From: AIPS 1998 Proceedings. Copyright © 1998, AAAI (www.aaai.org). All rights reserved.
The "Limit"
Domain
Erica Melis *
Universit~it des Saarlandes, Fachbereich Informatik,
D-66041 Saarbriicken, Germany,
melis@cs.uni-sb.de
Abstract
Proof planning is an application of AI-planning
in mathematical domains. As opposed to planning for domainssuch as blocks world or transportation, the domain knowledge for mathematical domainsis difllcult to extract. Henceproof
planning requires clever knowledgeengineering
and representation of the domainknowledge. We
think that on the one hand, the resulting domain
definitions that include operators, supermethods,
control-rules, and constraint solver are interesting
in itself. On the other hand, they can provide
ideas for modelingother realistic domainsand for
means of search reduction in planning. Therefore, we present proof planning and an exemplary
domainused for planning proofs of so-cailed limit
theoremsthat lead to proofs that were beyondthe
capabilities of other current proof planners and
theorem provers.
Introduction
While humans can cope with long and complex proofs
and have strategies to avoid less promising proof paths,
classical automated theorem proving suffers from exhaustive search in super-exponential search spaces. As
a potential solution of this problem, proof planning has
been introduced by Bundy (1988) for inductive proofs.
As opposed to classical theorem proving, proof planning employs high-level planning operators rather than
calculus-level rules and global control rather than the
more local search heuristics which are used for search
control in automated theorem proving, see (Melis
Bundy 1996). The first proof planner, CLAM(Bundy
et al. 1991), has successfully planned inductive proofs
and some proof planning attempts have previously been
performed in the OMEGA
system (Benzmueller et al.
1.1997)
This work was supported by the Deutsche Forschungsgemeinschaft, SFB378
IOMEGA
is an assistant system that has proof planning
as a central componentand interfaces several external reasoners, such as computer algebra and automated theorem
provers.
Based on ALplanning experience and on theorem
proving heuristics, we have gained a deeper understanding of the general needs of proof planning and extended
proof planning in (Melis 1997). The objective was
plan more and difficult proofs in several mathematical domains. One of the key extensions is the extension of the domain knowledge a~ilable to the planner.
This paper presents the knowledge of the limit domain
that is necessary to prove theorems about limits and
that includes operators, control-rules, and a constraint
solver. With the extensions we succeeded in automatically planning proofs of limit theorems as introduced
below’, e.g., LIM+and LIM*. While LIM+ is at the
edge of what today’s theorem provers and planners can
handle, LIM*was beyond the capabilities of other current proof planners and theorem provers.
Why Is the limit
AI-Planning?
Domain Interesting
for
For mathematical domains, as for manyrealistic planning domains, essential ingredients for a success are an
appropriate knowledge representation and means to restrict the search. For established mathematical fields,
this knowledge exists, however often implicitly. The
appropriate operators and control knowledge can be
pretty difficult to extract and to represent, however.
This is one reason why mathematics appears to be hard
for humans(Schoenfeld 1985). Therefore, our representation of the limit domainis interesting in itself and, in
addition,
it canbe interesting
fortheplanning
communitybecause
¯ thelimitdomainis prototypical
forplanning
proofs
thatinclude
constructions.
As a prototype
of a mathematical
domainthatis wellknown,thelimitdomain
is comprehensible
outsidethetheoremprovingcommunity,we hope;
¯ thelimitdomain
is partof a hierarchically
organized
theoryknowledge
base.Sucha hierarchical
domain
organization
couldbe usefulforan agentthatcan
plantasksin several
domains.
¯ Forthe limitdomainwe can demonstrate
a general
wayto design
operators
fromexisting
special-purpose
CopyrightO1998American
Associationfor Artificial Iou-Iligence(www.aaai.org).
All rights Teserved.
Melis
199
From: AIPS 1998 Proceedings. Copyright © 1998, AAAI (www.aaai.org). All rights reserved.
heuristics and to devise control knowledge and constraint solving mechanisms;
¯ several new ideas can be of interest for pla~ming
in general, e.g., supermethods, control-rules, e.g.,
iterate control-rules,
and some of the operators,
e.g., Focus.
The paper is organized as follows. First we introduce
the class of limit theorems and briefly describe the general operator format and a glimpse of proof pla~ming in
the OMEGA
system. Then two kinds of operators are
presented, followed by a set of control-rules that proved
successful and by a brief description of a domain-specific
constraint solver. Weconclude with results.
