Scope-Bounded Pushdown Languages
Salvatore La Torre
Università degli Studi di Salerno
joint work with
Margherita Napoli Università degli Studi di Salerno
Gennaro Parlato University of Southampton
Multi-stack Pushdown Automata (MPA)
•
n stacks sharing a finite control
– states: s, , , ...........,
1
2
n
– transitions :
• push one symbol onto stack i
• pop one symbol from stack i
• internal move: stacks stay unchanged, only control
location is altered
•
input is from a one-way read-only tape
•
model of concurrency
– captures the control flow of concurrent programs with
shared memory and recursive procedure calls
Visible alphabets
• Alphabet is partitioned into:
– calls (cause a push operation)
– returns (cause a pop operation)
– internals (stacks are not used)
• For n >1 stacks, alphabet is also partitioned
according to stacks
– the stack to operate is uniquely identified by the
input symbol (it is visible in the input)
What visibility gains for MPA?
•
Stack usage is synchronized with the input
– parallel simulation of multiple runs, cross product
construction, subset-like constructions
•
1-stack (VPL):
– intersection and determinization
– universality, inclusion and equality
•
n-stacks (MVPL):
– just intersection
– emptiness is undecidable: the runs of MPA are visible!
– checking for emptiness of MVPL equals to decide
reachability for MPA
Theme of the talk
• The formal language theory of visibly nstack languages of k-scoped words (SMVPL)
Visible alphabet (retns) --st1: a, a’ st2: b, b’ internal: e
matching relation (matching call/retn)
a e b
a a’ a’
b
a
b’ e b’
a’
scope of matching relation over S: max number of S-contexts
between matching
call/retn
word
is 3-scoped
– scope is 2 for GREEN and 3 for RED
A few observations....
• Interest for restrictions of MPA mainly comes from
verification
– bugs of concurrent programs are likely to occur within
few context-switches [Musuvathi-Qadeer, PLDI ‘07]
– efficient sequentializations of multithreaded programs
[Lal-Reps,CAV’08]
• Robust automata theories are useful tools for other
domains
– Automata-theoretic approach to verification (modelchecking)
– Pattern matching problems
–…
• K-scoped visibly languages indeed form a robust class…
k-scoped MVPA
• Closure under Boolean operations
• Det./nondet. models are equivalent
• Decidable emptiness [La Torre-Napoli, CONCUR’11], inclusion,
equality, and universality
• Logical characterization (MSO with matching relations)
• Parikh theorem
• Sequentializable: computations can be simulated with one
stack (rearranging order of inputs)
• Decidable temporal logic model-checking
[La Torre-Napoli,TCS’12] [Atig-Bouajjani-Kumar-Saivasan, ATVA’12]
More related work
• Visibly pushdown languages [Alur-Madhusudan J. ACM'09]
[Melhorn ICALP'80]
Restricted MPAs:
• Emptiness/reachability/closure properties
[Carotenuto et al. DLT’07] [Atig et al. DLT’08]
[Seth,CAV’10] [LaTorre et al. LATIN'10]
[LaTorre et al. MFCS'14]
• Model-checking [Atig, FSTTCS’10] [Bollig et al. MFCS’11]
[Bollig et al. LICS’13] [Bansal-Demri, CSR’13]
• MSO of multiply nested words
[Madhusudan-Parlato POPL'11] [Cyriac et al. CONCUR'12]
• ............
