Time Complexity
Appli"catlon to
Chapter
8. 1
8,2
8.3
8.4
RELATIVE COMPLEXITY
Results
Chapter
7,L
7.2
7,3
7.4
7.5
6:
90
6
L2L
Schenas
e MONADIC FUNCTIONAI
104
105
111
1L5
119
SCHEMAS
Introdr.rction
Evatuating Pushdown Automata
t23
t25
r!B-S.Autonata
Bounds on Succinctness
L27
L29
Conclusion
L34
References
136
i.
INTRODUCTION
To understand the meaning
have
of an executable
computer program rre
to relate its finite specification to its possibly infinite
distinct behaviours
when applied
general ways of doing
to different inputs. Finding
this is a prirnary
goal.
in the theoretical
study of computaLions. The specific aims of such investigations are
to gain insight into the computational behaviour of whole classes of
prograns, and hence to be abl-e to analyse instances of various
particular problems for any
program
within these classes.
The
recursive decidabiLity of the problems concerned is an important
criterion of the practical viability of
such
analysis: the existence
of an effective decision procedure capable of determining the truth
of any instance of a particuLar question for a given
cLass of
prograns, uleans not only that this analysis can be automated, but
usually atso that an a priori bound can be pLaced on its difficulty.
In contrast, if the
problenr
is undecidable for the class, that is,
there is no effective pr'rcedure for soLving it in general, then new
instances of it can always be found for which the solution is more
difficult than before
and requires
further creative effsrt"
The courpuLaurons we shaLl" study are those
that
can be carried
out by abstract deter,"ntinisti.e autocnieta rhat have onLy a
pr:shdown
stack and a fini.te*state control for storage. This is a formalisation
of the stack coniellt lrhi-cbE i.s wi.dei"y
for
exampLe i.n
used
in practical
wricing syntax anaJ-ysers, or in
progrannning,
implementing
reeursion in compi.Lers. Consequently our resuLts will relate to
sreas outside autoloata ttreory, such as the syntax of programring
nanguages, and cert.ai.n f,crmal"
wdels of recursive
programs.
2
The observation primaritr"y rnr:ti."uating r:ur worit i-s
these f,amiiies
of
autcmata are wide enough
practical computations,
behaviour can be shown
tlrat,
to be reievant
whil-e
t.o
of the important questions atrout their
to be dacid.abl.e. 'Ihis is a very rare
rnany
conjunction of, attrributes for comprrtational rnodels so far sttrclied.
It is generally found that for abstract
useful classes of
shown
programs e
models designed
to
describe
nearly all the key properties can be
to be undecidable. For exarnple, if
rse extend any
of
the
families we strall consider by introducing nondetern:inism, or
a1
bv
lowing a second staek, then rve already lase the deci.dability
properties we desire most.
The
actual
nnachines we
investigate are deterrninistic
automata (dpda), which have been widely studied
pushdown
beforei'2'3,
anci
cerLain restricticns cf them. These are all accePtors in the sense
lhat the input consists of a sequence of
symbols on
a Eape, and Lhe
output of ei.ther a tyest or a tnc,t depending on Lhe configurations
of the machine afrer the input tape has been scanned. Thus each
naehine defines a set
of strings of
syrnbols,
or a language, namely
that consisting of exactly the sErings that are accepted, i.e. lead
ro a ryes?. These l.anguages are all context-free, 'and have the
additional useful prcperty that they can all be parsed easily in
linear
cime.
We
note that this input-outpur behaviour is not qrrite
restrictive as i"t would appear.
superficially
syurbol
for
Problems
more general rnachines
each symhol read
t0
do
as
with related but
(e.g. transducers which output a
) can often be reduced to
questions about acceptors and languages.
correspcnding
3
The decision problem
these families
of
automata
with which rre are most concerned for
is that of equivalence, i.e. is there
effective test co determine whether
an
two rnachines perform equivalent
computations? The existence of such a test has the practicaL sig-
nificance that it provides a most convenient model-independent
mechanism
for verifying correctness.
which is optinised in
and we want
some
Thus
if
we have an automaton
rray, and therefore perhaps complicated,
to prove that it will perform exactl-y as it is
then we could
build a second
automaton which
intended,
is nore perspicuous in
structure but possibly othervise unsuitable, and test the
Ewo
for
equi.vatence. Other criteria of correctness would, in contrast,
necessitate that a new distinct language for describing the intended
computation be introduced and
The inclusion problem
related to the automata formal-ism.
(i.e. to decide whether one language is
a subset of another ) is related to the equivaLence problem in that
any procedure
for deciding the former would lead directly to
the Latter. Thus, as distinct from the existing analysis of
one for
such
classes as the regular sets3, bounded Languages4, and parenthesis
languages5,
for all of which both inclusion
and equivaLence are
decidabi.e, positive sol-utions
to the equivalence
with an undecidable inclusion
problem assume new significance.
Our main
results for the
problem
equival-enee problem
for families
are to
show
for
three distinct families of automata that they are in just this
category. The onLy
comparable
family we know of is that of
Ewo-tape
acceptors6rT. For rhe class of LL(k) languages8 (which include the
sirnpleg languages), atthough equivalence has been shown to be
decidable, the inclusion problem is currently open.
that both
Ewo-tape acceptors and
We
will
show
LL(k) languages correspond closely
t+
to particular restrictions of two of our farnilies, and hence that
the decidability of the equivalence problem for botl'r of them
now
also follows as corollaries to our results.
In addition to proving dec_idability results we are also
concerned
with giving
bounds
for the complexity of the derived
decision procedures. A bound on the time complexity of a problem
(i.e. on the time required by the best possible decision
in the worst case ) gives an indication not only of
may
procedure
how much time
be needed for an automatic anaLysis, but also ofthe possible
difficulty in an informal senserof resolving parricular instances
it.
The complexiries are expressed
in terms of functions
r,rhich have
the paraineters of the machine description as arguments. Thus if
have a polynomial ti-roe bound sn a decision procedure
for
of
we
one problern,
but know that an exponential time is necessary for deciding another,
then this will indicate that for: sufficiently large machines,it
will be easier to solve a problem of the first kind than one for
the
second.
Valid measures of absolut.e ccmplexity for decision procedures
are numerous, and may depend additir:nally on the nachine model on
which one int,ends to execute them. For this reason we are usuai-ly
eontent
with
answers speeifying
cnly the nurnber of levels of
exponentiationn if any, involve-d. ?hus tvpicaliy l,Ie may say that
procedure takes polynomial,
or
perhaps exponential time, without
giving further detaj-l-s. This kind of ciassification is
known
to
robust enough not to depend on which one of the customary machine
modeLs
is
chosen.lo
More
a
satisfying results, which are
now
totally
machine
be
5
independent, can be given, however,
for the relative complexities of
the various decision probLems. Thus we show that, in a specified
sense, certain decision problerns sust be equally
some
others must be at least as difficult
difficult,
and that
as these. Such results,
besides confirming our experience with known decision procedures,
also throw light on several specific currently
we show
open problems, which
to be at least as difficul-t to decide as a qLrestion for
the best procedure
ure
have been able
to find works in
which
double
exponentiaL time.
Although the
decidability
and complexity
yield considerable information, the techniques
results
used
themselves
in the
proofs
and the properties on which they depend are equally important in
givi.ng insight into the structure of trese computations. Our
presentation will therefore attempt to highlight informally the
ideas we believe are new and nost important.
of
We
shall omit details
arguments whi.ch are well. known and occur elsewhere
in the litera-
, for the sake of compleEeness, we mention resuLts not
directly related to our main theme, Ire shal-l, just sketch the main
ture.
W?rere
ideas from which a proof can be reconstructed.
We
suggest the fotr-lowing as appropriate both
for
preLiminary
reading and for providing rrctivation to our work from diverse angles:
A general exposition
of the concepts of
automata and
decidability is
given in Hopcroft and U11rnan3. Basic resul-ts about dpda are also to
he found there, as weLl as in Ginsburg and Greibachl. Various
grrrnrnatical eharacterisations
and
of fanilies of deterministic
their relevance to parsing are described by Knuthl1,
languages
Korenjak
and llopcroftg, Rosenkrantz and SteamsS, and Harrison and llarell
2.
An
introduction to progran schemas, and a description of the relation-
6
ship bemeen certain recursive schemas and pushdown automata can be
found in Patersonl3.
7
PRELIMINARIES
This first section introduces the definitions, basic notions,
and prel-iminary
results which we shall use in
subsequent chapters.
Fron the start we take a rDre complexity conscious approach to the
is available
subjact than
elsewhere
in the literature.
Consequently
the preLininary ideas we have to introduce, although rnostly well
known, have had
to be reformulated so as to give the rnore precise
constructions which we now require. In particular we have to bypass
those
of the widely
exponential explos
We
Our
that lead to
used standard constructions
i.orrs
.
first define the class of deterministic
criterion of
acceptance
differs from the most
is the nore natural for our purposes.
in the sense that for
pushdown automata.
The
any dpda specified
popuLar one3, but
definition is very broad
in
one
of the other
standard
ways, there is al-ways an equivalent one in our formuLation that is
not
much
nore complex. Further, by placing sinrple restrictions on
this class,
sha11
Iile can also define several important subclasses which
we
later study.
The size
or
compLexity
of a dpda we describe by means of
several of the parameters of its description, which \ile go on to
specify.
I.Ie then
when a machine
is
economise on one
In the
investigate the tradeof fs that can be realised
transf,ormed
of these
to another equivalent one in order
parameters
second chapter we
at the
expense
to
of the others.
give results about the complexity of
problens atready known to be decidable. I{e justify the clain that
B
these problens are 'easy'by
exist for solving
verifying that sinple decision
them, which work
in tirne
procedures
depending only poly-
nornially on the paraneters of the tested machines. In the course of
this ne define a normal form for dpda,
deriving
some
and il-l-ustrate
of the decidability results as trivial
its
use by
consequences of
its existence.
Throughout we judge the
relative
pohrer
of the various sub-
fanilies of nachines by comparing the classes of
languages
to
which
they correspond. For sirnpLicity, therefore, when this is not
otherwise confusing, we often identify a family of machines with the
cLass
of
languages
it defines.
Ctrapter
DETINITIONS
1
1.1 Deterministic
Pushdown Autonata
A pushdown automaton (pda) is an abstract device whose menory
consists of a pushdom stack and a finite state control. It
can
read a string of characters from a finite aLphabet from its unique
input tape, which can progress past the input head of the machine in
only one direction, and only once. At any step of the computation
the transitions which the machine can undergo are determined by the
state of the finite state control, the contents of the top of the
stack, and the character under the input head. An input word is
either
ttaceept,ed"
or "rejected",
d.epending on
the "confi.gurations"
the rnachine can reach after having read its l-ast character.
such automaton defines a Language
i.e. the set of words it
The languages which are defined by the class
of
Each
accepts.
pushdown automata
are exactly the context-free languages of Chomskyl4'3.
We
are here interested in the deterministic restriction of
this cl"ass, namely the
case where each coribination
of state,
top
stack symbol, and input character defines a unique machine transition.
The cLass
of
such deterministic pushdown automata (dpda) we caLl D.
More forma11"n
for a machine
Me
D, let I be its finite
input al"phabet {a, b, c, ...}, f its finite stack_4-p@. {A, B, ...}
and Q
its finite state set {s1, szr...}.
input characters by o,
rrr2
etc.
We
B,
etc.,
and
We
denote strings of
strings of stack
symbols by
define e and A to be the null elements of I* and l*
(,)I t
10
In addition we use the special symbol
resPectively.
$?
to denote an
eropty staek.
c = (s, o) is an eLement of Q x ({n} v I+) and
describes the state and stack content (starting from the enpty stack
A configuration
end)
of the machine at
some
instant.
is an element frour Qx ({A}uf)
symbol
of c.
is (s, 0).
Thus the mode of
The machine has
The mode
of a configuration
and describes the
c
state and top stack
(s, oA) is (s, A),
and
that of (s,
0)
a set F . Q* ({CI} vr) of distinguished
accepting modes.
The
set of transition rules A is a set of rules of the form
(s, A) -l* (st,
where
ei ther
or
,r)
r e x u {e}, with the additional restriction that
i.e. occurs on tteleft in just
t.here with an j* t.urr"ition.
(i) an
(i
i)
e-mode
a
reading upde i.e. has at most
a
e f,, and no i+ rule.
The machine M makes the move
one configuration
(s, A) i*
to the other if
(s,
oA)
and only
one
1I
-+
if
a
one
rule,
rule for
(st, ,rt)
one
each mode is
and
each
frorn the
of the rules is
(s', ,t)rwhere n e X u {e}. If n e I then n, the input
character currently under the input head on the tape, is eonsidered
to have been read, and the head rpves to the next position on the tape.
Otherwise the head does not nove.
ct is " sequence of moves starting
configuration c and ending with c t. rt is an a-derivation,
A derivation c
-+
from
"
*5
"t,
11
if the execution of it is
on
input tape.
Ehe
There
is a distinguished starting configuration c", which for
simplicity we assume t,o belong to
Qx
configuration ct i" rea"hable from c
c
by the reading of the word s
accompanied
->
cr
.
I^Ihen
we say
that
cl
(in)
u
r).
we say
iff there is
some
that
a
derivation
is reachable we mean that it is
reachable from c".
An input
string
whose mde bel-ongs
cr
is
to F,
accepted by M
and rejected
iff
"s
S
c for
some c
iff it is not accepted.
configurations c, c' are distinguished by o if a-derivations
take one to an accepting
mode
but not the other. The rank of
configurations c and ct (denoted by rank (",
Length
and
"t)r)
Two
can
two
is defined as the
of a shortest string distinguishing the two, if
- othervise.
Tr,ro
one exists,
configurations are said to be equivalent,
i.e. c f cf, iff rank (c, ct) = -.
Two nachines M,
Mt are equivalgnt
iff their starting configurations are equivalent.
I,Ie denote
the set of strings which can take the machine from
a
configuration c to accepting rmdes, by L(c). Then clearly c = c'
Ltt L(c) = i,(c'). For a machine M we denote L(cs) by L(M), and for
a class X of such machines , we wi.1l abbreviate
{L
(M) | U . X} to
L(x).
'
Thus, to summarise, each dpda (as well as the language it
defines) is completely specified by a sextuple M(I, l, Q, F, A, cs).
The foll-owing
iLlustrates the main points:
f
rl
t2
ExampLe 1.1
For machine M let I = {a, b, c}, f = {Ao, A, B}, Q = {s1, s2, s3},
F = Q x {o}, cs = (sr, Ao) and A the set of transitions:
j* ("r, A)
b, ("r, B)
(s r, A)
(sr, A) *9* (sl,
AA)
(sr, B) b , ("r,
BB)
(sr, B) -3*
(sr,
(sr,
Then
Ao)
1s2,
n)
B)
5
j+
e, B)
(sz, B) 3> (s2, A)
(s
(sz, A) 3* 1s2, n)
("s, A) 5
it
can be
(s3,
A)
r
A)
(s3r
A)
1s 3
easily verified that
L(M) = {anbman
lr, r:il
u
{anbmcn
lr, * ril.
For our notation we follow standard conventions for sets. For
a string x we denote the reversal of x by xR, the length of x by l"l,
and the concatenation
of n copies of x by xn. Fot Z a set of strings,
z*
its transitive closure under concatenation,
and Z+ rLLL denote
with
and
empty
without the nuLL string respectively, and I will- denote the
set.
The syrnbol $
wilL always stand for an input character
which, by convention, does not belong to the I being discussed.
L.2
Elementary Properties
We
first note that the trDre customary definition of
is by states rather than rnodeslt3. IE
fied (Hopcroft
and ULl-nan3, Lettuna
can be however
acceptance
easily veri-
I2.3) that the classes of
languages
defined by the two methods are identical. Further, as we shall
now
13
indicate, all the key properties
on which theorems
particular to
depend, are a consequence of determinism, and are independent
manner
of
of
dpda
the
acceptanee chosen.
One
effect of the deterministic restriction is to ensure that
once a sequence
is uniquely
of
e-moves has
determined by the
started, the subsequent
original configuration,
eomput.ation
and can depend
on the input tape again only when the E-sequence has Eerminated.
Thus
if, the latter never terminates, then it is
and can be
eliminated. If it is finite
cl-ear1y redundant
and gives
rise to the deri-
vation (s, urA) ", ("t, ootB; where (rt, B) is a reading mcde, then
such e-derivations can be eliminated by replacing the e-rule for
(s, A) by rules of the form (s, A) -3t (stt, r,rtrtt) for
("t, B) -g* (s", ,").
Thus we conclude
that
each rule
e-moves can
only
be
essential i"n a dpda for derivations which cause the stack to
decrease
in height"
Similarly, but for other reasons, having a multiplicity of
states cannot be essential in derivations where the stack is
increasing. State information can always be coded into the top stack
symbol and thus transrnitted
is ever rernoved.
Thus
in
indefinitely, as long as no top
Example
symbol
f.i-, while the distinctness of
scates s2 and s3 is essential, the information which sl carries'
i.e. that the stack is in
the stack symbols.
We
che increasing phaseo coutd be coded into
could therefore nndify M to obtain another
equivalent one with state set just {s2, s3}, but with an enlarged
stack alphabet {A0, A, A', B, Bt}, if we replaced cs by (s2, A6),
and replaced
(s2,
the transition rules involving st by the following:
A9)
a
->
("2, At)
("2, A') -g+ (s2, AAt)
L4
("2, A') 4
("r, n')
(sz, n';
(r2, B') -3+ (sz, A)
A
q"r,
(s2, B') -9> (ss,
Another consequence of determinism
BB')
A)
that is useful for
showing
that certain languages cannot be recognised by any machine in
a
given class, is the following observation, of which a formal version
is stated and proved by Ginsburg and Greibachl:
Observation
The
f.i
effect of input strings periodic in a given word a
(i.e. of the form on for n > 1) on a dpda is
either (a) to
cause the stack
to never
grow
larger than
some
eonstant hei.ght,
(b) to creete etacks which are periodic. in
or
e..'erywhere except
for parts of
some stack word
bounded length
at the top
and bottom.
Using this we can formaLly verify that e-moves, multiple states,
and a
mriltiple stack alphabet, are all essential- features of any dpda
reeognisi.ng the J.anguage
of
Example
1.L.
Formal- arguments
of this
ktnri are given by Ginsburg and Greibachl, and by Harrison and Havell2.
We
note that the normai- form given by Greibach for context-free
languages shows tha.t
in the nondeterrninistic case,
multi.ple state sets are inessential"iS.
e-moves and
15
1.3
Subfamilies
We
can define some important subfamiLies
of
D
by specifying
sinple syntactic restrictions on the machine descriptions:
One-counter machines (C) are those dpda
with just one stack
symbol. In these the stacks can only be used to count,
and hence
configuration is best described as (s, n) where s E Q, n : o.
a
Such
counters are closely related to the registers of Minskyl6.
ReaLtime machines (R) are those with
recognise strings in rea1 time, as opposed
e-modes. They
merely linear time"
Stateless machines (S) are those with just one state. This
state can therefore be omitted from the description of the configurations or transition rules. A consequence of the restriction is
that no information can be transmitted during stack decreasing
derivations.
Thar the languages defined by each of these three classes is
properly contained in L(D) follows from the previously
vation that the
l-anguage
urade obser-
defined in Example 1.1 cannot be recognised
by any machine in C, R or
S.
For alL such classes we can have the additional restrictions
that
(a)FcQx{CI}
or that (b) no rule is defined for modes in Q x
{521.
For the classes D, C, R and Srwe respectively define the sub-
16
elasses Dg, C0, R6 and So by imposing both
and D1,
each
Ct,
R1 and St
restrictions (a) and (b),
by imposing just the first of these. That
of the implied containnents is proper can be easily verified
by
observing that if X is any one of D, C, R or S then I* eL(X1) but
*
t" t L(Xo), and {anbn | *: n} e L(X) but I L(Xr). Alrhough X1 rhus
properly extends Xg, it does not use the stack in any more general
wa5 and consequently
many
of its properties
can be deduced
directly
from those of the corresponding X6. For simplicity, therefore,
we
shall only study the classes X and X6.