The Class
of Limit
Theorems
The class of limit theorems includes the weU-known
theorem LIM+from calculus that states that the limit of
the sum of two fimctions in the real numbers, IR, is
the sum of their limits. Other class membersare, e.g.,
similar theorems about differences (LIM-) and products
(LIM*), Composite that states that the composition
of two continuous functions is continuous, ContIfDeriv
that states that a function having a derivative at a point
is continuous there, Cont+ that states that the sum of
two continuous functions is continuous and a similar
theorem (Cont*) about products, UNIFcont that says
that a uniformly continuous function is continuous, and
theorems like LIMsquare: lira
x2 = a2. Bledsoe (1990)
z--~{g
proposed the following LIM+as a challenge problem
for automated theorem proving.
lim f(x) = LtAlim g(x) = L.2 --~ lim(/(x)+g(x)) Lt+L.,,
~r--+a
z.-~a
x---~a
which, after expanding the definition
ofZ-.~n
lim, becomes
Ye13,51Vzl(O
< t:l -’, o < $1 ^ [=L - al < 51 ~ [.f(tl) - LII < el)
v~-,3aaVz~,(o
< E2.-+ o < a~^ 1:2 - al < &, -+ IJ(z2) -/-.21 < ~’,)
.-. V~3aVz(o
< ¯ --. o < ~ ^ I= - al < a --, I(/(=) + 9(=)) (La + 1.2)1 < e)
Thetypical
way"a mathematician
goesaboutto prove
sucha theorem
is to (incrementally)
inventallinstantiation
of J thatdepends
on e. Thetextbook
(Bartle
Sherbert
1982)proposes
to construct
6 by estimating
rangerestrictions
withthehelpof auxiliary
variables
thatpropagate
certain
rangerestrictions
frome to 5.
In theremainder,
we use thefollowing
namingconventions:
nameswiththecapital
initial
letterdenote
Operators,
and nameswrittenin lower caseletters
denoteprocedures,
div,*, +, -, I.Idenotethe division,multiplication,
’addition,
subtraction,
andabsolute
valuefunction
in IR,respectively.
F~ denotes
theresult
of applying
a substitution
a to an expression
F.
Proof
Operators
in OMEGA
Let us first consider proof planning operators. The
following description of OMEGA’s
operators is a bit
simplified. Operators have the slots premises and
conclusions, application-conditions, and proof schema.
Premises are (annotated) sequents s that are used by
an operator
to logically
derivethe conclusions,
and
conclusions
are(annotated)
sequents
whichtheoperator is designedto prove.Froma planningpointof
view,roughly,
theadd-and delete-effects
in STRIPS
terminology
are indicated
by the annotations
~ and
~, respectively.
An operator
witha (9 premiseand
a O conclusion
is introduced
in planning
froma goal,
whereasan operator
with@ premiseand ~ conclusion
is introduced
in planning
fromthe assumptions.
For
moredetails
see(Sehn1995).
The application-conditions are formulated in a metalanguage and restrict the applicability of an operator
and the instantiations of the parameters. The operator
is applicable with an instantiation Z of parameters, if
for Z application-conditions evaluates to true.
In case a proof of the conclusion from the premises
is known, proof schema is filled with a declarative
schematic representation
of the proof. This proof
schema can be used for the expansion of the operator.
The lines in proof schemacontain a label, a sequent, and
a line-justification. Since a proof line can be justified by
calculus rules from Natural Deduction (NO) calculus),
by an operator, by invoking tactics 4, or by invoking
automated theorem provers such as OTTER(McCune
1990) the line-justification can be a nameof a NO-rule,
the nameof an operator, a tactic, or a prover, a metavariable, or OPENin case the sequent is to be planned
for. Additionally, the line-justification mayinclude supporting lines.
For instance, in the operator MP-bthe line L3 A iF2 (~E;L1,L2)
states that the sequent A [- F2 is derived from the sequents in line L1 and L2 by the NO-rule ~E. The proof
line name (e.g., L1) abbreviates its sequent.
Planning
Proof planning needs planning in a static and deterministic environment with complete information about
the current state of the world. Furthermore, no goal
interaction has to be considered at the object-level 2 because the application of a sequence of proof inferences
2Exceptbinding inconsistencies handled separately.
200
applied
by operators
doesnotdestroy
object-level
preconditions.
However,
thepotential
search
spacein proof
planning
cangrowprohibitively
largebecause
of very
longproofsandevenworsebecause
of thepotential
infinite
branching
duetotheneeded
instantiation
of existentially
quantified
variables
andto theintroduction
of
lemmata.
Therefore,
searchcontrol
is crucial
in proof
planning.
Satisfiability and Logic
aA sequent is a pair (A t- F) ~dth a set formulaeA and a
formula F. Its meaningis F is derived from A. A proof line
additionally containsa label and a line-justification, e.g., 01
A t- F -~E)
4Atactic is a programthat executes a numberof logical
inferences (Gordon, Milner, &Wadsworth1979).