Rest of the talk
• Determinization construction
• Brief comparison with the known MPA
classes of languages
• Conclusions
MVPL are nondeterministic
•
L = {(ab)i cjdi-j xjyi-j | i,j>0}
is inherently nondeterministic for MPA
[La Torre-Madhusudan-Parlato, LICS’07]
– j is arbitrary and needs to be the same for both
stacks
– a guess is needed when pushing both stacks
• L is not SMVPL
– For any j, (ab)k cjdk-j xjyk-j is (k+1)-scoped
a b a b a b ……… a b
c………… d x………… y
Determinization of SMPA
• Summaries of computations for SMPA
– Linear interfaces
– Switching masks
• PDA computing linear interfaces
– linear interface automaton (LIA)
• Simulation of SMPA by deterministic
composition of deterministic LIAs
(using switching masks)
View of runs by stacks
Input word (contexts)
w = g1 r1 g2 b1 r2 b2 r3 b3 r4 g3
Run (without stacks)
q1
g1
q2
q1
r1
q3
g2
q4
g1
b1
q2
q5
q2
q10
q4
g3
q6
b2
r1
r2
g2
q3
r2
q11
q7
r3
q8
b3
q9
r4
q3
q4
b1
q5
q6
q6
b2
q7
q8
q8
b3
q9
q5
r3
q7
r4
q10
q9
stack 1
q10
stack 2
stack 3
g3
q11
Linear Interface (LI)
• k-LI for a stack S just summarizes starting and
ending control states for k consecutive contexts of S
in a run (starting from stack S empty)
q1
q3
q10
g1
g2
g3
q2
q4
q11
q2
q5
q7
q9
r1
r2
r3
r4
3-LI for stack 1
stack 1
stack 2
q3
q4
q6
q6
q8
q8
q10
b1
q5
b2
q7
b3
q9
2-LI for stack 2
stack 3
Switching Mask (SM)
• a tuple of LI, one for each stack
• a function Nxt that links contexts of LI’s
stack 2
stack 1
q1
q3
q10
g1
g2
g3
q2
q4
q11
q2
q5
q7
Switching mask
q
9
r1
r2
r3
r4
stack 3
q3
q4
q6
q6
q8
q8
q10
b1
b2
b3
Nxt function given by purple arrows
q5
q7
q9
Simulating MPA with SMs (1)
• Control state (h, M)
h: current stack
M: switching mask
• Move within a context: just update LI of stack h
• Es:
q1
g1
q2
q2
r1
q3
q4
b1
q55
M:
q3
g2
q4
h=3
MPA transition from q5 to p5 on stack-3 symbol
q1
q2
q3
q4
q2
q3
q4
(h is not changed)
p5
Simulating MPA with SMs (2)
• Context-switch 1 (accumulated stack content needed):
add a new context to an existing LI
• Es: h=3, and MPA moves from q5 to p5 on a stack-2
symbol
M:
q1
q3
g1
g2
q2
q2
r1
q3
q4
q2
q3
q4
q5
p5
b1
q5
q4
then h=2 and the SM is
q1
q2
q3
q4
q5
Simulating MPA with SMs (3)
• Context-switch 2 (accumulated stack content
no longer needed):
start a new LI
• Es: h=3, and MPS moves from q5 to p5 on a stack-2 symbol
M:
q1
q3
g1
g2
q2
q2
r1
q3
q4
p5
q4
b1
q5
q4
then h=2 and the SM is
q1
q2
q3
q4
q5
q5
PDA accumulating LIs
Given a PDA P over an alphabet , symbols ,#
a k linear interface automaton (k-LIA) for P is a PDA s.t.
• input is over   {,#}
w11#w12#........#w1i1 w21#w22#........#w2i2 ....
• control states are h-LIs of P for hk
• on , simulates P on the last state of the LI
• on #, a new context is appended to the current LI
(provided that it is a h-LI with hk-1)
• on , a new LI is started and stack is reset
(a bottom-of-the-stack symbol is pushed onto the stack to avoid
the use of previously pushed symbols)
(k)-LIs suffice for SMPA
Theorem.
By restricting to k-scoped inputs,
h-LIs with hk suffice to simulate the
behavior of an MPA with switching masks
Thus, for each stack of an SMPA,
we can restrict to k-LIAs
Determinization of SMPA (1)
For an SMPA A
• construct the LIA Ah for each stack h
• construct Dh by determinizing each Ah as in
[Alur-Madhusudan, STOC’04]
• construct the deterministic SMPA D (equiv. to A)
– cross product of the Dh‘s
– parallel simulation of A with all the generated
SMs (subset construction)
Determinization of SMPS (2)
• a state of D is of the form (h, Q1,...,Qn, ) where
– h is the current stack
– Q1,...,Qn is a state of the cross product
–  is a set of switching masks
• within a context of stack h, D simulates Dh
(the Q
the
size of Dand
is all the switching masks in 
h–component
gets updated
accordingly)
• exp
in the size of A and
• 2exp in the number of stacks
• on context-switching from stack h to stack i (a
and
thei bound
call/return of
stack
is read),kD simulates in parallel
– Dh on either # or 
– Di on the input symbol
Comparisons
CSL
OMVPL
[BCCC96]
[MCP07]
[ABH08]
RMVPL
SMVPL
[LPM10]
VPL
TMVPL
[AM04]
[LNP14]
PMVPL
[LMP07]
CFL
Decision Problems
VPL
CFL
RMVPL
SMVPL
TMVPL
PMVPL
OMVPL
CSL
Conclusions
• SMVPL form a robust theory of visibly languages
(the largest among those closed under
determinization)
• Sequentialization is nice for analysis purposes
– Computations of MPA can be analyzed via
computations of PDA
– used in software verification
• Scope-bounded words meaningfully extends to –
words
– Describe infinite on-going interaction among
different threads
Theory on infinite words?
• Little it is known on MPS over –words
• visibly pushdown Büchi automata
[Alur-Madhusudan,J. ACM, 2009]
- the model is not determinizable
• emptiness for k-scoped Büchi MPA is PSPACEcomplete
[La Torre-Napoli,TCS’12]
• closure under union and intersection are simple