We
note that classes equivalent to D6 and R9 have been studied
by Harrison and Harzef 2 lwho caLl them strict, and realtime strict
deterministic respectively), and the class
Hopcroftg. To show that
latter,
we have
to
Sg
is just the rsimpler
demonstrate
that
This is obvious since stack symbols
can
So by Korenjak and
machines
of
the
in it.
with the property that A 5
e-noves are redundant
clearly be elj,minated from the rules, while e-moves which
n
cause
increases in stack height can be removed in the manner indicated
in
51.2.
L.4
Some F_amil"y
Pfoperties
There are some
and languages
further restrictions on families of
machines
that are related to the definitions above.
A language L c I* is said to have the prefix property iff
F
c
X+ and
it is
a e L *uB I
immediate
that
L"
these
?rom the
definition of D6, C0, R6 and 56
all have this property. For L(Do)
we can
17
sayr further, that it contains all the languages of L(D) that
this property. For consider
erty,
"a
and
an M e D
s.t. L(M) has the prefix
that no accepting derivation has an intermediate
new
proB-
nodify it to an equivalent Mr e Dg as follows: rntroduce
special 0-simulating symbol at the bottom of the stack to
uration.
have
Then
for
each accepting npde
of
ensure
empty stack config-
to
M introduce an e-move
state frnm which any stack wiLl be emptied by further
a
€-moves.
The modified machine clearLy has the required propertiesl2.
The quasirealtimel2- property
restriction.
nurnber
E-moves
is a relaxation of the realtime
are now allowed to occur, but only a
of tirnes consecutively. rf this
to n stack characters
can be removed
bound
bounded
is say n, then only
in a single e-derivation.
up
Thus,
by changing the sta.ck alphabet so that each stack is now specified
by symbols encoding bLocks of n + I old symbols (with a block of
possibly fewer synbols at the top of the stack), and changing the
transition rules so that these blocks are manipulated correctly,
old machine carr be simulated by a
neh/ one
with the same states,
the
but
with no e-nodes. A npre formal proof of a similar statement is
given by Harrison and Havel12. Thus we can conclude the forlowing:
Observation 1"2
The power
of
of the quasirealtime
machines
is no larger than that
R.
Finall"y
we
mention a syntactic
rui.es. A dpda is conservative iff in
restriction on the transition
any ru1-e (s , A)
-+
(s t , to)
.ltitnere
luJl I Ir 1t is the case that the f irst syrnbol of o is
A.
18
Lemna
1.3 will show directly that this is not a proper restriction
on the power of D, C or of the quasirealtime machines. However
and S9 are properly
respectively.
restricted to the symnetric l-anguages,
An innnediate property
S
Sy and
Syo
of these symrnetric machines is
while the stack is
that they cannot transmit information
even
increasing.
| " : 1}
Thus the language {an$an
cannot be recognised
by such a machine since the latter could not distinguish
between
being in the upward phase from being in the downward phase of the
computation. Consequently Ey6 g
Sg and Sy
g S.
1.5 Derivations
We
now introduce some terminology and
notation for describing
the geornetrieal rbvements of the stack, that are in addition useful
concepts
in talking about derivations.
The
height lcl of a configuration c is the length of its
i.e. l"l = lrl if c = (s,,,r) for
3,
stack
some s.
is a stacking derivation iff l"l . l"'l
and every intermediate configuration in the derivation has height
The derivation
"
"'
It is then written as c +(o) ct. The derivation
is a popping derivqtion iff l"l t l*'l and every intermediate con-
greater than l"l.
figuration in the derivation has height greater than lc'1. It is
(N.8. The notation t, * is used by
then written as c +(o)
"t.
Stearnd but with the converse rneanings.) It is a positive derivation,
written as c f
empty stack.
"t,
iff no internediate eonfiguration
has
1Y
We
define the slggElqg sequjlnge r:f a stacking derivation as
a
of nndes ui, Each ug is the nnode of the unique configuration
of height (i + 1) in the derivation that is not followed by any
configuration of height < (i + f) subsecruently in the derivation.
sequence
Note
that if (s,
ur)
+(o) (s', ,') then ui is not defined unless
lrl 5i+f S lr'1, and may not be defined for atl intermediate values
(e.g. if the derivation consists of a single move corresponding to
a
transition rul-e of the form (s, A) *
(s', o1) where lrt | > 2.1
The significance of pi for the configuration corresponding to itn is
that it contains all the information about the previous
that
may
computation
influence the subsequent part of the derivation.
lJe def ine analogously the popping _sequeEe
derivation to be a
sequence
of
,st.gte,s
oi.
Each
for a popping
oi is the state of
the unique configuration of height i in the derivation that is not
preceded
by any configuration with height < i.
*
e I' , j " i: o we define oirj to be the substring of o
An
s tar ting fron the (i + f)th character and ending with the jth.
For trr
index
seE
N
is an ordered subset of the positive integers.
N induces a natural- segmentation
and
of o into
words
oirj,
Thus anv
where
i <j
i, j eN.
w.r.t. index set I (Fig" 1)
are just the subsequences of these corresponding to the elements of
The stacking and popping sequences
N.
We def
c +(a)
"t
ine ttre segmenf"_ation of u in the stacking derivation
qr11rt, an index set N = {ii,
... ir}, as the unique
of input words o\ , ,... , cin_', i' with the properties
!
qip,
that therc exist
iI, oin, in+l whm ig +1 = l. | , inal+1= I "' | , s. E.
sequence
20
Trace of c t(g)
"t
i3
- l-o
-z
Trace of c *(cl)
lEI
"t
-
I
.. ct.
".
I
I
.14
I
*{
I
I
I
I
1o-
-
L
rig.
L
t13
27.
oinrin+1'
d - oiori10i1ri2
and that if cf has stack o, ther: for the stacking sequence
{(sr, Ar)} w.r.t.
inal},
{io,
(sr, Aa) +(airrir+i) (sr+i, rrrirrir*rAr+r)
f,or 1 : r : n.
In other words cirrir+I is just that part of
the
input stri.ng which takes the derivation from the last configuration
of height ir+L to the last
one
is fol lowed by c'enfigurations
of height iral+l" provided that neither
srnal"ler than themselves. We can
similarly define ui,3 for i < j and i, j e N.
Analogously the segmentation
can also be
of a popping deqivation w.r-jJ:-
N
defined. oi,j is then the part of the input string that
takes the derivatiorr from Lhe first configuration r:f height in to
the firse configrrratiori of height j.
r+here now
i > -j"
1.6 Complexity MeasursE
We
shall describe the size of a dpda in terms of the following
paralneterg
!
t = size of stack
alphabet
q = siae tlf state set
p = size af input alphabet
h = Length of longest stack word occurring in the transition
rules
u = number of transition rules.
22
We
machines
shall express the complexity
measures
relating to
these
in terms of functions with these parameters as arguments
e.g. X(t, g).
We
shall be primarily
values, as the arguments
monotonic
in
each
become
parameter.
concerned
with the
asymptotic
large, of positive valued functions
We
introduee the fol-Lowing notation to
describe such behaviour:
Definition
The function
X(x1r..., xn) is of order En iff n is the largest
integer such that
Tffi 1og(n+-I)x(y,.. .,
y-)rog y
Then,
for
exampte, x2 ,
wil-l be El, etc.
even
t) ,
o.
/*, *logx will be go, *y2/*2
For convenience rre shall refer to an expression X,
if it is of several variables, as poLynornial if it is
above
by
some
When
and xl
bounded
multinomial.
we are interested
in the leading exponent in an expression,
we use the following more detailed notation;
Def
ini tion
".. xm) is of order En(Y(x1, ... *n,)) iff
3 k, k' > 0 s.g.for all sufficiently large x1,
tnr
The function X(x1,
Expn(kY) < X < Expn(k'Y).
[ner" rxp0(x) = x, Expr*l(*) = 2Expr(x) tor r: o, and simiLarly for
Lo9, which we shall always take to the base 2.]
23
L,7 Tradeoffs in Desg:'iption
The dpda
definition allows equivalent
machines
to have very
dissimiLar specifications. For example any language recognised by
finite state
some dpda
We
auto{naton
with n states can clearly be recognised
with q = n and t = 1, and also by
some
with
Q
a
by
= 1 and t = n.
investi-gate an aspect of this flexibil-ity of description by asking
to what extent an arbitrary
alent one, so es to
dpda can be modified
econosiise on one
of the
to obtain an equiv-
Parameters
at
the
possible expense of the others.
I n3) as the class of languages specified
by dpda with q j nI, t j nz and h 5 ns. Then the questions rre want
We
to ask
define LD(n1,
can be
elasses.
We
n2
naturally phrased as inclusion
probl-ems
for
these
note. howet'er, that only the f irst of the fcrllelwing
lemras depends on deterrninism.
Lema 1. I
LD(qo
t, h) # tD(fl , tlz, n3) for an] n2, n3.
Harrison and Havel"l2 show that the set
{a%karlbk il":t
jn, 1<m}
can he recognised by a dpda
with n states, but not by one with fewer. I
24
Lernma 1.2
LD(q, t, h) c LD(q, th, 2) for t > 1.
Proof
We
give a simple construction for converting an arbitrary
to an equivalent one which simulates it closely, but
dpda
does not
require long right hand sides in the rules, or extra e-moves.
We
leave the state set unchanged but enlarge the stack alphabet to
consist of all strings in f+ of length less than h, each typically
denoted
by [as..C], in the following
Replace each rul-e
and
n
,_
2, by the set of
(s, [rl] g
>
(s
'
(s, A) i*
way:
(st, 81 ...
Bn), .where
n e I u {el
ruLes
,
[rn1]
[n2
Bn] ) ,
*,
I s.t. lrl < h - 1, and make the analogous
replacenents for the rules with n < 2.
one
for
each
o
e
ctearly the new stack al-phabet will be of size
t + t2 + .. . + gh-l " th, and if acceptance is defined in
Then
obvious way, then the new nrachine riill be equivalent
Note
the
to the old one.
1
The new stack syrnbols
that are not redundant in the
rnodified
of the form [uA] where Bu-ror'appears in
some rule of the old nachine for some B and ut. Consequently the
nachine are exactly those
Rew
stack alphabet is essentially of size at nost thu, and so really
depends
only polynomially on the original machine description.
Frorn
25
this it fol-lows that the
of the description,
cons
number
of transition ruLes, and hence all
need only be increased poLynomially
by this
truc tion.
Note
2
The construction prGservcs the conscrvativc
synurnetric languages have a normal forrn
with h -
property.
Thus even
2.
Lema 1. 3
LD(q, t, h) . LD"on"(qth, r, 2) for t >
1
and LD(q, 1, h) - LD(qh, 1, 2).
Proof
At the
ean reduce
expense
h to 2 and make the machine consarvative.
The new
Q
Each
u
of introducing ncn e-rpves and incraasing q we
state set consists of
i[t, ,J | " .
Q, o e
r+, l.l
< h]
transition rule (s, A) -I* (rt, Bt ... Brr), with n > 2,
we
repl-ace by the set:
(s, A) Jt 1["', Bz
(["', Bi...
(["',
Br,]
Br,], Br-r)
nrr]
, rr)
, Bi-ilj* ([r', ri+l
5 ("',
B"] , Bi-rBi) for r < i
Bn-rrn)
This construction merely ensures that instcad of a large stack
word
being added in one move, it is buiLt up in a succession of specially
<n
26
created E-rnoves. I
Note
The non-redundant subset
of the new state set
depends only
polynomially on the original- parameters and is bounded by uh.
The rnachine
polynorniai-
description as a whole is again increased only
ly.
lem+e-L.S.
lD
(q, t, h) e LD(qt, 2, h.ftogtl ) .
Proof
We
r1
llogt l.
code each stack syrnbol
into binary
words
of fixed length
For each old state of the machine, Ehe nerr one has to be
able to interpret the top coded word in the stack. To do this it
traces down, via e-moves, a path of a binary tree of depth ftogtl
while popping this top word. Thus the new machine needs a tree with
about t nodes in its staEe diagram for each original state.
Thus
the state set is now of size qt, and h has increased to tr.flogtl. il
I,le e,,:ncl"ude
and
h. By the determinist{c condition on the transition rulesr u is
clearly
mayr
bounded above
by qtp. On the other hand rhe input
al-phabet
in theory, be arbitrari.ly Large. llowever, the number of char-
acters whieh
ded
by reJ-ating the other parameters p and u to q, t
prardulee
distinct behaviour from the
automaton
is
boun-
by the other parameters in the fr:l-lowing way: Each input char-
acter can occur in at nost qt rules, each of which has one of
rnore than
qth+l different possible ri-ght hand sides. Thus the
no
27
machine can
ters,
and
only distinguish up to (qth+t)et aifferent input charac-
if there are more, then at least
identified with
some
of
them can be
each other without influencing the computations.
1.8 Family Relationships
lJe conclude
by su'nnarizLng in fig. 2 the retative. polter of the
classes of automata with which we shall be most concerned.,
the foLLowing relations for arbitrary classes X and
(i)
(ii)
X
->
We use
Y:
t only if V M e Y, I M' e X s.t. L(M) = L(l'l')
X ---+ Y
only if V M e Y, I M' e X s.t. L(M)$ = L(M')
where $
is a disti.nguished
syn$ol.
For the definitions of N0, T and dB - S, see chapters 3' 4
and
8 respectively. Al-1 the relationships in the diagram ae irnnediate
from the definitions or well known, with the exception of the
position of N9, which we prove in Chapter 3, that of 2-tape
acceptors, explained in Chapter 6, and that of S, for which we mit
the proof.
28
Family Relatiernships
D
dpda
rI
a
Do
f,l-acce rP
I
I
I
B-S
a
realt i
\tI
t
I
I
t
finite-turn
I
,
I
ce
I
c;
I
one-c0unter
I
'0
s
t
state ess
r
(two-tape
acceptors)
.- t
Fsa
./
a
finite-state
automata
Fig.2
simple
29
Chapter
2.L
EASILY DECIDABLE
2
PROBLEMS
Introduc-t,ion.
Here we shal1
discuss
some
properties of deterministic
languages which have been previousry known
to be decidable.
we give
decision procedures for these, and show them to be easier than the
procedures f,or equivaLence and
regularity that
we derive i.n
later
chapters, in the sense that they all require only polynomial time.
These
(i)
"easily decidable"
menbership
-
does
problems
for t e L(D) include
a e L?
(ii) eurptiness - is there sor€ o s.t. o e L?
(iii) finiteness - is L finite or infinite?
(iv) totality
- is there
some
o s.r. a l. t?
(v) equivalence with reguLar set Lt - is L
(vi) has L the prefix
= Lt?
property?
first three of rhese are decidable for context-free
Langrrages in general. but the last three rely on the determinist.!"c
The
restric.tion.
Techniques
for proving
such
decidability results
given by Bar-Hillel, Perl"es and ShanirlT and by Ginsburg
and
Greibaahl. I{ere we shalx concentrate on the complexi.ty of just
of thern.
we know
of
rro arguments
are
some
giving interesting lower bounds for
the time complexity of any of these decision probrems, and, for this
30
reason, we sha1l not
Ilowever,
measure
for
linger to give exact uPper
each problem there
bounds either.
is often an alrernative natural
of complexity. For some of these we can give upper and
lower bounds which
particular
differ by only a modest arnount.
measures
Examples
of complexity include, for the emptiness
of
such
problem,
the maximum length of shortest strings accepted by non-empty machines
of a given size, and, for the reguLarity problem, the maximum size of
the smallest finite-state automata equivalent to
some dpda
of
a
given size.
2.2
Emptiness
We
shaLL consider from two viewpoints the complexity
problem, which plays a
The proof
crucial rol-e in
of decidability of
PerLes, and Shami-rI7
of this
subsequent decision procedures.
emptiness given by Bar-Hil1el,
for context-free
languages depends on the
derivation tree3 of the gramnar for a shortest accepted string. If
the grarrrar has x non-terminals and right hand sides never longer
than
h, then in any derivation, starting with a non-terminal, of
a
shortest string, no non-terminal need repeat along a path in the
tree, which therefore can be taken to have depth < x.
lrle now define
the depth of each non-terminal to be the depth of the shallowest
derivation tree of which it is a root. Then, knowing which
terminal-s have depth <
i,
we can
find all those that
have depth
< i + 1 siurply by looking for productions in which all the
on the right hand side have depth < i.
list of productions at
most
non-
symbol-s
Thus by going through the
x times, and establishing the depth of
31
all the non-terminats,
has a defined depth
Thus
(a)
we can deterrnine whether
i.e. whether the
language
is
the starting
symbol
empty.
it fcllows that
there is a decision procedure for empciness which works
in time depending only polynomially on the gramar description,
(b)
and
the shortest accepted string can be no Longer than hx"
Ttre standard constructionr
for a pda with q states and t stack
synbols gi.ves a granmar with qzt non-terminals. Further if (and
only if) h is taken to be a constant, the nunber of productions in
the
graunnar
wiLl
depend
However, from the notes
rde can modify any pda
only polynomially on the parameters.
to
Lernmas
1.2 and 1.3 we already know that
to an equivalent one with h = 2, with only
polyn'cmial g'rowth. Thus we can concLude that:
Lenrna 2.
L
Emptiaess
time.
ft:r
pushdofrtn automata can
be tested in
po1-ynomial
tJ
Since Ehe standard construction of the graflurnr from the nactrine
does
nst change h, it
aLso foll-ows that:
Lerrlna 2.2
If
M has parameters
h > l, q and t,
tken 3 a e L(M) s.t. lol : hq?t. il
and
L(l'l) is not
empty,
32
We
can show
that for q = 1 the order of this
bound is achievabl-e.
Example 2.1
Let M e
S9
be the simple machine defined by
qh
Ai+AT+t
lj -3*
where the
forl"<i<t
n
starting
syurbol
is
A1 and acceptance
is by empty stack.
Let min(Ai) be the length of the unique string generated from A{.
Then from the productions
it is clear that
min(Ai) = 1 + h.min(Ai..1) for
Since min(Ag) =
1.
< i < t.
t-i
1, it follows that rnin(Ai) = f, hr.
Thus M accepts
r={
just one word, and this is of Length of order nl(t.logh).
I
For q > t we can find a class of rnachines which achieves
bound
of order fl(qt,logh).
Ex_ample 2.
2
sq}, stack symbols
Let M e Rg have states {rt,
{Ai, ... At-1, B},
and
transition rules:
(si, A3) e* ("t, a!+r)
1
j i : q, I j j . t -
(si, Ag-t) -3* (si+r, BAI) 1J i < q
(sqn As-t)
-3* (sq, A)
(si, B) -1* (si-1, A)
I . i :
q
1
a
33
Let the starting configuration be (s1, A1) and the unique accepting
mode
be (sr,
$l).
This is a generalisation of the previous example.
occurrence
of Ai proliferates to
Each
produce an exponentiaL number of
At-lts, but now instead of these being
renrcved
by a single input
character, they are each allowed to proliferate themselves all over
again for a different state. For any configuration the depth
reacheci
in this
sequence
of
renewal"s
is given both by the state and
the nurnber of r:ccurrences of a B in the stack. By a second induc-
tion process, similar to that given in the previous example, it
follows that the unique string accepted by M is of lenglh of order
nl (qt. logh)
We
.
I
leave unresol-ved the gap between this lower bound and the
previousLy derived upper bound
of El(q2t.logh).
2.3 Finiteness
Bar-Hillel-, Perles, and ShamiriT show that a context-free
Language
is i,nfini-te if
and onLy
if there is
some
derivation tree
which has a path atong which some non-cerminaL repeats and
in
so
doing generates distinct terminal strings. Thus to test for fin:,te*
ness nre produce a polynomial
size grarunar, eliminate redundant
non*
terminal-s (i.e. 'those that do not generate terminal words, or cannst
be derived frorn f.he starting symbot), and then
embeddi.ng
test for the proper
properEy. This last test can be done by looking at each of
the remaining prodr:ctions that have
some
terminal on the right
hand
34
side, and searching through the other productions to see whether a
self-enbedding cyctre of productions can be found for this given non-
terminal. As all this
can be done
in polynomial time it follows that:
Lerma 2.3
Finiteness for pushdown automata can be tested in polynomiaL
time. I
2.4
A Normal Form
There
is a well
known3 normaL form
for
dpda, which
is
very
useful for our purposes and fits in naturally with our definitions.
Def
ini tion
A dpda
finite
is
nunber
of
loop-freel iff every input word can be read in
machine moves.
This condition excludes the possibilities of either (i)
from some reachab,Le configuration being undefined,
or (ii)
nnves
there
oceurring an infinite sequence of consecutive €-moves, thus
preventing further inputs from being read.