From: AIPS 1998 Proceedings. Copyright © 1998, AAAI (www.aaai.org). All rights reserved.
operator:
premises
HP-b(FI, F2)
conclusions
appl-cond
proof schema
~L1, L2
eL3
subset(A1,
L1. A ¯
L2. A1 ¯
L3. A ¯
A)
F1 -+ F2
F1
F2
(OPEN)
(j)
(-~E;L1L2)
The annotations indicate that L3 is removed from the
planning state as a goal and L1 is introduced.
The LimHeuristic Operator
LimHeuristic
is a centraloperator
in planning
limit
theorems.
Depending
on the particular
problem,the
LimHeuristic
has to be applieddifferent
numbersof
times.Forinstance,
in planning
LIM+itsis applied
onceandforLIM*it is applied
threetimes.Ourdesign
of LimHeuristic
is basedon thelimitheuristic
implementedin the special-purpose
programIMPLY(Bledsoe,Boyer,& Henneman
1972).Froma high-level
planningview,LimHeuristic
reducesa goal
thm :[b[ < e,
Planning
in OMEGA OMEGA’s planner
has a
STRIPS-like algorithm with a goal agenda. The planto three simpler subgoals (1), (2), and (3). For instance,
in planning LIM+,the goal is A t- If(x) g(z) - (/ 1 +
ning process searches the space of planning states. A
planning state contains a set of sequents that is divided
12)[ < e and the subgoals are the sequents
into open sequents that have to be proved (goals) and
1. A ~- Ill <M,
closed sequents (assumptions). An initial state is speci2. If(X1) -!1[ < E1 [-If(X1) -/1[ di v(e, 2* M),
fied by the proof assumptions and the proof’s goal, i.e.,
the theorem to be proved. The planner searches for a
3. A I- Ig(z) 12
l < div(e, 2)
solution, i.e., for a sequence of instantiated operators
whose application transforms the initial state into a fioperator:
LimHeuristic
(a,b, el,e)
nal state that has no open sequents. Forward and backward search is possible. Similar to HTNplanning (Tate
premises
(0), e(1),~ (2),
1977), the planner expands the operators if possible as
conclusions
@thm
soon as a plan is complete. Eventually, planning and
appl-cond
recursive expansion leads to a ND-proof that can be
3k, l, a(extract(a, b) = (k, l,
checked for correctness.
(0).A ¯ [a[< el
(j)
For instance, the introduction of the operator MP-b
(1).A ¯ lk~[< S
(OPEN’)
(2).A ¯ la=l<div(e,2*M)(OPEN)
intotheplandeletesa goal(A [- F2)(L3)from
¯ [!~.[ < div(e,2)
(OPEN)
proof schema (3).A
stateand addsa goal(A }- F1 --~ F2)(L1)instead,
L1.
¯ b = k¢ * as + !~.
(CAS)
provided
(A1[- F1)(L2)is available
as an assumption
thny.1 ¯ [b[ < e
(fix;Ll(l~
in thestate.Theexpansion
of HP-bintroduces
thein(2),(3))
stantiated
proofschemaintotheproofplan.
In OMEGA,domainknowledgeis storedin a hiThe application-conditions require that the proceerarchically
organized
mathematical
theoryknowledge
dure extract returns terms k, I, and a substitution a.
base.Theories
mayhaveparents
the)’caninherit
from.
extract(a, b) works as an oracle that tries to compute
Forinstance,
thetheoryordered-field
inherits,
among
terms k, 1, and a such that b can be represented as a
others,
fromthetheorybaseanda parentof thetheory
linear combination of a, i.e., b = ka * aa + la. extract
limitis ordered-field.
A theorymaycontainaxioms,
returns ± if it did not succeed in finding such k, l, and
definitions,
operators,
control-rules,
anda constraint a. In this case, the operator is not applicable.
solver.
The proof schema contains a schematic proof [b[ < e
from b = ka*aa+laand from (1), (2), and (3). The
Operators and Supermethods in limit
justification "fix" is an abbreviation for a fixed subproof
that proves the sequent in line thm from that in line
In this section, some operators are presented that beL1 and from the subgoals (1), (2), and (3). ’~I’
long to the limit domaintheory and to its parent theoND-rule implication introduction. The line-justification
ries, respectively. Note that LimHeturistic is the only
CASnames a computer algebra tactic that can justify
operator that is used exclusively for planning proofs for
the equation b = ka * a~ + l~. During the expansion of
limit theorems. All other operators described below are
LimHettristic, CASruns and returns a proof plan for
operators widely applicable at least for planning probits computation.
lems in ordered fields.
Each application of LimHeuristic suggests the exisSimilar to the mathematician’s behavior described
tence
of a new object Mwhose range is restricted by (1)
above,eachapplication
of LimHettristic providesa
and
(2).