T,enrna
a
2.4
Every dpda M has an eguivalent loop-free Mt
stack alphabet, and just one extra state"
with the
sane
35
Proof
I^Ie
f,iret
au$nent
1"1
by introdr-rcirrg I
transition rules (s', A) ..3t (st, A) for
without changing
each
aeI
mode we add
and
new
and A e
iin
the rules (s, A) ", (u' ,
(s, A) --4, does not previously oecur.
stopping, the
s'
F.
For each reading
whenever
new staEe
maehine
will
now continue
A)
Thus instead of
to read the tape and
reject it.
To exci-ude the otber
possibility
mentioned above, we
to note that in any infiniie e-derivation there must be
first
some
configu-
ration which has a smallest stack. If the mode of this is (s,
then hy replacing the Eransition
(s, A) :S' (nt, A; for
each
have
A),
rule for this by the set
a, we again ensure that
bire r*:sL of, the
tape is read and rejected. These rmdes can be effectively found by
appropr!"aL'. emptiness
teses. il
A11 the neru rul-es introduced serve the same
do not create additional comptexity
eonsequently r*e
role
in any significant
and therefcre
sense.
shall frosi now on assume that the loop-free
machine
i.s nf eseentialXy the same size, in every way, as the original one.
Def
ini tion
A dpda i-s
in
pormal_rEgmrr
accepting mede i"e a reading
iff it is loop-free,
mode,
and every
36
Lenuna 2 .5
For each M e D with q states and t stack syurbols, there is
equivalent Mt in normal form with the same stack alphabet
an
and
2q + L states.
Proof
}Je
"t
first
make M
loop-free by introducing a distinguished state
as in the previous leqma. For each other state si E Q we intro-
duce an
sitt.
The
role of si" is the
same
as that of si, except that
the former indicates in a derivation that an accepting
been passed since the
Thus
for
mode
of
M has
tast time an input was read.
each accepting e-rnode of M, we replace
(si, A) - r* (sj, o) by (si, A) - 3+ (sj' ,
u:)
"
For each readiug rule (si, A) -3* (s5, ur) we add the extra rule
("i", A) -3+ (s5, rrr), and for
the extra rule (si", A) -:'
we
(sj,
M and accepts
the
(si, A)
ur) we add
(s3", ,l).
This new machine clearly simulates
strings if
€-n
each e-ruLe
define the new set of accepting
same
modes by
F' = 1(si", A) | (si, e) is a reading mode)
u {(si, A) | (si, A) e F and (si, A) is a reading mode}.
I
37
2.5 TotaLity
A,s corrol-lari"es
to
Ehe lemna above,
decidability proof,s for
properti.es whi.ch are known to be undecidable in the nondeternninistic
case3, can now be easily derived.
CorolLarv
2.I
For each M e Dr3 M' e D with about twice as many states, which
accepts exactl"y the cornplement
of L(D).
Proof
We change M
to M" in
normaL
form.
Then we
redefine the
accepting:,rnod,eg
to be jr-lst those reading modes which are not
aceepting nodes
in
U"
"
[J
Corollarv 2.2
Total"ity ferr dpda can be tested in polynonial time.
Proof
ConstrucE ttre eomplement machine and tesE
cqrq.l legy
!.
for emptiness. I
3
For M e D, and Mr a
finite state automaton, the following
can
38
be tested in time depending polynomially on the parameters of
M and
M': L(M) = t(M'), L(M) n L(Mr) = /, L(M) c L(M'), L(M')cL(t{).
Proof
rf
Mr has
state set Qt of size qt,
and M
state set
Q
of size
then both machines can be simulated together by a dpda which
that it has state set Q * Qt and is able to
mirnic Mr in the finite state control at the same time as simulating
M. Provided that M is in normar form, each of the above questions
resembles M, except
can be tested by defi.ning acceptance
and testing
for emptiness. I
suitably in the new machine
q,
39
rr
THE EQUMLENCE
I.Ie have aLready observed
of
PROBLEU
that an equivalence test for a class
automata can provide a method
of verifying the correctness of
a
given machine by eomparing it with another of the same kind. In
addition, we can attribute other useful properties to classes for
which equivalence is decidable, that give us increased confidence in
the possibility of systematicalLy analysing and handling particular
machines contained
We
in
them.
first notice that for aLl automata with a decidable
membership problem, inequivalence
can enumerate
is partially decidable, For
all strings over the input alphabet and test
we
whether
the two machines behave differently in each one. Then if the
machines are inequivalent, we are sure
to recognise this
when we
reach the first offending input string.
We
then observe that all the automata in a syntactically
defined class, such as D, can be enumerated in some lexicographic
manner. It foLlows that if we can decide equivalence, then for
automaton
in
Conversel-y,
such a class we can enumerate
if
we can enumerate
all
all other equivalent
machines equivalent
to
any
ones.
any
partieuLar one then, since inequivalence is partially decidable,
can decide equivaLenee by running
uri.eh ehe
this
we
enumeration simultaneously
partiatr decision procedure.
Thus we
arrive at the following intuitive interpretation of
the signif,icance of this problem: equivalence is decidable if
and
only if for any automaton in the classr r{e can en!.merate al-1 other
40
\rays
of doing equivalent computations. Undecidability then means
that the relationship
between the automaton descriptions and
computational behaviour has a
and we can no longer hope
different
ways
their
distinctly higher order of complexity,
to be able to describe adequately all
in which the
freedom avaiLable
within
the
Lhe class can
be used to perform a given task.
The
partial decidability of inequivalence has also
the
following imnediate consequence for all the classes we shal1 study:
Observation
If equivalence is partially decidable, then it is decidable.
The decision procedure consists of simply running the
decision procedures, for equivalence
and
partial
inequivalence concurrently.
Thus, except when we can derive meaningful results about the
complexity of an equivalence test, we shall, for the sake of sirn
plicity, be content with proving decidability by showing partial
decidability.
This section consists of three chapters in each of which
distinct subfamily of D is
shown
a
to have a decidable eguival-ence
probl.em. There is an essential unity in the methods used for aLl
three, for which the basic inspiration
StearnsS. They show that
for
comes from Rosenkrantz and
dpda recognising the
LL(k) languages,
there is an easiLy derived canonical forn with the property that
two equivalent machines must,
after reading the
sarne
input
rdord,
any
41
have stack heights
differing by less than a ce.rtain constant.
such machines can therefore be simulated by a single dpda
Two
with a
t'{ro
track tape, and equivalence verified by a suitable emptiness test"
Wtrat we sha1l show
is that
even
for
m<rre
general classes, where
such close relationship between the st.ack movements need occur,
suitable
machines.
pushdown automata can be devised
to simulate equivalent
no
-
42
Chapter3
A REALTIIfl FAI.{ILY
3.1 Introduction
The syntax
of
progranrning languages
is customarily defined in
part as a general context-free granmar. Most frequently, however,
the languages generated by such granmars are not only deterministic,
but also expressible as Rg languages, i.e. those recognised by
realtime deterministic pda accepting only by empty stack, and even
subclasses
of these.
such subclasses as the simple languagesg, and
the LL(k) languages8 which generalise them, have been studied with
exactly this nnotivation, and equi"valenee tests have been found for
these particular ones.
principal restriction on LL(k) recognisers is that they
require essentially only one state, and therefore cannot transmit
The
information during stack decreasing derivations. Thus, for example,
in Al9o1 60, after some expression'((.. (( Expression ))..))'with
arbitrarily nested matching brackets is
remenber whether the contents
of the
gcanned, no such machine can
innermost brackets was
an
arithmetic or a boolean expression. since'if ((..(( Expression ))..))'
has
distinct valid
sequeLs depending on
just this condition,
we con-
clude that no essentially one state machine can recognise arithuretic
expressions as specified
Here we
in that
language.
shall define a fanily
N6
of quasirealtime
dpda such that
L(No) properly contains the tl(k) languages. N6 machines may have
arbitrary
number
of statcs,
and can
easily
accormodate the above
an
43
mentioned example.
languages
In addition, however, they extend the LL(k)
in a much more significant
wave thus
indicating that
generalisation of the equival.ence test is an important one.
procedure
for LL(k)
our
The
languages given by Rosenkrantz and SEearnsS relies
on the fact that nin(c), the length of the shortest string accepted
from a configuration c, is a very well behaved function of the stack
c(rrents.
deduced
The value
of this function for any configuration
can be
to within an additive constant from the contents of
the
stack atone, and, further, can only change a bounded arnount from one
configuration to the next in a derivation. This property does not
hol-d
for
concept
scme languages
in L(No), such as {anbcn} u {ande2n}. The
of tthicknessr of stack words, central to the proof of
Rosenkranta and Stearns,
is no Longer applicable here.
ldhile the inclusion pr6blem for sinnple and LL(k) l.anguages is
at present open, we can already prove this to be undecidable for Ng"
In fact we strali show that L1"L2 is undecidable where L1 is a simple
language and !,2 e L(Ng) l'ras ja"rst two states"
We
eonjecture that the property on which the validity of rhe
equi-velence
test fnr
Ng depends hoLds
betieve t-hat a n:r*of of this
?he enphasi,s 'i"n
this
"*rc,r-rtd
c.'ixapter
itse1"f,, rathex'tlran the elass
Eire iucreased
generality sf
l-{60
also for R6 in general,
and
be illuminating'
is
ern
the derision procedu::a
which !ile use merely
o,+rr E,eehnique
to
demnsEraee
over previous ones.
44
3.2 Nonsingular
We
Def
Automata
defi"ne the cl-ass Ng of nonsingular machines as fol1ows.
inition
M e D9
Vr^,,
is nonsingular iff
3m>O
ot , f*, s, st e Q where lrl
s.t.
>
L(s, urto) = L(st, ,t) + L(st, ,rr')
m,
= y'.
Theorem 3.1
(i) LL c L(No) whcell, is rhe class of all LL(k)
(ii)
languages.
L(N6) e L(Ro).
Proof
(i) For a configuration c let rnin(c) be the length of a shortest
string in L(c). Then from Lernma 8 of Rosenkrantz and stearnsS, we
can deduce
for eertain canonical LL(k) recognisers,
when translated
into our terminology, that there exist positive eonstants .c,
s.t. for all s, st, o, and rrtt,
(a)
| nin(s, o'o) - min(st,
(b) 0', lrl t *itt(",
r,r)
rrrt)
- min(s,
,^,)
| : l,
and
2 lrl,
provided that min is defined throughout.
It follows directly that
any dpda
L'
with these properties
,
45
belongs to N6, with nonsingularity constant m equal to 1,.
that for the sinple9
We
note
machines this constant can alwavs be taken to
be sero.
is proper, consider the following
To show that containrnent
the
dpda M recognising
A:
a
(sr,
A)
i,
A)
-->
(sr,
A)
(s
cs = (s1, A),
In
M
l-anguage {anbcn}
u {ande2n} for n > 1.
(sl
r
AA)
(s2
r
A),
(s2, A) - r'+
-$ (ss,
A),
(s3, A) - 3'+ (s4,
-)
b
d
(sz,
A)
A),
(s4, e,) 5
(s3,
A)
F=Qx{CI}.
for any i, j
and m, n where m f n,
t (sir An) # L(s5, An).
Thus
!{ e N0. However, it is inplicit
abova:
that :\e
in conditions (a) and (b)
value of min can only change by up
to a bounded amount:
from one cnnfiguration to the next in a derivation. For any dpda
accepti.ng che above language,
ubves
this cannot be true for the set of
in which a synbol d is read.
[,Ie
therefore conclude that
r.(M) g LL"
(ii) nf a rnaetline is nonsingular, then the appropriate constanc lri
clearn"y sees an !"ipper bound
raeion can
eharage
in
two eonfigr"u:ra&iones
in
erra
Lhe arount the height
clnc csurse
of any sequence of
of a configu-
e-moves,
for
any
sueh a derivation must necessarily be equiva-
Lent. Thus, from an argunen8 given in 51.4, it follows that the same
l"anguage can be aeeepued
by a realtime
machine.
46
To show
that the converse is not true, consider the
language
ianbcn) u {andczn}, which clearly be1-ongs to L(Ro). Let a dpda M
recognising this language reach the equivalent configurations (s,
and
(str rt) after reading a2nb
n.
Then from Observation
and and respectively
1.1 (gf.2) ve
for
.rr)
some large
rhar lrl is abour rwice
loltl , and further that the effect of ck for some "small" k is to
reduce (st, ,t) to a configuration whose stack is periodic right to
know
the top. Thus for an infinite set of distinct values of n,
string ek takes the two equivalent configurations to
some
new ones with
stacks stil1 differing in height by about n, but with one srack
being a prefix of the other. Thus M I No.
3.3 Alternate
We
Mn,
I
Stacking
now describe a way
for simulating
Erro
of constructing a single stack
machine,
realtime dpda M, fr together.
A configuration of M' can be specified by a pair of states,
one from each
segments
of
M and
il,
and a stack which consists
of words from the stack alphabets
two machines. The simulated configurations
recovered by taking the corresponding
of
distinct) of
M and
fr
the
can be
state component, and concaten-
ating the appropriate set of alternare
to obtain the original stack.
(assumed
of alternate
segments
On reading each
of the stack of
input
symbolrM'
simulates M and M by sirnultaneously manipulating the two topmost
segments
in its stack according to the respective transition rules
of the component
nachines.
Mt
n
47
More
formally, distinguishing the notation for
M and
ll
by
over-lining everything concerned with the tatter, so that M' has
stack alphabet f u T and state set Q * Q, we describe a typical
configuration of Mt by
t |
-1
([s,
sj,
where only the
olo2o3or]
...
uozn-262rr-lo2n),
last segnent, i2n,
may
be enpty.
hrithout loss of generality we shalL assume that the tno topnost
non-nuli. segments of the stack ar€ o2n-t = oA and i2n = ;f.
If
the
traneition'rulee of M, il specify that
(s, A) t, ("tr
lrof
)
and (1,
f) tr (6r, ii'),
I if rrrt is non-nul.! and nefu{A} otherwise, and sinilarly
ror i) then for input a in M' , rhe segment urAil will change ro
,niir'-.t, and the state rrom [s, i] to [rt, ir].
(rhere n e
An iuoportant
inplication of this notation is that if, for
exauple, ii = A, then the words un and r,lt wilL merge into
one
segnent si.nce they are from the same alphabet.
3"4
Main tr('esults
The alternate stacking eonstruction
can ensurc
is only useful to us if
we
that the simulating naehine is itself a pushdown automaton"
Definition
Alternate stacking for a pair of nachines, for a given set
48
of input strings is said to
succeed
iff the simulation of
them
together for these strings by the above construction produces stack
segments
We
of only
bounded size.
define an input- string o to be live for a dpda M iff it is
the prefix of
some accepted
string,
and a configuration
c to be live
iff L(c) * l.
Theorem 3.2
For M, M equivalent nonsingular dpda, alternate stacking
succeeds
for all live inputs.
Proof
We
shall
shsw
that in the simulating
machine
described in F3.3, no input a that is Live in M, il
M' constructed
"at
as
lead to
a configuration
([s,
s] ,
oriz ...
u,;)
with lrl > z(22 + m). Here z is the maximum of the lengths of the
shortest strings accepted by dpda of size no larger than M and M,
and m
is the larger of the nonsingularity constants of these two
machines, which are taken
to be realtime,
Let us assume the contrary. Consider the configurations e and
E
of
the
M and
fr respectively at the moment when the bottorn symbol of
segrnent
il was actually placed there. Let s be the input
read
since that time, which has taken u, fr to the present live configura-
49
tione cr, Er respectively. By our choice of E and a, all the
configurations in the a-derivation in fr ha.re height 3 l;1. Also, by
virtue of the alternate stacking construction, all the configurations
in the a-derivation in
M have
height > l"rl, for what is finally the
i segrent in Mt has been in existence throughout the o derivation,
and hence
the
segment below
it cannot have been increased in
the neantine (Fig. 3).
Let
;l
B be
a shortest string in
l"rl = l;1.
f.(ir),
and
let
B
=
BtB3 where
lBrl : lrl uy the realtime
property. 'Also B nust be a shortest string in L(cf ). But then B is
+(Br) E2 and
Then
the concatenation of segments induced by the popping sequence in
each one
taking
some
(q,A) to
sorne
(s2, A). If
such segment must also be nininaL and
Thus
Bl consists of a
sequence
of
B
is minirnal,
of length no rDre than
such
nininal
M,
each
z.
segments terminated
possibly by a proper prefix of another such segment. Let Bzt a
prefix of 83, be the completion of this last segment.
A^
A^
ca and -c2 4
cg. ClearlY
c2 4
Then 1et
l"rl-l"gl:lsrll".
Now consi"der
a shortest string y taking
M
from
cto ca.
that lyl : "
+
l82l
7
the def initions of E2, E3
Let c --Lt ct g.
(a)
"tt",rte
Then
22.
Then
l.i - l"'si :22.
But if M and
But from (a), (b)
we conclude that
l'1
and
(b)
are eguivalent then by construction
the observations that
l"l : l"tl
ca
and
_l
- e
ler
|
3.
:
l;l
50
M:
It:
fig. 3, Derivations in Proof of
Theorem 3.2
51
l"rl : l"'.1 + 2z - lillz.
Thue
if lil t " -
2z > m then we have
a
contradiction to the
non-
singularity condition.
l{e therefore have to eoncLude that for equivalent nonsingular
autonata, alt,ernate stacking succeeds for alL Live inputs.
n
Theorem 3.3
If alternate stacking succeeds for all live inputs for all
pairs of nachines M=E in sore class Ic D, then the equivalence
problem
is decidable for
X.
Proof
ig permitted by Lema 2.5, that M, M are in
Assume, as
form. First
can grow
llr for
that
we know the bound
to which stack
in the alternate stacking sinulating nachine for ll
Then we can
t(ld') is
Buppose
normal
segments
and
il.
effectively construct a pda Ut' with the property that
ernpty
iff
l,t =
il.
M" nimics the
M, E by encoding the top segrent
alternate stacking nachine
of Mf in its
f
inite state
controL. As Long as this top segment never gets larger than
the
given bound, M" accepts iff exactly one of the configurations it is
eimulating is in an accept mde. I.lhen the bound is exceeded, M"
proceeds nondeterninistically
to ninic one of
M
or il, and acceptsas
the appropriate machine would.
By assr:mption
for live inputs,
, if
M=
Ii then alternate stacking will
and the stack segment bound
succeed
will only be exceeded
52
once nothing rncre can be accepted by M and M. 'Thus l-t" will be
empty by construction"
Conversely, if M" is empty, then clearly no input can produce
different behaviour in
M and
il, which are therefore equivalent.
Thus, if we have an a priori bound, \Je can test equivalence by
constructing this pda and testing it for emptiness. However,
if
we do not know
this bound, by enumerating
and
even
testing for empti-
ness the possible candidate machines, we can obtain a
partial
decision procedure, That is,we construct pda of the form M" for
assumed segment bounds
of L, 2,
successively. If M, M are not
equivalent, then none of these constructed nachines can be
while if they aren then one of
We
them must be.
therefore have partial decidability and hence, by
earlier observation, decidability.
Corol
empty,
an
tl
lary
The equivaLence problem
for nonsingular
machines
is decidable.
Proof
Immediate from Theorems 3.2 and
Note
3.3.
D
l.
Nondeterminism
in the siurulating
rnachine
is an inessential
convenience, which can be avoided. M and fr couLd have been preprocessed
to recognise instantLy whether a configuration
reached is
53
live, and to enter a distinguished
dead
state if it is not.
A simulating deterninistic pda could then be easily constructed.
Note
2
I{e can show that the order
of the bound on the
segment growth
derived in Theorern 3.2 is achievable for simpl-e urachines. For these
we have already observed
that the nonsingularity constant m is
zero,
and, in 92.2, that z is El(t.Logh). Thus the bound z(22 + m) is of
order El (t.logh) also.
Let A1 be the starting
Example
ht.
syrnbol
for the simple grarmar of
2.1 generating the singleton string {an} where n is of order
Then consider the simpLe languages generated
by B and C
resPectively:
B -5 nrBz
n2
-3, n1f2
c j'lrcrcz
C2
3, Cp2
n2J*41
Czj*n
Bl-3+A
C1
3n
set {arban+r I r 3 1}. However, if
s - srx$6n, then t l* E , but C 3t Al. Thus in an alternate
sEacking rnachine fon these tvo, the top segment can gro\t to size n,
Both B and C generate the
which is of order EI(t.logh).
Note
3
To illustrate a s}ightly different way in which the success of
alternate stacking can be used, we outline a proof that the equivalence problem for symetric dpda, Syo ( 1.4)rcan be solved in
polynomial tine.
54
We
recall from the notes to
machine can be transformed
Leruna
1.2 (91.7) that any such
to one, only polynomially increased in
size, with h = 2. Since alternate stacking clearly succeeds,
and
the enptiness test is itself polynoniaL, it remains only to prove
that the simulating
rnachine
is itseLf of only polynomial size.
The
key observation here is that, instead of having to mernorize the
ruhole top segnent,
to
remenber
it is
now
sufficient for the finite state control
just the top symbol of the next-to-top
segment, and to
treat the top segment itseLf in the normal stack-like way.
The
property of Syg machines from which this can be derived is that
any
rcve that causes the stack to increase, also eauses the value of the
function min( ) to increase.
We
note, honrever, that for the class 561 and therefore also
Syg, algebraic properties can be derived9 which give nnre detailed
insight into these restricted classes than our more macroscopic
approach. Thus for Sy6 grarnr,ars with a fixed terminaL alphabet
with h = 2, an equival"ence test on the lines of Korenjak
can be obtained, that works
and
and Hopcroftg
in tirne cubic in the number of
non-
terminals.
3.5 Undecidability gE_Inclusion
FriedmanlS prolr"s the undecidability
for
dB
-
S schemas (chapter
of the incl-usion
problem
8) by showing that for each instance of
the Post Correspondence Prob1em3, a pair of appropriately related
dpda M1, M2 con be constructed
iff it has no solution.
with the property that L(Mr) c
The construction
is valid for certain
L(Mz)
5.'.,
classes (e.g. dB - S) which accept
b:y
enpty stack'buc are essentially
not real-time, and also for rea.Lcime fanil-ies that do not tiave this
empty stack
We
restriction (e.g.
use a
S).
similar formulation of the problem but, by taking a
refinement of the correspondence prnblem that is implicit in the
customary proof
of its unsolvability,
undecidable even
Because
we can show
for a real-time class with
of the slight novelty in'roLved,
we
that inclusion is
enrpty stack acceptance.
shall digress here
to
give this undecidability proof.
Theorem
_3..4
For M1 e S0, M2 e Ng ir is undec,idable whether i"(M1) . L(Mz).
Proof
The instances
of the
correspondence problem
just those obtained directly from Turing
standard proofs
shord
that for
of its undecidability.
each Turing machine
that we take are
machine computations
in
the
Thus Hopcroft and lJl1man3
a correspondence problem with the
foll"owing properties can be constructed:
Let X = xl, .". xk, Y = yle .,r yk be the
words over a
finite alphahet X.
kle def{ne
a
sequences
sequence
of
non-nulL
of positive
5 k) to be a solution (or partial sotution
respectivei.y) for X, Y iff x:xil .".
(or a proper
"in is equal to
integers i1,
io
(a3"1
prefix cf,, respectively)
)llli1 ... yirr. Then the construetion is sr.ech that a solution exi.sts for X, Y if and only if the
!:he stri"ng;
Turing nachine has a terminating eomputation, but
no
partial solution
56
for Y, X exists under
any circums tances.
lle construct two dpda M1, M2 which both reject strings not of
the form aRB$ where c is a string over the integers {1, ... k},
*
8eX and$is
a
distinguished terminating character. Both
initiall-y
a
special syrnbol A at the bottom of the stack.
M1
crRB$
place
machines
is a simple (Sg) machine which accepts exactly the strings
where B
is the word indexed in
Y
by the sequence o. Thus M1
records aR in its stack, matches it with B, and accepts iff the
natching never fails and the $ is read when there is only the
symboL
A Left in the stack. It rejects al-1 other strings by placing
a
special symbol on Lhe stack which permanently freezes its motion
once matching
M2
fails.
also tries to match o and B, but
If natching is successful
reached
and the $
specified by
X.
is read when the syrrbol A is
at the bottom of the stack, then rejection occurs. Other-
wise, if matching fai1s,
M2 goes
effect of any X-input is to
Wtren
now as
into a second state, in which the
pop the stack without changing the state.
the A is reached in this second state, all input synbols leave
the configuration unchanged, except for $ which pops the A. If
acceptance
is defined by empty stack then clearly the length of
a
shortest string accepted frorn a configurafion is just the height of
the configuration. Thus it follows that M2 e No.
If there is a solution to the
chosen instance
correspondence problen then the inpuE
of
the
string which specifies it is,
by definition, accepted by Mr but not by Mz. Thus L(Mr) - L(Mz)
implies that it has no solution. Conversely, if it has no solution,
57
and oRB$
is any string
accepted by
Mt, then there can be at most
partial solution specified by an initial
segment
of a.
a
However, by
our particular choice of X and Y, we have ensured that at the point
where the
syrnbols
*
c,
B
faiL to
match
in M2, the stack of
M2
has fewer
left than that of M1. Since M2 proceeds to accept any string
from I-$ that is long enough to empty its stack, it must accept
in particul-ar, since
"Rg$
the remainder
M1
will take aE least as long to
accept
of this string as M2 will take to reach the bottom of
its stack. Thus if there is no solution then L(Mr) . L(Mz).
Since
for
any Turing machine rde can construct an instance
of
the
correspondence problem, and hence also the machines M1, Mz with the
above
properties, it follows that if
we could decide
inclusion for
such dpda, then we could decide whether Turing nachines had
terminating computations, which, howeverr Ite cannot. I
3.6 A Conjecture about R0
We
conjecture that Theorem 3.2 hol"ds for M, fr e R9, and hence,
by Theorem 3.3, that the equivalence problern is decidabl-e for
$ere we shall-
outline the proof of a property of
not Btrong enough to prove the eonjecturer
maY
R9
R9.
which, though
throw some light
on it.
Lema
There exists a function F(Q,
in 9, with the property that for
t, h, 9), aslmptoticall-y linear
t:tro machines M,
il e Rg rdith
58
par:rmeters appropriately bounded above by
q, t,
and
h, if c, i tt"
equivalent configurations of M, il respectively, then
if c t(a) cI nhere cris live
Jcr' s.t. c f(cr') c1
and C-->
and
(l
l"tl - l"l
t
11 where
2 e., then
l"t | - l;l > F(q, t, h,
Outl-ine Proof
Choose
the crr to give the shortest stacking derivation from
c
to c1. If ct, c" are any two configurations occurring in this
| 7 zt (where z is as in
then it is easy to verify that min(c') > min("").
derivation with l"' | segmented
segment
1""
Theorern
Thus o,t can be
into lengths of no npre than 22, such that
in i.t takes c,
and therefore also
with increased values of min.
The
i, to
3.2),
each further
new configurations
nain argument then is to
show,
using this segmentation, that no subderivation of the or derivation
in fr can produce a stack drop of nore than G(q' t, h),
El function. This is
where G
is
an
done by assurning the opposite and inducing on
the number of states that can be reached at this lowest level by
popping derivations from the previous configurations
in the
sub-
is that if min(s1, ,r'; '
nin(s2, rrrott) for some sI, s2, o, ot, ,tt, such that ltt l/lr"l 2 zt
then 3s s.t. (s, ,rl) is not reachabLe from (s2, otl"). Since the
reading of the successive segments of qt must lead to a set of pairderivation.
The observation used
wise inequivalent configurations, the statement of the lemma then
easily follows.
I
P.\ "
59
Chapter
4.L
FINITE.TURN
4
AUTO},IATA
Introduct,ion
We
consider a family of deterninistic pushdown automata on
which the only restriction is one on the movement of the stack.
Using a technique related
we show
to the
that the equivalence
one given
probLem
in the previous chapter,
is decidable. fuain
a pda to try to simulate the tested machines, but
now
we
this is
build
an
intrerently nondeterninistic one. Since two equivalent machines in
this family
may have
totally unrelated stack
movements, a deter-
ministic simrlation by a pda is no 1-onger possible.
A speciaL case are the
l-turn machines.
the inclusion problem for these is
in our proof of
Theorern
The
wel-l- known, and
undecidability of
is also implicit
3.4. A further restriction gives a class
which is intimatel-y related to the tr*o-tape acceptors of Rabin
and
Scott6, for which equivalence has already been proved decidabLe
Bird7.
From
by
our proof, therefore, another equivalence test can be
abstracted that is not directly related to that of Bird and, while
possibly less efficient, involves a technique of apparently
general-
more
applieability.
Although the finite-turn property is essential to the proof of
the effectiveness of our main construction, we know of no pair of
equivalent dpda for which a simulating machine of a broadly sinilar
nature cannot be found. Thus it is possible that, when more
detailed knowledge about the structure of dpda in general becomes
50
availabLe, our methodology nay be extended to prove decidability for
the unrestricted case.
4.2 Definitions
We
istic
define the class T c D of finite-turn dpda as a determin-
anaLogue
of the nondeterministic class studied by Ginsburg
and
Spanier I 9.
Definition
A derivation
it
in a dpda is a stroke if either no single nove in
decreases the stack
(i.e. an upstroke) or if no single move
increases the stack (i.e. a downstroke).
Definition
A dpda M belongs
to the class n - T, for
some
n > O, iff every
derivation in M fron the starting configuration can be
no nore ttran
segmenred
into
n + I strokes alternating in direction.
Definition
A dpda M e T iff M e n - T for some n >
In other words, the restriction
O.
we impose on T
is that in
the
set of all derivations from the starting configuration of a machine,
there is a bound on the number of times the direction of the stack
movement can change.
51"
Clearly the language {arbr I t:
l}m belongs to r,(l2rn - t] - rl
Further, -in general, L([2"] - 11 =
but not to L([Zr - Z] - rl.
r,([Zn - f] - T). This is because after an even number of Lurns the
stack is inereasing, and so after its last turn a [2"] - T machine
can only proceed
to
recogni.se a regular
such a machine cannot be essential
set.
Thus the
last turn in
for recognising the language. In
particular we note that L(O - T) are just the regular sets'
lJe
give
Erilo
properties of f inite-turn autornata which can be
derived from analogorls properties given by Ginsburg and SpanierI9.
Ttrey both depend on
l'lt
the fact that for any pda M, an equivalent
way and,
in addition, to
in its state set lthether the derivation it
has been doing
can be constructed
remernber
one
to minic
M
in every
is in an upstroke or a downstroke.
Observation 4.1
It is decidable for
M e D whether M e T.
Proof
For
M
we eonstruct a nondeterministic pda M" wtrich mimics
the stack movements of
M
by
e-moves instead
of reading the
norrnal
inputs. In addition M" remembers in i"ts finite-state control
direction of the current stroke,
this changes.
Then M e
and reads
Thus
the
a character $ whenever
T iff L(M") has strings of only
length (i.e. iff L(M") is finite).
all
to test
turn property, we test L(M") for finiteness.
D
M
bounde<l
for the finite-
52
Definition
AmactrineMen-Tis
ordered iff
(i) its state set ie the disjoint union of sets Qg,
(ii) a state
reached
is in Qi iff
the
Qnr and
in a derivation from the starting configuration
derivation has undergone exactly i turns.
Definition
A state s
of
an ordered nnachine
is of order i iff
"
e Qi.
Observation 4.2
For any M e n - T we can construct an equivalent ordered
Mt e n - T.
Proof
l{e first nodify M so as to be abte to remennber the stroke
direction,
and then make
n + I copies of the transition rules,
each
referring to a distinct state set Qi, for O : i _. n. We obtain
by slightly nodifying this ners set of rules. Mt starts in the
starting configuration corresponding to
way except
that
whenever
QO, and mimics M
in
M'
every
it is in a state s e Q1 and the current
stroke is terminated by a nove in the opposite direction, the
appropriate transition occurs to a state of Qi+i, instead of Qi.
The computation continues
rules, with
changes
within the new copy of the transition
acceptance defined as
in M, until further directional
occur. Mr is cl-early deterministic
properties.
u
and has the required
63
We
notice that we can test easily whether M e n - T for
a
particular n, by buil,ding the candidate ordered M' t [n + t] - t
equivalent to it, and testing whether any derivation in Mt can reach
a state of order n + 1. The latter can be done by suitably
redefining acceptance in M' and testing for emptiness.
4.3
Proof Srrategy
We
shaLl prove that equivalence is decidabl-e for T by showing
that it is partially decidable. In particular,
for
each
pair Mi,
M2 E Tp
a faurily P of
we shal1 show that
pushdown automata can be
constructed with the properties that
(i) if L(MI) = L(Mz) then for
some M
(ii) if L(Ml) * L(M2) then for all
Thus the enunneration and
M
I
t e P, L(Mt) 6,
=
and
eP,L(M)*1.
testing for emptiness of all the
in P constitutes the required partial decision
machines
procedure.
For sinplicity of presentation, we shaLl- form a machine M from
M2
as f ollows.
initially reads a character from the input tape and,
depending on
the dis joint, uni.on of the two tested machines M1,
M
what this is, moves to the starting configuration of one of M1 or
which it then proceeds to simulate precisely. Clearl-y, if
then M e T
also.
now rre need
The advantage we gain by
MI
1
ll2 e T,
this construction is that
only talk about the equivalence of configurations in
single finite-turn
M2r
a
machine.
In our constructions
we
shal1
further assune that this
rnachine
64
M
is:
(i) in
normal. form
(ii) ordered
(SA.Z1.
can be transforned
We
4
and
2.5 and Observation 4.2 ensure that any finite-turn
Lenma
4.
(52.4),
into this
autornaton
form.
shall present our proof in a nurnber of
stages.
Paral'lel Stacking
We
pda Mt
shaLl first outline the general form of the nondeterministic
that is to sinnrlate M.
A configuration of Mr has a stack which can be thought of
as
having a Left track and a right track, the top of each one being
associated
with a state of M.
of
size by special syrnbols, called 'rceiLings", occupying
bounded
The stack
is
segmented
into
lengths
both tracks. The finite-state control is able to manipulate
directly the top segrpnt (i.e. the part of the stack above the topnost ceiling).
rn each segnent both tracks contain stack
words
of length
greater than one. rnto each ceiling is encoded the following
information about the previous history of the computation:
(1)
a quadruple (st, At, s2, A2) which states that at the tine the
ceiling nas created, the two tracks
(sz, Lz) respectively.
had nodes
(st, At)
and
65
(2)
an inlic,ator
IgLr from {L, R}2 specifying the connecEion of
the tracks above the ceiling to the ones irnmediately below.
For example, (L, L) will mean that both the tracks above are
to be associated with the left track
Each configuration
to two configurations of
below.
of Mt is to be interpreted as corresponding
M
in the obvious way, i.e. the M configu-
rations can be recreated by taking each track in the top segment of
M' and concatenating it with the appropriate words in the
below as specified by the indicator pairs
in the ceilings,
segments
and by
adopting th'e corresponding state.
The basic operations
of Mr are to try to urirnic simultaneousLy
for all inputs the transitions appropriate to the two simulated con-'
figurations of M. rn order to be able to do this, while at the
same
time maintaining an upper bound on the length of the segments that
can
arise, the machine Mr can also on occasions,
depending on the
contents of uhe top stack segment, do one of, the f,ollowing additional
operations without reading inputs:
(a)
if
one
of the tracks in the top
segment
is empty, and
the other contains a word from a certain set of tshortf
words, then the ceiLing below it is rernoved, and the
tracks formerty irunediately below and above this ceiling
are fused tc form one segnent, in the manner specified
by the indicator pair.
(b)
if both tracks
have more than one symbol, then a
is placed to be jr:st below the top symbol of
The indicator pair (L, R), and the quadruple
ing to the modes of the tracks are
ceitins
"
encoded
ceiling
each track.
correspond-
into this
66
move (a):
gt
st
(1
c
c
) (RR)
rl
ll
A
B
A
B
A
c
A
A
A
A
A
A
')
(
) (rR)
) (RR)
rcve (b) :
st
lc
(sBsfC) (tR)
B
__-:_l
A
A
A
(
C
)(
)
s4
rpve (c) :
Ar
(r) t
4l
(t
(s A s2A2) CtE
(erAt e2A2) (LR
1A1s2A2) (LR)
Fig.4
67
(c)
if one track is
eurpty
or has just one syrnbol, while the
other contains a word from
some.
speci.fied set of tlongl
words, then Mt has the foLlowing nondeterministic choice
of moves: A replacerent stack word, with the same mode
and
indicator as the large track, is introduced to
reptace either one of the tracks in the top segrent.
The simulation
is then to be alLowed to continue to
compare the newly introdueed configuration
with which-
ever one of the old ones is left.
The replacement word
is tshortr in the
is uniquely specified
above sense, and
by the Mr configuration.
Examples
in fig. 4.
We
its quadruple
trpveg
of
each
of these three kinds of
moves
are illustrared
notice that once a ceiling has been created,
cannot be modified,
of type (a) or
(c)
its indicator
can be changed by
.
I{e define acceptance
to occur in Mt when both tracks of
top segment are in reading nndes, exactly one of which is
accepting rcde of
4.5
though
the
an
M.
Existence of a Verifying Machine
l{e have
to shor, that if
M has
two equivalent configurations,
then there exists a simulating pda M' of the kind just described,
which starts with these configurations in its tracks and accepts
input string.
no
58
Clearly any pair of eonfigurations of M, reached via the
input string from a pair of equivalsnt configurations, will
equivalent.
we have
of step in Mt,
back
to
show
be
that when the tracks get too much out
we can, by uraking appropriate replacements, get them
within a finite
bound
while still naintaining the equivalence
of the pairs of configurations simulated. Further,
that the replacements, and the whole simulation,
by a
same
tre have
can be
to verify
carried out
pda.
[,le nors
Q
define the function
Rep over
* f xQ x f xQ x (fu {n}) * Q' f x {1,
2}
Eo determine these replacements.
Definition
Rep(sl, Al , s2, A2 : s3e A3 3 s4, A4, 0) = ar,where ur, if
defined, is a shortest non-nulI stack word with the properties that
(i)
(s
r, Ar) --t (sq,
(ii) yol,
t^l2
. f*, if (s1,
then (sa,
arlurA4)
,
oA4)
and
tr:1A1)
= (sf ,
=
(s2
,
w2A2)
rr.rUA3) .
I,lithout having to construct this function or to determine the
arguments
for which it is defined,
we
a finite domain, ne can define p to
be
can observe
that since it
has
the maxirnrm of the lengths of
the stack words in its range.
, if the top ceiling contains (st, At , s2e A2) and (L, R),
and the right tsack above contains (s3, A3), and the left one some
rn
Mr
69
(",., ,'Aq) where (sq, Aa) is a reading mode, then we make the nondeterninistic
replacement
by (s,*, oAr+) whgeRep(s1, Al, sz, A2 :
83r A3 ! 84, Aq, 2) E o, provided that lrl . lr'1.
This is illus-
trated in fig. 4(c). If the indicator were (L, L) then the vatue of
for 0 -
Rep
1 would have been appropriate,
were interchanged we would
treat
them
while if the top trackg
in a sinilar
It is easy to see that under the
way
circr-rmstances
by sp@try.
specified for
the replacement, the function Rep nrst be defined, for ro' is itself
a guitable candidate for its value: I{e first notice that the
sinpleet way of realising the necessary conditions is if there is
Some
c 8.t.
(sr, Ar) *
(sz-, Az)
Then tor
t"u,
,,,'Aq)
and
* t"s, As).
clearly has the properties required of a value of
(except poesibly ninimality).
In addition, all other
Rep
ways
of
reach-
ing the specified situation in M' (i.e. if nondeterministic replacementa have been rnade
in the meantime) invotve only valid
eqrrivalence
preserving replacetrpnts which are defined completely by the ceiling
concerned and subseguent computations, irrespective
of the stack
contents below. Thus by induction on the nuniber of such replace-
rnts
made
during a derivation in
repLacement
is available
whenever
Mf
,
we can deduce
it is
that a valid
necessary.
A finite-state control can carry out these replacements since
Rep
is finite in every senae, provided that it can always appropria-
tely nanipulate the whole of the top segtrent.
Thus
it renains to
prove that a borurd x exists such that no track segment can
become
70
longer than x in any computation.
It is here that
che
finite-turn
property is required.
I^Ie f
may
irst consider in turn the various eventualities that
arise:
(1)
If both tracks are steadily increasing then new ceilings
will be frequently
(2)
added and no segment
will
become 1-arge.
If one track is increasing steadily in an upward stroke,
while the other one is stationary (i.e. with stack height not chang-
irg) then after a stack increase of q2thin the one, a valid
ment musf be
possible. For there
repLace-
must be two intermediate points in
this derivation at which the mode of the growing track and the configuration of the stationary
one both
repeat.
Thus
a replacement
word can be obtained by cutt,ing out a segment from the growing track.
Therefore, for however long this track is trying to grow, the effect
on Mt
will be to
repeatedl-y rnake nondeterministic rpves so as to
keep the segment lengths bounded.
(3)
If one track is decreasing then situations can arise in
which the length of the segment created depends in a bounded way on
the size r:f the prerriously existing segnents, and is not a priori
bounded
by p + 1". There are just tlro ways in which a Lrack can grotl
"out of bounds" without imnediately being arrested by a nondetermin-
istic
replacement:
(i) If one track in the top
segment
is empty, but
the
other one is not Eqrite i"ong enough tcl qualify for a replacement, then
the ceiling below may be rennoved by atr (a)-move. Thus the resulting
fusion may create a top
segment which
is suddenly longer by y
than
before, where y is the previous bound on the segments occurring in
VL
the configuration.
(ii) A track may grow larger
replacement being possible,
and larger without
if the other track
does
not
become
enpty or a singleton in the meantirne. This second track can
assured to be steadily decreasing since, as
a
be
in (2), stationary
periods do not contribute to growth. Thus if the decreasing track
is initially
of height y, then this
may cause
to arise a segnent of
length no more than q2t2hy.
The
crucial observation for both (i) and (ii) is rhar rhe gain in
length achieved by each nethod can only be exploited to achieve
further gains by the
sarne method, once
the simrlated configurations of
M
the order of at least one of
has increased (i.e. after a turn
has been nade).
More preciselyrwe
define the order of an M' configuration to
be the pair (i, j) when i, j are the orders of the states of rhe left
and
right track respectively.
Thus the
effect of a replacement on a
configuration of Mf of order (i, j) is to create one of order (i, j)
or (i, i) or (j, j), while an ordinary simularing move would lead ro
one of order (i, j) or (i + f, j) or (i, j + 1) or (i + 1, j + 1).
we have
to enslrre that the sinulating nachine cannot enter
any ttloop"
which could cause the segrnents to increase in size indefinitely.
The ease we have
to consider is that of derivarions from configura-
tions of Mt of a given order to others of the same order, via
of differertorders. clearly these must involve
kind that substitute for the
empty
or singleton track.
definition the rcdes of the tracks are the
repl-acement,
repLacements
same
ones
of
the
Since by
after any such
either both tracks will be in an upstroke, or both in
downstroke. rn the former case ir is obvious that before the
two
a
72
tracks can again extribit dd.fferent behaviour, turns must be rnade by
both tracks. In the latter case the height of the top segment can
be exploited by method (ii) for further growth, after a turn
been made by
just
one
of the tracks.
Howeverrthe new gains
has
will
only be achieved in the track that has undergone a turn, and cannot
be exploited for further gains untiL a turn has occurred in the
other track also.
Thus the times when successive gains can be made
of
segments
in
excess
of the bound p,
in the size
can be regarded as occurring
at periods"during which M' has configurations
whose orders form
a
npnotonic increasing seguence under the ordering defined by:
(i, j) r (i', j') iff:
1r J >
,l
l"
,
and j
or i > it
If
M can
paLrs.
>J .f t
j',orjtj'andi:i'.
Hence we conclude
size of a segment that
Consequently a
that there is a bound on the
may occur
in any computation of
Mt.
finite-state conErol is sufficient to specify for
each stack word
moves
J
only make n turns, then this sequence can have no more than
about n2/2
maximum
or 1,
that nay arise in a track of the top
of type (a), (b) or (c) are appropriate
segment, whether
and what form these
should take, and to carry out the normal simulation with inputs
otherwise.
4.6
The Decision Procedure
If a machine Mt is constructed for the tested
automata
Ml,
M2
73
as described above,
with an arbitrarily
and segment bound, and
guessed repl.acement function
if it is found to be enpty, then we have a
verification of equivalence. For let us assune to the contrary that
the starting configurations of M1 and
each
point in the conputation of M'
M2
are inequivalent. Then at
we can folLow the shortest
string distinguishing the sinulated configurations.
Then as each
input character is read, the rank of the sirmrlated pair
Wtren
one
decreases.
nondeterministic replacements are made then, clearly, at least
of the
new
pairs has rank no greater than that of the old
Further, the rgplacerents
one.
all occur in reading modes to ensure that
further progress along the shortest distinguishing string can be
nade
imediately.
The only exception
to this is the situation that
occurs when successive replacements without intermediate inputs have
to be made because of a long stack decreasing e-derivation.
this
mr.rst once come
conclude
to an end since the stack is finite.
that if l\ I
142
However
Thus
we
then for some input string we will reach
an
accepting configuration of Mr after a finite nurber of steps.
Conversetlr !t€ have already shown that if Mt = Mz then some M
exists which aLways simrlates pairs of eguivalent configurations of
M, and therefore accepts no string.
Thus enumerating such siutulating machines
for
M1, Vl2,
for
aLl
possible replacement functi.ons and appropriate fi.nite state controls,
and
testing each f,or emptiness is a partial decision
f,or equivalence.
The mai.n
result therefore follows.
procedure
74
Theorem 4.1
Equivalence
of deterministic finite-turn
pushdown automata
is decidable. I
Corollary
Equivalence
of deterministic
Ewo-tape acceptors6
is
decidabLe.
Proof
For a Ewo-tape acceptor M' with input aLphabet Xt, and
(cl, g) on its input tapes, construct a dpda M, with f = X',
X=
is
X'u {$}, that accepts exactly the strings
accepted by Mf" The oR part
is read by M,
oR$B such
words
and
that (c,
B)
and stored syrnbol by
synbol on the pushdown stack. Once the $ is read the finite state
control treats th.c input tape and the stack exactly as M'treats its
pair of input tapes.
whichever
M can be made
of the various criteria of
and Scott6, nird7l
to accept whenever M' accepts,
acceptance
is taken (e.g.
. l,t is evidently a one-turn machi.ne. !
Rabin
_/{
Chapter
5.
5
ONE -COIJNTE
R AI'TOMATA
1- Introduction
Counters have heen snudied
in a variety of
computat,ional
contexts as simple and naturat mechanisms for unbounded storage.
A counter can be convanientl-y regarded as a stack with an aLphabet
of just one syml"'ol".
The gain :i.n simptric!"ty achieved by this
restricfii"ar. is that 6 eouat€r llae eesentiall"y just two behaviours,
depending on whether
it is
enapty
or not.
Despite the apparent severity of this Lirnitation, it is well
known
that
severaL basic probLems that are undecidabl.e
for
automata
with staeks, remain so even when these are restricted to counters.
Such ',:nderi-dability resuLte r,sa be- deduced from rhe theorem
cf
1A
Minsky'" that any Turing machine can be sinnulated by a 2-register
machine" 'fhisr tratter can be regarded as a deterrninistic automaton
with twc counters, a finire sEate controL,
and
transitions reading
no i,nputs. The computation executed by such a machine can be
described by a sequence
empriness *ond{ei.ons
configurations
of triples,
cf fhe
resrchecl
.
each giving the state and the
trvo counterso
Si.nee
of the
successive
a deterrninistic one-counter automlton
(doca) can check whether successive triples correspond to valid
transi.t!.ons, and alsa whecher the ,:veralt aetion induced in one of
the tg{o colmters f s *ons.i.stent with the sequence, the terrnination
prcbleu for Z*register mactrlines (arad hence Turing machines) can
reduced
to bhe nullity of -{ntersection
this latcer
probJ.en musfi he r-s+decidabLe
probLem
for doca.
Thus
be
76
Since the class
of doca CC) can be
sholin
to be closed under
corylenenration by the siune arguments as used in 92.41 52.5 for
dpda, it can be easily deduced that totality, and hence
equivalence, for nondeterministic one-counter autonnta, and,
more
pertinently, inclusion for the deterministic case, are alL
undecidable.
Against this background we are neverthelees able to develop
a detailed analysie of the structural properties of the computations
of doca,
and hence derive a decision procedure
lJe have seen
that any
for
doea can be transformed
equivaLence.
into
normal
form (52.4) with h = 2 (91.7) with no more than a polynonial
increase in q
We
shall
assume
that the machines are all in
thie form, so that rre can e)rpress all the derived properties in
terms of just rhe one paraneter q.
The decision procedure takes the form
of a nondeterninistic
sinul-ation, as in the previous chapter, but now the properties
deduce
are eufficient to speci.fy exptricitly the nondeterministic
replacenents regr-rired,
nondeterminisrft here
Vire
note, however, that, as in Chapter 3,
is a convenience that is not essential.
procedure could also have been obtained
number
we
of deterru.i.nietic
sinmrtations.
A
in the form of a finite
77
5.2 Preliminary
Results
The function S(q) (which also gives rhe maximal order
in the syrunetric group on q elerents) will. be the
factor in the bounds we derive.
element
of
an
dominating
Definirion
S(q) = nrax {{,.e.m.
Lenora 5.
inr} I Xri = {, ni
> 0}
1
-_
s(g) =
"6Toaq'1(q)
where 1(q)
*
1 as q *
-,
Proof
(outri.ne) uaing etandard
nuurb.er
theoretic resulte, this
aslmptotic functi.on can be easily obtained for the case where
} in the definition is additionally reetricted to be a set of
prines. To show that the sane bornd is valid for arbitrary {nr},
it remains to prove that the possible extra contributions from
{n.
is not significant. Details of this
given in Valiant and patersonzo. I
prime porrers
argument are
For a deternninistic one-counter automaton M, we describe a
configuration c with state s, and counter contents n (n > 0) , by
(s, n). For this c, we u6e c + m to denote the configuration
(s, n + m)r provided that n + m > o.
78
Definition
The input word B
is a stan{$ffS,eqg
for the configurations
c, ct iff
(i)
B
ie a shorrest string
(ii)
g
= Blg;83
(iii)
where
B
such
that c ?
lgr8sl. q2, llzl
c',
= q, and r > o,
Forgomestatese andwrd>0, forall-vs.
RnV
-L*2
c ---:+
(se ,
w
-
t.
O
and
<vSEr
vd).
Legrna 5.2
For a doca with q states there is a positive integet Z,
greater than s(q), such that
c?
if l"l
ct, then there is a standard
loop drop d divides
l"'l ,sequence
q,2,
l"'l
> q2
no
"r,a
for c, cr in which
the
Z.
Proof
We
define the efficieqcy of a state s to be the maximtm vaLue
(possibly infinite) ot d/ fyl, where, for all sufficiently large n,
the derivation (s, n + d) *
(s, n) existsrbut repeats no state
except I at the beginning and end. Clearly d, l1l S q.
Suppose
o ie a shortest string such that
" *
"t.
We mark
the laet occurrence of one of the states with greatest efficiency
af\
in ttr^is derivation, Let this state be sur
y,
simpl-e loop be generated by
and
ret its efficient
and carrse a dr:op of
d, where d > c,
excise from this derivation a set ot (not necessarj.lv sirnple)
Now
disjoint loops of
to
due
is a multipre r:f d,
thern
preserved"
maximal tertal
length, suctr that the totaL
ar:.rl
that rhe
trte ean show
drop
fhe rnarked occurrence of, so is
l-ength m of the remaining derivation
is no more ehan qi * t* brl firsfr rgs4lpring that at 1,east
t = [(m + Z)iq - el disje,int simple loops, nor containing the markecl
su in thei.r interiors must occur it.
some
But
k > d would imply that
non-null subset of these 1"oops accounts f,cr a total drop ttrat
is a nultiple of d. This subset eould therefore
have been removad in
the originsl exe.i.sion contrary to the maximality conditiou"
(nc
+
2)
I
.q
* 2<d-
1-, anel $o
Tll
<
d < q. By a simitar arglffient ir
qi * i follows
c,en
Ilence
inrnedi.an:elv {.f
be shsr,n that t?le Length of
the derivssli.on *f,ter the malr,.*C s* i"s i*-cs than q(q - 1), if d < q,
Let
g*
*$* be Ehe input stri,ngs fnr the parts of this remaining
derivation before and after ihe
choseR oecurf,enee
'thene cLea::?"y, for some integer
r
of se reepectivel.y.
rtR
P1i',*
e
..''--'-'
'
,"1
'
+ l-ci
,
+
Eut
l*l* l*'i
> r'?
>
m
::rn ! i e{ r.ha;:
'!t
I
i:-,
r :, {";.
t
f.
since the bounds derive'C f;r the rarts of this derivation in the
d . q, and eiraple
opeciaL argtlF"le::lrs
that the eounter is at
na
in the trivial
case d =
e'
case
€osur€
strlge e'*pt:y. Also, since we have repl-aced
80
arbitrary loops by ones of at
l-east
the sane efficiency, BrttB,
nust stil1 be of ninimal length.
I,Ie not,e
that
Ehe caee
of a null
Y has not been excLuded
this argunent. t{e aLso note that a sinrpler
if the bounds of
q2 were relaxed
To obtain a value
to
argument would
in
suffice
2q2.
for Z we investigate the set of possible
values of d in the above construction. For each state s
we
sel-ect, if possible, a maximally efficient sinple loop Ehrough s'
and denote
the set of states in this loop by Loop(s). ClearlY' if
st e Loop(s), then the efficiency of st is greater than or equal to
the efficiency of s. Alsor any standard
sequence whose
principal
loop is based on s couLd be replaced by one based on s' , by
applying the construction to the derivation of the old sequence,
in which st
must occur
sincer
by
definition, r > O.
Let s t > I be the transitive cl-osure of the relation defined
by en e Loop(a). Defining s, sr to be equivalent iff st > I
and
s 2 st, the relation > becomes a partial ordering on the
equivalence classes" Let
one from each clase
slr..r8k be a selection of representatives,
that is
maxinaL
in this ordering. It is easily
verified that the corresponding loops
must be
disjoint,
standard seguences can always be based on some such
drope due
z = 1,. c,n { <la a. is the drop due to
I
Thus Z
toop.
to atrl these must add up to no more than q,
divide Z where
< s(e).
I
Loop
(sr) } .
and that
Then the
and each one must
81
That thie bound for Z is achievable can be verified by
examioing Exanrple 7.3.
As we are concerned only
with asytnptotic bounds, and as
SCq)
clearly dominates any fixed polynomial in q as q becomes large, it
will be sufficient for our
rather
we
etran
derive.
technique
purposes
to prove the existence of,
obtain specific expressions for, the various polynomials
The
proof of the foLlowing
Lerma introduces
a useful
"
Lemg 5.3
Ihere is a polyaonial F3 such that for any configuration c with
l"l > nr(e),
(i)
and any positive
uank
(ii)
nultiple Y of
Z,
(er c*Y) +Y/q<rank {c+Y,, c+2Y) <rank Cc, c+Y) +Yq,
c = c+ Y iff c +Y = c + 2Y.
Proof
Assming that c and c + Y can be distinguished, there nnust be
nrinimal dietinguislaing sequellee S6*
a
whene
.: bt (s, q-)
;-
for
eooe
ensures
s. Provided that p, is sufficiently large, Lema 5.2
that I
noay
be taken to be in the forn of a standard
OUOfOr. Let the drop due
ta 3, be d,
where
d > o. Since
Bd
sequence
82
dietinguiehes c and c + Y,
clearly grul * t'u
uro distinguishes
c + Y and c + 2Y. since lgrl < q and d > 1, the right
hand
inequaLity follows.
r > 'Ild,
to be a rninimal string distinguishing c + Y and c + 2\.
Then
8rl'r- t'uuro distinguishes c and c + Y. But lBzl > 1, for
In a similar fashion lre can choose
OrO!016, where
otherwise, since d divides Z and thus also Y, the sequence would
not distinguish c + Y and c + 2Y, The Left hand inequality then
foLlowssincedsqalso.
Statenent (ii) is an imrndiate consequence of Ci) . n
5.3 Propricty
I.le now eetabLish some
retationships that hoLd for periodic
sets of configurations,
Definition
A configuration c is impgrper iff c = c + mZ for all integers
m (not necessarily
positive) such that
l"l
Len'na 5.
+ nz > nr(e).
4
If c = c + rnZ for
funproper.
eome m
> O and
l"l
> nr(e), then c is
83
Proof
It ie easy to
'
eee
that for any set of configurations
{crr...1crr}1
rank (cor
Hence,
t .ili
"t)
'
1s5"
{rank ("3,
if rank (c, c + fl) = or
"3*1)}.
then
rank (c, c + z) > min {rank (c + iZr c * (i + l)z)i.
13i<m
By Lemra
5.3(i) it
foLloqrs
that these ranks ngst all be infinitet
and therefore atso, by Lerma
5.3(ii)' that c is inproper.
D
Definition
A conf iguration is propel if f it is not iryroper.
That the period Z is optimal for the propriety condition
can be seen from Example
7.3, where for any configuration c in
the
starting Btate, c r c + i iff i is a nultiple of Z = S(q - 1).
Lenma 5.5
is a polynomial p5 such that if l"l
and c = ct, then c is improper"
Ttrere
> Pr(9).2,
l"'l t
q.2
Proof
rhat c is proper, and fet ErBlBr6 be a string distinguishing
c and c + Z, constructed exactly ae in the first part of the proof of
Suppose
84
Lenrna
5.3, but for the case Y = Z.
We
define
"rr, ".1
for n > 0
by
B.
B:
g',
"r,
and ct
Bl
"ir,
where, in the case of e{oves, naxinal derivations ci.e. to readi-ng
Eaken. If c = cr, then cLearly .r, = "rl for all n' The
polynomial p, is chosen to ensure that r is sufficiently large for
rnodes) are
the following argr:ment to work: Either in cip...pc'(rr|) sourc
configuration repeats, or else eome cl in this set has height not
less than 2q3. In the latter case it is easy to verify that for
some
for
we
i1 j such that i , 3 < 2q4,
oj -i
-i+JJ1 cl and cl=c-*w,
"'.'?,
w>
some
o. In either case, for
some
i, j
such
that i ' J =
2q4,
have' putting g = (j-i).21
sl+wZ
"i*1, =
where now w >
rank
o.
By Lemna 5.3(i)
(ci, clas) < rank ("i*o'
However,
if r is large
Leurna 5.
(i)
3
and
trivially if
w = or
cj.+2g).
enough then from the
propriety of c'
and
'
rank (cr,
T1li-s
if w > 0,
"i*.Q,)
> rank ("i*0,
"i*2t).
contradicts the assrmtption that "* = "rl for all n' Thus c
be inproper.
D
must
65
l{e can now derive, as a consequence of this result, the
property of equivalent configurations on which our decision
procedure depends.
Definition
Integers m, n are (x, :r)-rational,ly related iff there exist
integers a, b with 0 < a, b { xr such ttlat
lna-nbl sy.
Lerrma 5.6.
exist polynonialr p6, il6 such that if c = cr, l"l t FU(e).2,
and c is proper, then l"l, l"'l are (c2, e5,(q).2) - rationally
There
related
"
Proof
that c = ct, that c is proper, and that l"l , FU(g).2,
sufficiently large for the fotr lowing argument to work. Choose
Suppose
for f.
'b
81rg2r$316, and define
"rrr
ni as in the previous lemma. Let !, be
leaet n such that
miur
c' I
cr, +
lc;l) .
q2.
infinite strictl-y
c* ia proper for all n s l, and therefore
l, mugt exiet, for otherwise
decreasing sequence.
(i*,r1,
ti"*l]
Z if l"rrl > nr(e).
would be an
the
86
rf l";l . q2 tb.n by
Le,r,rna
5 and the propriety of co,
we have
rbar l"gl = p5(q).2.
Alternatively auppose l"cl . q2. For
O
some
i, kt
< i < i + kt 3 q, the states of clr clalt are the
.i.*t, * ci
such that
same'
l"l = q'. Then if k is a large enough
nultiple of kf to ensure that l"o-nrl t pg(q), the propriety of the
cn-cives
e
Bay' where
t,-kzl tu-t, + dkZ = tL-zkz'
Hence
,L-vz
t .L-nr,
that is
"i,*"tz*ei+Zekz'
By lernrna 5,
l";l
<
nr(e).2, for othemise lcil
woula be proper,
contrary to the previous statement.
Thus
for a suitable choice of p;
we have
in both
ll-l - ,rsl . pi(q) .z and ll"'l - ellkl '
Since O < d < q and 0 < e/k n q2,
lIcl"e/k
it follows for
- l"'l.al
< nu(e)
ni(q)'Z
soroe
,2.
n
cases
p, that
af
5.4 Decieion Frocedure
Using the
result of
Le'smE
one-counter automaton Mt which
5.5 'are construct a nondeterminiscic
is abie to sirnulate, in a certain
sense, the computations of a pai.r of equrvalent doca. By taking
the disjoint union of the star.es and transition rules of the
two
rnachines, we can regard the simulation as maintaining a representation
of pairs of equivalent confi"gurations, c
and co,
of the
cornbined machine.
point in the si"mulation alcl - Ul"'l = f ,
for some arbrf such that 0 < a, b. q?o arrd if l . pg(q).2. Then
We
ensure
that at
each
Mt can represent c and cu h;l hol-ding alcl in i-ts counter
remernbering f
,arb,
and the states
of c
and
and
co, in its finite state
controL.
that
The action
of n' is as
wheneveE m
or n is greaier than po(q),2,
f'ollows
" Let
Fo be some polynomial such
and they
raLionaLl"y reX-ated, then they are rel-ated wittr respect
,
are (g-, lU(!.).2)'
to only one
admissible rational ratio a/b" Whenever l"i,
l"'l are both Less than
lo(e).2, their values are stored in tire finite state control. When
a simuLation step is about to exceed tlris bound, the finite state
control det.e;mines the coefficie&8s arb, ri any, and sets up the counter
for the appropriate represenfiagion.
ia]"rten
a sj.mulation step would reach
a
pair of configura;i.r:ns not ratr-!.l.a.lii lr:X.:rr*erln s.!)'c ie jusC too large for
ctn then by Lerflm.l5.6, i"f q1 :: {i:'1 ':[len c must be improper. Instead of,
cont"inuing the sinulaticn wiLti
either to
(c, c 'Z),
tr:
o
*t), a nondeterministic step is
ic * Z, e') g to the simulation of
Ttren, i-f c = r{ a.:ld eo also c = c - Z, tl:.e simulation
Ehe simulatior: c",t
continues to be one for equi'v'atent cgnfigurations in either case.
made
88
Since the originaL doca are
in normal form,
we can easily
define acceptance in llr to occur if and onl-y if exactly one of
the simulated eonfigurations is in an accept mode. Thus if the
starting configurations are indeed equivalent, our discussion
shows
that no string is accepted by Mr. 0n the other hand, if they
are
inequivalent, we can show, as in the previous chapter, that
stri.ng must be accepted, For if
some
some
d distinguishes the starting
eonfigurations, then either both derivations wiLl be
simuLated
directly to their different concLusions, or else the rational
relationship nust fai1" In the latter case, if (c, ct) is
reached
c I ct, then the remainder of a distinguishes one of the
new pairs created. The assumed normal fora guarantees that any
where
long e-derivations wiLl steadily reduce the counter. This, in
turn, ensures that further progress
a finite
number
of
a1-ong
moves, and therefore
o can always be made in
that o will- eventually be
accepted.
The construction and
testing for enptiness of the sinnrlating
machine described therefore
constitutes a decision procedure for
equivaLence. The nuniber of states of this nachine need be no more
than p(q),2* for some polynoniaL p, where q is the totaL nuriber
of states of the tested machines, and Z is bounded above by
'__---;-q.
S(q) * evg'roge
Assr,rrning a fixed input al-phabet, and recalling
Lenma
2.1 we conelude that
E9
Ttreorem 5.1
The equivalence problem
for
doca
is decidable,
and there is
a decision proeedure whieh, for q state machines, has a running
time bounded above by
,r,G.-ffi
for
some
constant
k.
fJ
90
III
CONTAINMENT PROBLEMS
Certain restricted classes of deterministic pushdown automata
have gome important basic properties which do not hold
for the whole
cLass. For exanpte, finite-state machines recognise a class of
languages
that are closed under the
Boolean operations, one-counter
nachines, we have seen, have a rigid periodic structure, while for
simple machines eaeh configuration can be related very
directly to
the language it generates. Furthermore, within these and
some other
subclasses we know how
to test for equival-ence. It is also plausible
that even for
for which decidabil,ity
problems
can be proved
in
the
unrestrieted case, easier decision procedures ean be found for these
subclasses than
for the whole
c1ass.
For any of these reasons we may want to determine whether for
a given dpda there exists an equivalent one beLonging to a particular
subcLass. Formally we ask the foLlowing containment problem: If
X, Y are
Ewo elasses
such that L(M) =
of
L(M')?
automata and M e
I,Ie denote
X, then is there an Mf e Y
the containment problem for X,
by (x:Y).
Emptiness, f i"niEeness and
containment probLems, Testing
seen
total-ity can all be phrased
for the prefix property
as
we have
in 51"4 to be equival-ent to the problem (D:D6). 0f the
difficult
probLems mentioned above,
the onLy one known to
rnore
be
deeidable is that of regularity, i.e. (D:Fsa). A proof of this
hae been given by Stearns2. Without resolving the remaining open
prnblems we
shall nevertheless throw sone light on their
expected
Y
:t
difficulty by relating
following
them
!.
ro the regularity
problem
in
the
way.
I{e shaLl define a very generar notion
of reLative complexity
with which one can compare the inherent difficulty of various
containment problems. ResuLrs expressed
in terms of this are of
wide applicabiLity. For example, our result that testing for
enptiness, and for the prefix property are, in che defined sense,
equally diffieult,
irnplies that if the time conrplexity for deciding
these two probLems on any machine modeL are polynomiaLs in the
parameters
polynomials
of the tested automata, then the leading terms of
E
i11 differ by
on1-y
a multiplicative consLant.
The result$ we then prove using
containment rroblems
these
this notion are that
such
as (D:C), {n:t1, and (n:Sg) must be, if
decidable at arL, at least as difficult
to
de*:ide as regularity.
with this as one source of urotivation, we then pr.oeeed in the
following ehapter to investigate the reguJ.arity probLem in detail.
For a naturai particular nreasure of complexity used by stearns,
improve
his
upper bound from
rde
a trebLe to a dou"nle exponential level,
which now closely approaches a
knorn"n
double *xponential
l-or*rer bound"
As a consequence r/* can a1"so si,g*l'r"ficantLy improve the upper bound
on the tine complexi.ty of ttris prob!.em.
we observe
that all the
eontainment problems
in which rde are
here intereeted become undeeidairl* if D is replaced by ND, the class
92
of nondeterninistic pushdown autonata. This can be deduced from
theorem
of Korenjak
of the Post
and Hopcroft9*ho
Correspondence Problemr
a
construct, for each instance
a context-free grailnar over
a
terminal alphabet r u {$}, with the properties that
(i)
if the PCP has no eolution the
language generated is
,*$,
(ii)
if the
PCP
has some solution then the tr anguage is
not deterministic.
From
this we conclude that for any class
(ND:Y)
Y c D such
that [*$
is undecidable. It therefore folLows that (M:Y) is
undecidable
if Y is
DorSorTrCrFsa, etc.
e L(Y)
'
93
Chapter
6
RELATM
C0MPLEXITY
6.1 Introduction
I{e
e:(press
wilL
shall define a partial ordering on decision problems to
the relative difficulty of solving them. Thus Pt > P,
nean
that ifr on any rnachine model, the
probLem
P, needs
Eine
x to be decided, then P, will require at least about the sane time.
Furtheraore, similar concLusions can then aLso be
made about sPace
reguirenents, and various other measures as weLl.
We
could define guch an ordering very sinply by saying that
tl = P2 iff any procedure to decide Pn is effectiveLy able to decide
P, also. For our appLications to dpda problems, however, it is
convenient
to relax this condition slightl-y.
We
denote by
D(nrrnrrn3rn4rn5) the cLass of dpda whose parameters grtrhrPru
are respectively bounded above by nlrtr2rt3rnO and nr. t'le define
a
transformation of a machine description to be direct iff it can be
carried out by an aLgorithm that
transition rules,
a finite
amount
of
(and
nodifies
memory
makes
only one pass of the
each one as necessaryr) requires
on1-y
additional to the capability of recognising
accepting modes, and increases each parameter at nost linearly.
We
then say that a procedure can decide problem P directly for a cLass
X, iff there is a direct transformation which takes all rnachines in
X to a form in which the application of the procedure effectively
decides P.
Definition
Pl > P2 iff
3
k > O s.t.
any procedure
to decide P, for
94
D(knrrknrrknrrknOrknr) can directly decide P, for D(nrrnrrn3rr4,r5).
Definition
P1 = P2
iff
P1 >
P,
and
t2 = Pl.
It is iarnediate from the definition that for the classes of
autooata XrXr,Xrr the foLloning reLation holds among the containment
problems:
(i) Xf c
Xrr
I{e obeerve
+
(11":X)
>
(Xf :X),
that, in order to decide any global property of
machine, lre require operations
a
at least as difficult as a direct
transformation. Thus the relaxation of our definition of the ordering
does
not endanger its validity for our
6.2
ResuLts
purposes.
As ie per:mitted by virtue of Lewra 2.5, we shall aseune here,
for convenience, that all
dpda are
in normat form.
Theorem 6.1
(D:T) > (D:Fsa).
Proof
Let L = t(M) c X* where M e D, and let L'= (Lf)* where
95
$
f t.
We
shall
show
that Ll = L(l,tt) for
some
Mt e D which is
only slightly larger than, and easily obtained frorn M, and, further,
that Lt e L(T) iff L is regular. Thie is clearly guff,icienr, for
then any M can be tested for regularity by testing the appropriate
t(X') for
containrnent
in t(t).
create Mr to recognice Lr by modifying M in the following
We
Iray" For every accepting mde of
to a nelr epecial state that
M we introduce a move on
causes the stack
$ input
to enpty and the
starting configuration of l{ to be reetored, all via an e-derivation.
If
we make
the starti:rg and accepting configurations of Mf to
be
the starEing configuration of M, then clearly L(Mt) = O$l*. To do
all thie
and we
we need add no
tpre than
one
will at lmst have doubled the
state,
nunber
Furthermore, Mr can be obtained from M by a
To shon that L regular +
L regular + (tf)*
and one
input
s;rmbol,
of transition rules.
direct transformation.
Lt e T, we sinply observe that
regular'+ Lt e O-T =' Lf e T.
To ehow the converse, rre
recall a coment
made
in 84.2 that
L(O-T) = L(Fsa). Ttrue, if L is not regular then in any recognising
machine
for (tf1*,
turna may have to occur during the parsing of
each eubetring between eucceesive $
machine cannot be
finite-turn.
D
characters. In that case the
96
Theorem 6.2
CD:C)
>
(D:Fea).
Proof
Let L = L0{) c E*, where M e D, and ter L,
= il9,$1"$r",
n=1
f, $, I r. we shall shon thar there is an Mr e D recognising
L' that is only slightly larger than M, and that any procedure for
testing Lr e t(C) will autornaticalLy decide whether L is regular.
where
I{e construct Mr
to simulate M repeatedln
and
to count the
number
of stringe from Lf read, by keeping a string
bottom
of the stack,
where A
is a new symbol.
An
at
the
Thus as
in
the
previous theorem, whenever an accept mode of M is reached and a
$
inmediately follows, the staek is emptied, but now onty up to the
toprrost
A, an extra A is added, and the starting
on top of this.
oode
is restored
$, characters are read, they are checked
one by one against the Ate, and acceptance occurs iff they are equal
in nrsber.
liltren the
Such an Mr can
transforrEtion,
and
It renaine to
clearly be produced from M by a direct
will not be much larger.
show
that Lr e L(C) iff t is regular. Clearly
if L ie regular then a recognieing rnachine for Lt exiets which only
uses
ite stack to store the Ars.
suppose
L is not regular.
Thus
Lr e L(C).
Any nachine recognising
Conversely
Lr
must have an
infinite set of pairwise distinguishable configurations
reached
97
via input strings terminating with a $. If it is a l-counrer
machine and L is not regular, then to recognise some words from L
from any such configuration, the machine will have to enupfy iCs
stack, for othenrise, regularity would be inplied. Ilowever
once
the stack is enpcied, all but a finite anount of inforrnation about
the nrsber of inst:urces of L$ atready parsed, is tost.
Thus
Lf c LcC)" 0
Definition
ForalanguageLcX*:
Sinit(t) =E*-LE+.
Lenrna 6.1
If L has the prefix property then:
L regularc+ Sinit(t) is regular.
Sroof
Ci) Since SinitCL) is defined by regularity preserving
operations, if L is regular then so is SinitCt).
(ii)
If L has the prefix property then it consists of
just those lrords in Sinit(L) ttrat are not proper prefixes of other
worde
in Sinit(L).
Thus
a finite state automaton for L can be
obtained from one for Sinit(t) by sirnply removing all- states from
wtrich any
further strings
accepting states. 0
can be accepted, from the
set of
98
Theorem 6.3
For any cl.ass Xo c Ro such that L(I's.)$ . LO(O),
-
CD!x^)
> (D:Fsa)
Proof
Let L = t(M) c X* wtrere M e DOr and let
Lr = Sinitct)$, wtrere $ I l.
x-transitions from accepting
I,rle
rnodify M by reptacing
to
rnodee
all
ones co a special
reject
moder ard
by adding a $ traneition for all reading nodes (other than the
reject
nnode)
to a nelr node which is
trcde. Then this
new machine, say
now declared
the sole accepting
Mr, recognises Lt, and so L(|,tr) is
regular iff t ie regular (from Leuna d.1).
If L(Mr) is regular rhen, by definirion, LCM') e LQ(o).
However' if L(uf) is not reguLarr then sinit(L) is not regular
so requires
arbitrary large
st,acks
to occur during recognition.
But from any such live configuration we e:(pect acceptance to be
possible in M' with a further input of only a single character
Thue
L(M') not regular
=t
and
$.
L(Ur) I L(Xo).
we can therefore concLude
that L(ltt) e L(xo) iff t is regular,
and hence that
@o :xo
) > (Do :Fga).
99
Ilowever, from property
(i) in 96.1 we know that
CD:xo)
>
(Do
:xo).
Also, eince any test for the regularity of t$ for any L c E* can be
used as a
tegt for the regularity of L by the now farniliar
arguuent,
we al"eo havc that
(Do
By the
:Fsa) >
(D
rFaa).
transiti.vity of the ordering the result follows. I
Theorem 6.4
(O:enpty) = (Dltotal) = @l:Do), where enPtYr total
refer to the class of nachines accepting nothing,
and x*
reepectively.
Proof
From
the argument in Corollaxy 2.1 to
Lenma
2'5 it is
imediate that a direct transformation exists to nodify any
machine M in normal form, to one that recognises exactly the
conplenent
of L(t{). It follows that
equally difficult
To sholr
emptiness and
totality
are
to decide in our sense.
that (D:enpty) > (o:Dg)
we
recalt that the latter is
equivalent to teating for the prefix property. For any M e
D
100
lte can construct en Mr consisting e88entia11y of two copies of the
transition rules for
M
with distinguished state sets.
These rules
are mdified so that transitions from accepting rnodes in the first
to the appropriate states of the second copy' The
starting configuration of ilt is defined to be that of the first
copy of M, while the acceptingnodes are those of the second copy.
copy lead
Clearly Mr accepts just those strings of t(M) that trave proPer
prefixes in t0{).
Thus
testing Mr for emptiness is equivaLent to
testing wh.ether L0{) e L(Do).
To see
can be
that (D:D') > (D:erytY)
we observe
that
any M e
mdified by replacing all- the traneition rules from
D
each
aceepting node by a reading rute leaving the mode unchanged. Then
testing this nachine for the prefix Property will effecCively
test
M
for enPtiness. I
Ttreorem 6,5
(DrFsa)
> (D:finiee)
>(D:enptY)
Proof
Let L - L(l{) c X* where }t e D, and let L' = L$*.
clearly M can be transformed to hecome a recogniser for Lr with
in size. since L' is finite if
and onLy
if I is
little
change
emptyr
L can be tegted for emptiness by testing Lr for finiteness'
101
inequalitlr tre use the observat,ion that for
a dpda M', LOi') is infinite iff Mt has sone live derivation
To show the other
that repeats a configuration, or has a repeated node in its
sEacking
sequence. I{e introduce some nen stack synbols, an E, for each
Ai . I,
1T
(s,Ar)
and
a special B.For each mode (srAr)
bI Cs{) 3 (sl,w)
we replace each
rule
$_
rute (s,Ar) i 6r,{),
$.,
and also, if it is an acceptingrgd", the rule (srAi) -* G*rn)
Cst,w)
and add rhe
s-a i.s a nelr epecial state, €rnd $, $., are nelr input characters.
$" " 'l
I^Ie also add the transitions (srA)
-5 (srl) for any mode with
where
t = r., or A = B,
Aseuming
can be
and add (surCI)
to the set of accepting
rnodes.
that Mt is in a fo::sl never requiring to eupty its stack, it
verified that the mdified nachine jusr described will
recognise a regular set iff L(M') is finite. I
6.3
Comentg
We
have only obtained theorems
However numerous corparable
neans
for (X:y)
results can be derived by sinilar
for X equal to various subsets of
Thus we can
where X = D.
D.
relate the realtime property,
and the stateless
property to regularity. If we define T1 to be the class of
ordered deterministic one-turn machines, with only one state of
ord,er zero, then we can show that (T:R) > (TI:Fsa) and that
1
(T:S) > (T-:Fsa). To do this, the construction we need for
Mr e tl is of an M e T that accepts the language
each
LO2
} u
oy'
{ "$o$r1 |
"$s$v I ey'
rthere each cr, B is a turn*free prefix of LOtf).
{
L(M')
L(M')
We
}
note that
despite the restricted nature of tl, our analysis of regularity
(Chapter
7) gives no indication of this
easier to decide for T1 than for
The equivalence problem
problern being substantially
D.
is not a containment problen.
Ilowever
the above mentioned kind of argument shows imediately that it is
no easier to decide for D9 than for D. For to test for LL =
L2
in D, we could test for tr$ = lr$ in Do. Even more trivial-ly we
notice that since enptiness is a particular instance of equivalence,
it
cannot be more
difficult to
decide.
I{e conclude by sumarising our results
in the following
diagram, in which Pl* P2 inplies Pl = P2, and each language is
assumed to be specified by a dpda in normaL form.
103
finite-turn?
ieL
one-counter?
sinple?
I
ieL
regular?
isL
finite?
J
Fig. 5. ReLative Complexity of
Containment problems,
104
Chapter
7
THE RXCULARITY
PROBLEM
7,L lntrodgction
Ste"trr"2 shows that the problen of whether a dpda accePts
regul-ar set
a
is decidable. To do this he proves that if a dpda of
a certain form, with q states and t stack eymbols' accePts a regular
language
L, theo L is also recognised by some finite-state
the statee of which nrnber no more than
I'leyer and
Fisch.t2l gi.r. an example to
circumstances a
some E3 expression
show
automaton'
in q and t.
that under these
finite-state autonaton of n2 size nay
indeed be
necesSary.
reeult in this Chapter is to reduce the upper bound
given by Stearns for this, by an exponential level, to an E2
Our main
function which differs from the lower bound of Meyer and Fischer
by only a rnultipticative factor in the leading exponent. As a
consequence
ne can also derive an E2 tine algorithrn for testing dpda for regularity.
By
siuilar analysie for the regularity of
and S6we obtain
each
of the classes R,
1
distinct E' expressiong, the orders of which in
case rre can show
each
to be valid as both upPer and lower bounds.
Alnoet all the ideas we sha1l use can be for.nd in stearns'
paper. llowever, in addition to the improvenent in the final results'
the following differences are notertorthy.
Our proof
right
is fot the general case allowlng arbitrarily
hand sides
in the transition rul-esr while Stearns
long
considers
C
105
only conservative mechines with h - 2. Thus, although for
dpda q may have
to
grow et(ponentially when it
is
reduced
some
to the h = 2
form, we shal.1 show that this is not a oource eontributing an extra
e:cponential
in the main result. Furthermre r our analysis will
directly applicable to' for example, stateless
(Chapter g), for wtrich equivalent consetltative
be
and dB-S machines
machines do
not exist
in general.
Ttre notions
of nuLl-transparent and C-invisible
segments,
introduced by Stearns, and the proofs of their existence in
sufficiently large stackc, remain at the centre of the argurnent.
tlowever, we have r.rnified the proof,e
of theee
theorems
Cl,errma
7.2)
by applying the technique used by Stearne for the one Cnulltransparency) to obtain the now improved reeult for the other
(f,-invieibility) also.
Our
definition of the latter is a generalisation
of that of Stearns, that contributes a further enaller iryrovement.
In addition lre are rather mre e:cpLicit about the
that eonespond to regularity
and
phenomena
irregularity respectively.
![e
exhibit the fact that our main construction picks out a fanily of
conputations that reeernblee a one-cotmter autonaton in structure
and tras the sane behaviour
We
vie-a-vis regularity.
shall use the notation introduced in
51.5.
106
7
.2 Null-transparency
dnd
Sinvisibil-ity
It will be convenient here to say that, in a derivation
c i (c) ct, r*rere q = o1ct2o3, "the o, subderivation
segment
for
pcips the
urU" in c iff c * (of) ,"ruoJ) and (srtlo,5) * (oZ) (sr 'oqi)
some
B,
B
t.
Definition
(srto) + (q) ct is a i-deritation w.r.t. index set N iff there
are fewer than j pairs (nrn) of consecutive elements of N with the
property ah.t ,rr'
is
popped
by a non-null subsequence of d.
Definition
the
segrnent trtf
i" $!1g!g!!!g in
(srutttttJ")
w.r.t.
index set
iff for any sr, and any l,-derivation (srutt^ttto") + (cl) (sr rtttlt),
it is the case that (strttot) * (e) (s',o).
N
In other words, the existence of the seguent or can only
be detected in the configuration by derivations which pop by non-null
input etringe at least t, of the
segments
of o"
induced by N.
Definition
iff for all- s e Q'
o it @
(s ,o) + (e) (s' ,0 ) * (s' ,o) * (e) (s' ,o) .
The stack word
LO7
A null-transparent seglnent therefore has the property that
if (ero) + (e) (et$) I then for alL n > l, (rrrt) + Ce) (st,n). Thue
e-derivatione which PoP sequences of a null-transparent word are
incapable of distinguishing different numbers of occurrences of thern.
Lema 7.1
ie null-transparent, then for all sr 01, no string a of
length n can distinguish (srrtfl frorn (srttrt ) for mrm' > n.
If
rrl
Proof
s derivations from the t\to configurat,ions. If these
have at no stage popped the top n + 1 copies of r,r in the stack, then
clearly they cannot be distinguishing derivations. However, if they
Congider
have, then at least one copy of
trl
must have been popped by an
e-subderivation. But then by null--transparency' the rest of the
o seg@nts'rould al.so have been popped at the same time, without
leaving any trace of their number. Thus we conctude that no a of
length n can distinguish the given configurations' I
I{e now prove by an inductive argument the existence
kinds of segnents in sufficiently
tr
of
both
arge configurations.
Lema 7.2
For a configuration c with stack r.l and an index set N of
etements
all lese than
lrrrll
fr'
108
(i) fr't q! + some segrnent of o induced by N is
null-transparent
(ii)
F t l(gq)q +
,
some segmenr
of
ur
induced by N is
0-invisible in c, provided that Lrj. > 2.
Proof
For each part ne produce an inductive aesertion of the
form
A(Pn'rNm)
N, . N.
for
We show
m=
0r1r2..., wherep* c a finite set p,
that A(PorNo) is true for
Po =
/
and No
and
= N,
that, it -% is eufficienrly large and A(prrNr) holds, then
either Nalready induces the required segment, or else we can
m
and
find
Pm+l and
N*,
such
that A(P*lrNrn+l) is true also.
that the induction terminates,
ensure
We
and thus guarantees to
produce the required segrnent, by showing
that
Pm
qE
t*L
that -N--'
Then by
m+Iis greater than sone given function of F_.
mpicking N large enough initially, lre can ensure that, though the
and
sete N, nay get successively snaller, they will always be large
enough
to enable the induction to continue untiL P ie
We
take P to be Q, and the assertion A(PrrNr) to be:
irj e Nr,
Then A(PorNo)
(i)
exhausted.
"
. Pr* (sro-) * (e) (srn),
is trivially true.
We agsume
that A(PrrNr) is true, and that k
r the
srnallest and largest elenents of N, respectively, are distinct.
and kr
109
Thcn
if
e_
r_r
KrK'
is null-transparent, the result is proved. Othemise,
by definition there erist srsr such that
(g)
(s,r,rn,n,
) * (a) (s' ,0) ,
but (b) not (c' ,\,1r ) + Ce) (e' ,o) "
Thcn
let N*, be the
eubset
of
N,o
indexing the most frequently
occurring etater Bay stt, in the popping sequence for (a) w.r.t,
P_
No state in this popping sequence
N , and let P_",
lEtI = m u {e"}.
$q4n belong
to Prr for that would iryly, by the inductive assertion,
that I| = srr . Pr, which would contradict (b). Thus it follows
that Pr€ Pm+l, and then aleo that Fr*1 =F*/(C-m), for m ( g.
A(u*rrNr*1) ig then clearly true. Also, if fr'= Fo t Q!, the
induction can continue, if necessaryr until P, erhausts Q, without
"r"t
F-m < 2 occurring. This
(ii)
now
We assume
completes the proof
of Ci).
that A(Pr,Nr) is true, and thst k
and
kr,
the srnelleat and eecond smallest elements of N, respectively,
are dietinct.
Then
if utrk, is 0-invieible in c, the result is
proved. Othenriee, by definition, there is
some
a-derivation
that i3 an .0-derivation rcndering it visibl€1 i.€.
(a) c * (o) (stroorkf),
but (b) not
(s t ,rrto
,U,
) * (e)
(s t ,t^ro
,n)
c derivation extract the e-subderivation that poPs the
rc8t Begrnnts induced by Nrr and let Nt*l b" just those indicee
inducing theae segmnts. Then, by definition, Ft*, > tF -flll.
From the
110
in the popping sequence induced by fr'rr*t can belong to
Prr for that w,ould contradict (b). Let s" be the nost frequently
No etate
occurring Btate in thie sequence, let N*, be the set indexing
theee occurrences, and l-et
Now
P*, = pr, {s"}.
if fr'= f^o > 0([q)Q irriti"lly,
Then
Nntl > Ntr*l /(g-a),
the induction will continue
eucceggfully until Q is extrausted. [l
Note
1
The bo'nd
can be ghown
for (i) of q! is the
same
to be optimal by looking at
as that of stearns. rt
Begnents
that are
e-popped
fron all stateg thereby performing peruutation operations on them,
and regarding these as elements
Note
of the symetric
group on q e1ement6.
2
rn the main theorem we sha1l be interested in c-invisible
segnents, where.Q,
is e:(ponential in g. rt is here that we gain our
eignificant iryrovenent, by obtaining a bound of order (gq)q
as compared with one of oc+9 given by stearns. rn general, if.0 is
most
of larger order than q,
we can ehow
that our bound is of optimal order tn
the fotlowing sense. A dpda can be derived from rhe proof of 7.2(ii)
with the properties that for some configuration and index set of
size (llq)9, no .t-invisible segment can be found.
t11
7.3
Main Theorem
Theorem 7.1
rf l'{r a dpda in
- stack words
and
of
norrnar.
l-ength
form, has q states, t stack syrnbols,
at
most
h in its transition rules,
if t(M) is regular, then L(!"1) is recognised by sone finite
with fewer than X(qrtrh) states, where X is of order E2,
and
automaron
Proof
shall prove that there is a function ycq,t,h) of order E1
that, if any reachable eonfiguration of M has height greater
I{e
such
than Yr then either we can cut a segment out from the stack to
obtain a small-er equivalent reachable configuration, or el_se
there are input strings 6r$, s.t" the configurations reached after
inputs of orol for m = Lrzr... are all pairwise inequivaLent,
if L(M) is regular, the first possibir.iry musr always hoLd
for configurations larger than y, and consequently M can onLy
Thue
be using up
to qtY painvise distinguishable configurations
in recognising the language. fhis gives us the required result,
To
prove the existence of Y we consider an arbitrary derivation
cI -*c= (errrr), lrhere l*l = n > y(qrtrh). tle let
N be
the set of
f"ntegers indexing the most frequently oecurring rnodes in the
staeking sequence
of this derivation.
Then cLearly
f > n/qth,
tL2
I
I
I
I
I
I
f
t
L
I
I
I
c2l
t
I
j
I
I
I
I
crJ
I
I
nu11-transparent
segment
r
clt
I
I
(
t
I
I
I
I
t
I
cir
rY
I
I
o
I
I
I
I
t
I
I
I
o
I
.,
I
I
I
I
I
I
I
I
I
I
I
I
)
1
I
I
invisible
t
segment
r
I
(
I
I
I
I
I
I
I
a,
K TIII
o.
KrE
iterations of
nul 1-transparent
ct
K TID
ct
K
c,
K
Fig.6.
segment
rlll
rlll
Constructions
in Proof of
Theorem 7.1
I
113
rf N is large enough, then by
Lenma
7.2 (ii) , we can find a segrnent
o.:,
in it that ie (qgl)-invisibl-e in c w.r.t. N. Furthermore,
1J
by
the choice of N, the configuration ct = (sroorirjrr) is reachable
from c^ via the input string o^ 10: _. I^le shall_ prove that if t(M)
I
ura jrD
is regular, then c lct. For siruplicity we shaL1 not nodify the
indexing of the segments of the stack in the translation from c to cl
(e,g we shall still
c I cr, and let I be the shortest string distinguishing
Suppoee
them.
Then
refer to the top s)mbol of ct m rrr-r,rrr.
by the construction of ct, for some yrr1 s.t. B = yl,
c * (V) (strt'lori)
thie is not a (qq!)-derivation. It foll-ows that
where
an Nf c
N of at least ql + t integers
between
j
and n
we can pick
with
the
properties that
(a) no
segment
of y,
(b) the
of
tir induced by
Nt is
popped
by an e-subderivation
and
elements
sequence
of N' all index identical states in the
popping
for y.
lle now pick a null-transparent segment ,Or, irdrced by Nr in
,jrrrr as guaranteed by Lemra 7'2 (i).
We
to
in c
do
with the popping of this
c
segment
* (trrrr) "2, .z * (v*rn) "1, c' *
Since yn
define sone nerr configurations
and cr:
(trrro,)
.i, .) * (v*,n) ci.
is a minimal distinguishing string for c, ct, yr,jn
must be
and ykrjn for ca, .i.
Since, by the
"2, "Zr,
construction of N', yrrk is non-nu1l, ylrj .rU t*rj rr"a be of
a mininal one for
different lengths.
Ilence
it is
impossibLe rhar both
c, = c,
and
ci = cj.
114
I.lithout loss of generaLity we shall
assume
that c, *
c2,
define the farnily of configurations'{rclr > oi by
,: ) r".
- c-s f 1o
'*o ,k*k
rrn
By the choice of NrNt the top part6 of the stacks of these will consist of
We
iterations of the null-transparent word. If we let {, = orrryrrr{,n
forr=0r1r...,
then
Er
r+1
'-c-]c2
Thus
. rc *"1
and
Er
forr)0.
c, I c, =t t" l t*1" for all r > O.
transparency
of ,krr,
From
thie,
and the nu1l
we can now deduce that
t" I l*t" for all r
>
o, I
> o.
othemise for some t, L, and conaider the effect
the configurations x'c+r" of inputs olr, and onrr, fot successive
For
if
ate aseume
values of x of
Orlrz...,
x'c*t"
=
then lte are led
to
on
deduce that
t*1" for all x ) o.
rc rnd *l*t*l"
=
7.1, the shortest distinguishing string for
t", t*1" cannot distingui"h *0+r", xf,+r+l" for sufficiently large x.
Howeverr by Lenma
c f ct has forced us to the
conclusion that the configurations'{rc} (or a corresponding set
To sr:cnmarise, the assuqtion
constructed from ct r)are all reachable and pair*dse inequivalent.
Thus
for regularity it
must be
that c = ct, which is the result
we
want.
From Lema
7.zGD
we
recall that for the above construction
it is sufficient that fr b" gr"ater than (e2q!)q*l.
Therefore
1L5
Y(q,t,h) - qth(q2q:)qnl
is the function
and hence
Note
we
requirc. This is of order fI(tog 6 + 1og h * q2 lcg q),
X(qrtrh) is of order r21tog t + log h + q2 log q)" il
1
Meyer and
Fisch.r2l
q = t and h = 2t
states.
for the
utl
"hoo,
that for a certain one-turn
equivatent finite-state automaton requires r2(q)
The bound obtained by Stearns
case
dpda wittr
for X is of order s3(q * log t)
of h = 2.
tilote.l
We
notice that the family of configurations {rc} exhibits
phenomena
strongly reminiecent of the notion of propriety in one-
counter automtte. In particular, they are either all equivalent,
or all pairwise inequivaLent,
7,4
Bounds
for Subfanilies
!,le now
give irnproved upper bounds for the function X for three
restricted families of dpda,
hound
and show
in
each case
that the order sf
ttee
ie achievable.
For the classes Sg and R, since eltpves no longer play a part, the
above analysis becomes
trivial.
From
the definitions in 57.2 it is
imediate in such cases that N > L is sufficient to induce a nuLl-
116
transparent segment, while N
one. Substituting these in
Corol-lary
-*
7.
>
l, + 1 guarantees an l,-invisible
the above argument gives
the fol.lowing:
1
For the class Sg there is a bound X of order fl1ht.l"og t).
CorolLarv 7.2
For the class R there is a bound X of order nl(nq2r.1og r).
From Len'rna
5.4 it can be easily
rcsniction h = 2, the bound
dominant
deduced
q. (S(q) + pf
that for
(q)) suffices,
factor, S(e), is of order El(/alog-q ).
a eimilar
bound rnore
C
We
with
the
where the
can obtain
directLy as fol.lows.
Corollarv 7.3
For the class C, with h = 2, there is a bound X ^, q2.S(q),
**:ngre
in- = q},
S(q) = lrrx {i"c,ru in,}l
11'
Proof
Ler {xri be the set of net staci( drops due to loops of
e.ru*ves
in
the di.agrann
for non*enpty counter transitions of
the
LL7
M. It follows i&nediately that any stack segment of
machine
length
x = max {g.c.m.{x.},
will be nuLl"-transparent, where,
q}
rnoreover,
x
S S(q) since the
e-loops must be disjoint.
can then obtain the claimed bound from the construction in
I^Ie
7,L Lt we also observe that any jderivation (w.r.t. the
Theorem
positive integers) popping a segmenr longer than jq must fini.sh
within an e-loop" I
We
now ehow
that the order of
each bound
is achievable in
both of the genses defined in gl-.6.
Exanple 7*L
Let
M
be a sirnple machine with t = aU, fB, where
fA =' {A.lr = i < m}, l, = {Brlo * i < m}, with
x = tailr s i < rn] u {arlo e rf,}, wirh
", = Bo, and rransirions
a
B, $
sBr.,
Bm -9--t
A
J
J+T
a,
A" *l-+
A
J
C<
to occur. But there
s1
lsj<n
Clearly the stack has to grow to height
and the turn
j (
nm
+ L for B, to be
reached
reachabLe
"r" rtt distinguishable
configurations of rhis height. Since L(M) is evidentLy regular,
l_
this
shows
18
that X of order fI1nt.log t) is achievable. !
Exanple 7.2
I.Ie
construct an Mf e & by generalising the previous
in the following way.
eugment
We
I by a synbol g,
{(s"rfl)},
l"eave
exarnpLe
introduce a state set Q = {.or...rk},
repLace
c, by (sorB.)
and F by
the input alphabet unchanged, and replace the
transition" rules by
a
(s'Br)*,tr*l,r'rBl)
osi<k,
a
o < i < k, 0 < j * r,
(ri,Bj) --l!-* (si,oB5+l)
a
(s*rB*) tr
(so,Br)
(sorcorBl)
a1
--:>
(sk,A)
6.
Th:i,o
(sirA3) *-J* (si,A)
O<i(krO.j.m,
(s1C)
o<i<k.
---9'* (s.-1,4)
again is a l-turn machine. After the turn is made in
ctrer:h"ation, the rernaining
a
string i"s accepted iff it natches
the
staek contents, and there are exactl.y k occurrences of C in the
stae[a,, Thus the turn has to be made when there are about k2rur
symbols
in the stacko since each e is
added
after successive
segments
length kurr" fhus L(Mt) is regular, and Mr has order f11nq2t.1og t)
distinguishable eonfigurations " I
of
1"
19
Exanple 7.3
Let {xrr.".**} be the Fartition of q * 1 with the greatest
least comnon nnulti"ptre. Let M e C read a string frcrn a* and
increment the counter by one
starting state"
e-Loop wi.th
Then
x. $rates
for
each
a, while staying in
the
if a character i frorn {1,"".n} is read, an
causi-ng
a stack drop of x' is entered.
A.ceeptance occurs
iff the contents of the counter at the turn was
divisible
Such an M wi.th
bY
xn'
q states clearly exists, and recognises
a regular set. lloweveru after each input of
n(nod
am
it
needs
to
know
y) for y = *tn..rxn. This requires S(q) distinguishable
configurations. fi
7
"5 Tigq Camql.qxitv.
To test- a language accepted by given dpda
for regularity
we can
constrrrct the candidate finite-state automaton speeified in the proof
of
fheorem
last test
7.Lr say Mr, and test for their equivalence. Since this
can be done
E- size, to
show
^2 timeo it
E-
::earei:r,s
.*2!;
I.n
M
polynomial time
(92.5), and since M' is of
that the regulari.ty test i,tself takes no more than
to
show
that
the construcf
ion of I'lt can be done
tlme.
The states
of
in
of Mr *orresponcl to the no nore than
of height less
we need are those
Lhan
Y. To construce Mr the on1-y new transitions
that specify for
Yr the (qq!)-invisible
E2 configurations
segments
each configuration
just larger
than
with respect to the appropriate index
sets (57.3)r that can be rerreved from their stacks. For each segment
L20
or of o, lre can determine the set of pairs (srst) with the property
that (sro') * (a) (st rO) for o f e, and also those for which cr = e.
The poasibl-e ways
of reaching each point in the stack by (qq!)-
derivations can then be deduced, and hence the invisible segments found.
To do
all this for all the tr2 configurations requires only
In a eimilar way, the bounds for
R.
E2 time.
and C give single exponential
time tests. However for Rg1 and hence aLso for Sgr regulariry can be
tested
much more
easily,
and
in only poLynomial- tine. If in
M
e Ro
an aecepting derivation goes through some configuration c with height
a
greater than hq't, then for
some
pair of levels repetitions
in both ehe stacking and popping sequences respectively, of
must occur
the
derivations before and after the occurrence of c. This would irnpLy that
accepting derivations can go through arbitrarily Large configurations,
and hence'
by the restrictions particular to the class R0, that L(M) is
not regular. Since the converse of this is obvious, we conclude that
bte can
test for regularity by testing whether,
once a stack leveL
.,
exceeding
hq-t has been reached, any further inputs lead to
acceptance.
This requires only an emptiness rest on a polynomiaL size machine.
121
IV A}I
schemas
A?PL]-CATION
TO SCTM},IAS
are direct formaLisations of coinputer prograns,
and
closely resemble them in syntax. Their essential characterisation
is that the meaning of the
of
schemas
conrnands
is left undefined.
The theory
relates the description of such a formalised program to
its possible computational effects
once interpretations
of various
kinds are given to its uninterpreted comands.
The
relaticnship between certain
schemas which have
register3 and automata with one-way input tapes, is
now
just
one working
well known.
t,
Rutledge" has established a elose connection between one-register
flowchart (Ianov) schefiras, and fi"nite-state automata. When such schemas
are
augmenteetr
deterministic
by a pushdown sta.ck, lhe correspondence transfers to
pushdown autornata"
This corresponclence is such Lhat, as
il"l.ustrated by Patersonl3, an equivalence test for a subfamily of this
class of automata leads directly to a test for strong equivalence (in
the sense of Luckham, Fark and Faterrorr23; f,or the corresponding class of
schemas. Thus, for exampler our result in Chapter 5 implies that strong
equivalence
is decidable for lanov
trn an analogous way, the
of whether a
to
one
regularitl'
tranov schema
with an auxiliary counter.
problem corresponds
to the question
with a pushdown stack is strongty equivalent
without a stack.
As the translatabiliny
autonata to
we
schemas
schenas
of
is direct,
such
decidability results
and depends on a
shall not pursue these further here"
from
well estabtrished techni{r"l3,
t22
In contrast, holrever, the complexity
problems do not necessarily'translate
a canonical dpda for
measures
for
directly. For
some such schema may
these
exampLe,
require to be of
exponential size in terms of the parameters of the
schema
description. For this reason rre shal1 investigate the coupLexity
of just one of these problems, for a case which is of special schema
fheoretic interest.
Monadic functionaL (deBakker-scott) ."h"r.r13 ,24 t25 are
a
f,ormalisation of recursive programs with a single working register.
Faterson26
h",
shown
that for
there does not exist
some such schema
an equivaLent flowchart scherna with any
finite
nuriber
of registers.
This ean be interpreted as corroboracing our intuitions about the
increased poner
of recursive notation. Other examples are known
whieh can be flowcharted, but not with a single
now elaborate on these
ftrnetional
schemas
register.
relationships by considering those
that
do have equivalent ranov schemas,
tr{e
shall
monadic
and giving
a measure to the subetantial succinetness with which soup large Ianov
sehemas can be re-enpressed
In particular,
by equival-ent srnall ftrnctional ones.
we shaLl show
that
sorre
functional
schemas
require
an
equivalent fLowchart to be of a double exponential (r2) size in terms
of {es original parameters. Further,
we
shall
show
that they may not
require larger flowcharts than this order.
Thus we shalL be
our f.ntuitive
giving a theoretical resuLt to correspond to
knowLedge about
the considerable econony of description
that can sonetines be gained by recursive notation.
L23
Chapter
UONADIC FUNCTIONAL SCHE}{AS
8
8.1 Introductisn
In our definitions
we shall-
in the main follow Ashcroft '
Pnueli24, tho give iLLustrative examples and rather
Manna and
more
detaiLs.
e d!:S_gg@_ has a finite set F,of, t., monadic fungFion
variabl-es
{F-.
t.
} (with a distinguished initial fqnglrg4 F^), a finite
set f, of t monadic function constants {f,}, a finite set of
rncnadic !f,e4:Sgggg.
{pi},
and an
g
individual variable x. A term is
a cornposition of functions applied to x' e.g. ttFZ(f ,(x))) which is
for short, omitting -the brackets and the x. A
written aB f,F^f.
i_23
conditional ter:n is any finite expression of the forrn
if pi then t, else t,
where
11
specif ied
,
,2 are terme
or conditional terms.
The schema
bya set of function definitions, one for
itself is
eac.h F.
, of
the
form
F.1 € r
where
t ie
some
tern or conditional. term. It is useful
LU
reserve
the nane 1 for the idenrity funttion (I(x) = x) r and l.-, for the
undefined iunction (F* e F-)
The schena
fo(x),
"
is evaluateC in the
expected way by
starting
from
and appLying the rightmost function variable each time to
the argument (i.e. to the string frorn f*x to its right).
The only
1"2t+
further information lre need to define completely
such an evaluation
is to specify the
Ehe course of
val"ues taken
by the
predieates each tirne a conditional term has to be evaluated.
Thus an
for
interpretatiog specifies the values taken by the predicates
each argument, and we assume
that it
does so
uniquely.
Thus
in eval.uation steps in which no function constant is applied to
the argument, the truth values of the predicates cannot change.
In other words, hre ean regard an interpretation as a function
from f*x to 6 =' {-, +}9, where - and + indicate falsity and truth
respectively.
Since each interpretation
an evaluation,
for a schema uniquely specifies
it defines a string (possibly infinite) of the
60111
f. 6. t.
o1
where
form:
....
6, is the truth vector defined by the interpretation for
the
J
ergunoent
t.tj-t .... f.tl t..to x.
Conversely, each such
string describes the step by step eval-uation
of a function, as well as a set of interpretations.
The schexna
for a partieular int,erpretation is said to have a
defined vaLue if and orrly if the eval-'-retion terminates producing
a
germ containing no more
function variables. This term, consisting
of only function constants applied to x, is then the defined value.
L25
8.2 Eval-uating Pushdown $rtomatg
A dpda M can be easily eonstructecl to evaluate such
functional schema. At each step M keeps all the
a
unevatuated
functions (i.e. everything to the Left of the argument) i"n its
pushdown
store, with the rightrnost function variable at the top.
It reads input
words from
(6f)* i.e. strings of al,ternating truth
vectors and function constants.
Wtren
there is a function variable at the top of the stack
a truth vector has just
been read, the variable
is replaced in
and
the
staek by the term specified by the corresponding function definition
for the truth values read.
state control, and these
The vect,or
is
remembered
repl"acements continue
it, until a function constant first
This is checked against the next
via a F- replacement if it
does
appears
symbotr-
in
in the finite
accordance with
at the top of the stack.
on the input tape, and rejected
not match. Otherwise, the function
constant is popped (and can be regarded as being output and appLied tc
the previous argumentr) and the next 6. is read. If further function
*en-stenes then appear
c.hecked
at the top cf the staek, ttrey are similarLy
against the input tape! with the intermediate truth vectors
on
the tape bei.ng ignored"
The dpda
M
by enpty stack.
ctarts with
F
0
i"n
its stack,
and accepts strings
Then elearly any tape accepted by
the vaLue taken by the
the tape. Also, the
schema
it wi1l specify
for all interpretations consistent with
language recognised from an
arbitrary configuration
L26
of
M
wilL relate in the
corresponding term
Fron
this it
in the
way
same
to the values taken by the
in the schema for different interpretations'
fol-Lows
that two configurations of
automnton sense
if
and only
M
are equivalent
if, for any interpretation,
trro corresponding terms in the schema either evaLuate to the
value, or are both undefined. Thus if
M has
only a finite
X, of pairwise distinguishable reachable configurations,
the
scherna can
same
number
then
be rewritten as a lanov schema with X boxes.
By examining the dpda M we have
it
the
can be constructed
just described,
we
find that
to have the following parailEters:
t=t+ttc
v'
where trr is the l"ength of the
l-ongest term in the function
h=hr
definitions,
p=28*t",
q = 28 + 1'
I{e
notice that the only kind of
that of being able to
suhclass
of
Dg
remember
memory
M
M needs
the Last input character read.
with thi.s property we shall
that the evaluating machine
capacity Ehat
denote by
dB-S.
I^Ie
is
The
note
is a special form of a dB-S machine,
sinee a subset, f, of its input alphabet need not be distinguished
in
nernory, and also no element
can be e-popped. That dB-S
verified by examining the
of f,
regarded now as stack syutbolst
is itsel-f properly contained in
language
{ anb"t
}u
{
anac?n },
Dg can be
L27
8.3
dB-S Autonata
The above mentioned
that for
dB-S nachines,
restriction
on memory implies
directly
states eaRnot change in the courge of
e-derivations. A consequenqe of this is that a simplified
analogue
of
Lenma
7,2
car. be derived.
Lema 8.1
For a configuration c of a dB-S machine, if c has stack
and N
is an index set of
(i)
(ii)
Any segrnent
fr-
elements
all
Less than
of o is nulL trensparent,
til,
l.,rl, ttet
and
If F > 2.Q,t, where l, > 4 then some segment of
(,) induced by N is .Q,-invisible in c.
Proof
(i) Trivial.
(ii)
Using the sarne notation as
in the proof of
Letma 7.1'
I{e apply the induction
chooee P
principle there stated.
= I and the inductive aseertion
However, no!il
A(PmrNn)
to
we
be
irj e Nrr A e Pr+ A does not occur it rirj.
Ttren A(PorNo)
is trivially true if
Po
=6
ar.d No
= N. hre then
assume
that A(PrrNr) is true, and that rrrt, the smallest and
second
snallest elenents of N, are distinct,
tt ,rrr, is
L28
0-invisible, then the desired result is proved. 0therwise there
must be some o-derivation
that is an l-derivation rendering it
visible i.e.
(a) c * (o)
(s t rr^ro., r)
but (b) not {e|,oorr,) * (e) (st,oorr).
We
notc that this second condition
now
necessarily iurplies that
o-ErE
' ort'-,).
-r cannot be e-popped at all fron the configuration (strur^
Let N*, be the Largeet subeet of N* induced by the popping sequence
of an e-eubderivation of the o derivation.
Then
fr.*, > (Nm-l-)/1,.
This e-derivation must eventualLy terninate when some Af in the
stack is reached. Furtherr this occurrence must be at a higher
level in
ur
than r, whieh means that Ar I Pr" Atso, Ar cannot occur
in any Begrnnt induced by Nnrl, for then that occurrence could not
have been popped
in the sane e-derivation. Thus, if Pr*l = P*, {At},
then A(Pr*lrNrn+l) must also be true. To ensure that the induction can
proceed, if necessary, untiL l ie exhausted, it is now sufficient
that
fr' >
zr,t.
n
In the light of this
we can rework Theorem 7.1
to obtain the
fol"loring.
Lerma 8.2
A dB-S dpda recognising a reguLar set can have no more than
7
X = E'(t.1og q + 1og
h) pairwise inequivalent reachable configurations.
L29
Froof
Wc
use the same notation as
Flav$.ng ehosen
l{,
we now
in the proof of
Theorem 7.1.
require oni.y a q-invisible segnerrt indrrced
by
!i. Fcr then, usi"ng the shortest distinguishing string g, assuming
8rlaf. i"t exj-sts, we can find a nutt*transparent segrnent induced by N
such rhat the B-subderivation
nsf
lre'e's
cause a
require* anei,
of
net
change
that
pops
it reads
sone inprrt, but
in steee. This is the construction we
frenn Lenmna 8.1.n
it clearLy works for any confi.guration
hei-ghe greater than YrwhereY
is of order 2hqtqt. This gives the
required trotnd for X. I
m-s"
Th{"c shows
that the dB-s restriction
reduces the q-dependence
*f, the l:o'urd x from a double to a singl"e exponential expression.
8,1* Bgqnllq*g rsuccinctness
Freim Lenuna
8.2 and the observations of 58"2 we imediateLv
e'l.1.Vg
Sbes.gss"
-E-*L
For a dE-S
terrns
sohema
wi,th t funeti.on syrnboLs, g predicates,
and
of !.ength no more than h f,n its fr,nction definitions, if
l-anov schema
strongly equivatenc to it (i.e. under all interpretations)
exisBsr then the !.atter
howss,
[,-l
a
need have no more than
order n2Gg + 1og h)
130
It remains to
show
that
an E2 order
of size rnay indeed
be
neceSaary.
Theorem 8.2
For each positive integer n, there is a dB-S
schema
with
3(n+1) function synbolsr f, * 1 predicate symbols, h = 2, and with
total description linear in n, that has strongly equivaLent
schenas,
lanov
hut only of size at least n2(rr).
Proof
I{e use the same idea as Meyer and Fisch.t2l
corresponding resuLt
even
in this
more
for dpda, and show that it
restricted
'
{f*, f-}
can be made
to
work
framework.
For each n lte construct the following
variables {For
,rr" for their
schema
tir...tlr ti,...ti, !t,...1r,},
with
fr.nrction
fr:nction constants
and predicate"' {prp1r...prr}:
Fo € lf pt then (if p then tl to f+ el'se ti to f+1 else 'f
p2
else if p' then (if p then F+ to t* else F- Fo f*)
eLse
++
F-t f
.
+
F: € i.f p, then I else "f p2
. . else if pi-t then I
else if pi then (if n then \
eLse
if pi*l th"t Felse I.
f+ else I)
. . . . else 't
Pn then F-
131
F. € if pl then I else "t pZ
. " el"se if Pi-l then I
else if pi then (if p then I el-se F. f )
. eLse 't Pn then F
etrse if pi*1 then F* .
el-se L.
&.
* if p, then I'- eLse 't pZ
. . else 'f pi then
F
else I.
Ler *= nin' i j I rrttf*tj) is false for 1 < i < n ] for
a particular interpretation. Then the corresponding evaluation
will consist of
taro partsp
definition,
of the F_
o
The
m
first part
the first one consisting of
and the second only
of appLications of the others,
produces an unevaluated term which
over rhe function variables irj,...tl,
m appLications
is a string of
F;,...t;),
l-ength
whiLe the second
nonotonically shortens this, unless an F- occurs.
Changing tG our dpda eerminology, we notice
machi"ne
that
we have
a l-turn
which, depending on the inputl cdrl read any word from
-+
Flr...trr}* into the stack on the uPstroke. Denoting by
{t;.r...Frr,
b *ny trr-lth vec!:or such that p1 = Erue, but p. = false for j ' k, we
observe
that
each
fl1-Ior Fr- as e-popped by k if k < i,
to rejection if k.'i.
If k = i bgt rhe sign *i F.
and leads
does
directly
not rnatch the
tru8h vslue *f p, then the syr:bol is again e-popped. However, if there
J"s mlat,ching, ,;hen
the apprcpriate one of f+ of f is "evatruated"
and
the next trurh veetor is read from the input tape (i.e. the "interpretat{.or:"'} '
Leaving q *t the top of the stack, The rol.e of li. it to insist that the
next vector be k for k > i.. Thus in the downstroke no more than n truth
u-reeeffr's,
aud
n furrction
rii:rrr$tiri"its e,nn he readu from whieh we deduce that
L32
there are only a finiCe number of inequivalent configurations in
such a machine.
To see
show
that this
number
is,
however,
of order E2(rr), t"
that for any of the 22t r,rbr.ts of {r*, f-}o, there is
some
configuration which can evaluate, depending on the interpretation'
to everything in the subset but to nothing in its
that any
complement. We observe
such subset can be represented by a binary
with branches
marked
by a sign from {+, -}.
tree of depth n
Furthermore any such
tree can be represented by a string from {t;r"..F*, F;r.'.F-}* in
the following polieh notation: Each Fl represents a Left branch
from rhe (i-l)th
to the i-th leveI of nodes,
and
f] a siurilar right
branch.
fire string itsel"f is the expansion of the tree from the O-th l-eve1 (the
root) by the foi-Lowing recursive
process:
<tree.> + <branchl> <branchlt, O < i < n
111
<tree > +
n
A
<branchl> +
<branchl>
L
Any
A I tl*, <tree.+l>, o s i < n
* A I' rl1+I- <tree,1+I-),
O<
i
<n
string generated from a!t€e6> defines the set of paths in
tree that go from the root to n-th
that the eval-uation of
such
l-eveL
the
nodes. It is easy to verify
a string by our
schema corresponds
exactly to
tracing a path in the tree it specifies, and that the possible final
values such a string can take correspond to the paths in this tree. I
133
trle have
therefore esrablished rhat the sr.lccinctness
we are i-nvestigaEing involves two l-evets
rueasure
of exponentiation'
lle
note, however, that, in the above example g and t cannot be varied
independentLy. Thus to obtain more deuailed results for
leadi-ng exFonent
for the different
combinations
of
uhe
parameter value$,
furEher analysis is necessary. For exarnpteo in the schema above,
we eould have economised on the ntrnber
of predicates
aE Lhe expense erf greater comptications
used,
in description' to obtain
a sii.ghtly better result for one partieular such
c:t6€.
134
COI{CLUSION
l,Ie have shown
pushdown autonrata
that the various deterministic families of
are rich in decidable and potentiall"y decidable
properties. In doing so hre have also indicated areas outside
automata theory
to which our results relate.
Several weLl motivated decision problems have been
left
tmresolved. Moreover, even for those shown to be decidable, the
derived procedures usually require at least exponential time.
we do not know
of any arguments to
aLgorithms do not
show
that
exist for these, important
lrle can, however,
single out from
As
polynomiaL time
gaps remain here al-so.
among these open questions
the ones which appear the most inurediate.
Finding an equivalence test for the unrestricted class of
deterministic
pushdown automata nas
the primary unachieved goal
of our work. Although the existence of
one can perhaps be
now
conjectured with considerabLe confidence in the light of our
results, a proof of this would still be of great interest, for
additional insights it
techniques related
may
pro'.ride. It appears plausible that
to our parallel
and alternate stacking
constructions, and our simulations by nondeterministic
automata, wi1L play a
suggests
the
pushdown
part in settl-ing the probLern. Our work
that the resolution of our conjecture about
aLternate
stacking for the class \ r and the finding of an equivalence test
for the class S, My be significant next steps to that end.
A
positive solution to the latter problem, which we have not investigated,
rj5
also appears to be a prerequisite for finding a test of strong
for deBakker-Scott
eguivalence
schemas
-
For the regularity problen, since we have improved Stearns I
test, to a near optirral level, to achieve further improvements
new approach
between
this
is necessary.
and a number
cf
The
a
relationship we have established
containment problems which are
currentl-y open, can be interpreted as a trto sided challenge. In
the abse:ree of a more efficient regularity test, it hints that
if these other
them
problerns are decidabl-e, then
it
may be
difficult tc
prove
to be so.
We
because
have
not given
much
attention to the inclusion
probtrem
of ttre welL-known negative results concerning it.
However,
since sre ha"rq proved its undecidability for even a very restricted
case of
seems
R6
, a resol-ution of this
probLem
for the sinrple machines
Sg
mcst t.imeLy'
Thus our work suggests
that there
remain numerous
distinct
fealures of fhe strlrcture of these classes of automata yet to
be
uncovered, and indicates some speaific directions along which
these might be sought" The rer'rards E:f
this
search
will be'
we
betieve" to increase ottr understanding of these computations, and
to i:ender particutar instances of them more susceptible to practical
analysis.
136
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BrRD, M"R.
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HARRTSON,
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139
APPENDI X TO SECOND TI"IPRESS ION
;.j.trce this
iaesl-i.r1,.s
rcport
was
ha',.e been ob';ained
that
first
Lo its
relai.e directly
f.iere we drarv att.ention
conLents
issued several
to some of these: -
frr Chapter 2 iL was shown that an arbltary
dpda can be converted to one in normal form by
Al'r-rilhnr
q-.rv
t!\JrltL<aL
yvt.:/lrrnaai-l
]_>lliL q^rrrv
;lnalogous exponential
1
U1.l man-
).
It
*iqa
urr:is
-a-
uPpvrEu
^nnncod
time construction
l-n
Lv
tho
Lrrs
of Hopcroft
and
has since been shown by the author that the
conversion
can be done in linear
nachine.
This strenqthens
given in Chapter 6.
I^le note that
Rosenkrantz (Computational
and context-free
an algoriLhm requiring
by H.B. Hunt and D.J.
paralleIs
between the regular
languages, Proc. 6th ACM symp. on Theory
of Computing, SeattIe,
I974)
The construction
irr r-h-rnlsy
time on a random access
t.he force of the reductions
quadr:atic time has been reported
,arl\r)r
\/
an
3 (i.e
.
used in the undecidabilitv
Theorem3.4)
has been used by
I
ii.P.
l'riedman (Inclusion
Eth Princeton
LC)7 4
Problem for Simple Machj-nes,
Conf. on Information
) to show that the inclusion
and Systems Science,
problem f or simple
lanorracres is also undecidable.
A modified
4 can be found in:
for deterministic
Control,
formulation
Valiant
finite-turn
25, June L974,
L23
L.G.,
of the contents of Chapter
The equivalence problern
pushdown auLomata, Inf.
-f33
and
1/^
The results
wiI I appear as :
Val iant. L. G. , and Paterson
one-counter
Determinisuic
automata,
The nain reductions
substance of ChapLer 7 will
i;or:rrla-ii-rr anrl rol:ied
r.uJuLqL
Lvl
;ru Loma t-a
,
proofs
in Chapter 5, with revised
JCSS
M. S .
,
, L975.
of Chapter 6 and t-he
appear in:
L.G-,
Valiant
nroblems for deterministic
pushdown
r!v+
, L97 5 .
.j-AC14
In relation
of chapter B it
to the monadic functional
schetnes
has been shown by E. P. Friednan t.h.at t-he
decicl"ibil i ty of equival-ence f or these would imply tlie
of equivalence
decidability
for the whole class of
pushdown automata.
deterministic
L.G.V.
tlnirznrsitv
of
:je:pLcnber,
1974.
I.eeds