Mis used real number propagate range restricnew auxiliaryvariableM on which6 dependseventions between the knownconstants and 6 as described in
tually.LimHeuristic
reduces
certain
inequality
goals
section. Howdoes the planner handle Linfliettristic:
to inequality
goalsthat containS. As opposedto
LimHeuristic,
operators
suchas SOLVE<band SOLVE*
¯ If a goal from the planning state matches thm,
proveinequalities
without
introducing
auxiliary
variLimHeuristic’s parameter are instantiated
by the
matcher.
ables.
Melis
201
From: AIPS 1998 Proceedings. Copyright © 1998, AAAI (www.aaai.org). All rights reserved.
¯ application-conditions
are evaluated. That is, the
procedure extract(a, b) is invoked. For instance,
a is g(X2) 12andb is f( x) + g(x) - ill 12), t he
extract(a, b) returns the list (1, (g(X2)-12), [x/X2]).
If extract runs successfully (i.e., yields a list
(k, l, a)), then the operator is applicable and the variables a~, k~, l~ are bound to terms resulting from applications of a to a, k, l.
¯ The application of LimHeuristic, removes the goal
thin and introduces the new goals (1), (2), and
The Solve
Operators
Tile operators
Solve<b, Solve*, Solve*<b, and
Solve<f (b for backward and f for forward) handle
goals and assumptions, respectively, that involve linear inequalities.
The operators Solve<b, Solve*<b,
and Solve<f call the function tell that provides an
interface to the constraint solver LINEQ(Melis 1997).
We describe Solve<b only. For the other operators
see (Melis 1997).
operator: Focus(F,pos)
premises
@L2
conclusions
OL1
S
= term.at_position(F, pos)
appl-cond
F1 = term_replace(F, S, focus(S))
(j)
L1. A ~- F
proof schema L2. ~ F Ft
(j)
The application conditions of Focus merely instantiate FI by a formula that results from the formula F
by replacing the subformula S of F at position pos by
focus(S).
Supermethods
In proof planning, a hierarchical decomposition is desirable because it can restrict the search space by planning at a higher level. Furthermore, a hierarchical presentation of the proof plan is easier to grasp by the
user, as shown in (Leron 1983). What a tedanique can
serve this purpose? HTNplanning (Tare 1977) replaces
operator: Solve<b(a, b)
an abstract operator by one of its predefined reduction schemas, but this is appropriate for proof planning
premises
only for abstract operators that have a proof schema
conclusions
ElL1
that provides a fixed expansion such as the operators
presented above. For other operators, tile right decomappl-cond
--occurs(a, b) & tell(a < b) tr ue
(solverCS)position may be computed from the planning situation
proo[schemaL1. A ~- (a < b)
rather thazl being one the set of predefined schemas.
Therefore, we introduce a class of operators, called suThe operator Solve<b is applied to a goal (a < b)
permethods.
and can be described as follows. In case tile occurs
These supernmthods have two faces: one that excheck for a, b falls (i.e., --occurs(a, b)), tell is invoked
hibits the features of an operator and another that
and tells its argument Ca < b) to the constraint solver.
amounts to control knowledge for computing a subIf (a < b) is consistent, with the current constraint store,
plan. Therefore, supermethods are operators that have
tell returns true and the operator is applicable. The
premises and conclusions and at the same time provide
operator removes the goal (a < b) from the planning
control knowledgeon howto build the expansion of the
state. The operator Solve<b has no preconditions, i.e.,
supermethod by planning with a particular set of opit produces no subgoals.
erators and control-rules given in the slots submethods
In proof schema, the line-justification solverCS names
and control.
a tactic that can recompute (a < b) from the constraint
store. During the expansion of Solve<b, this tactic
The expansion of supernmthods works as follows. A
runs and provides a proof plan for its computation.
new problem is created that contains one current goal
only. For this problem the planner is called with the set
Other Operators
of operators given in the slot submethods. Instead of
the usual ba~rktracking in planning, the supermethod’s
The application of 8P-b has to be controlled strictly
planning stops when no operator is applicable. It rebecause it can always be applied and produce new goals
turns the resulting changes of the planning state.
(Fil --+ (Fi2 ~ (... F2)...)) infinitely often.
Our most interesting supermethod brNWRhPItYPhas
Another operator, Focus, helps guiding the search in
the
submethods
hndE,Skolem-f,and Backchain.
Its
proof planning. Focus "colors" a subfornmla S of an
application
condition
requires
thattheassumption
F to
assumption F in order to provide a focus of attention.
whichUNNRAPHYP
is applied
contains
a focus.According
Actually, the operator does nothing else than replacing
to
the
control-rule
attack-latest,
UNNRAPHYP
decomS by a colored S. This focus of attention can be used
posesthelatestproduced
assmnption
thatcarries
a foby control-rules for guiding the search for assumptions
cusuntilan "unwrapped"
assumption
is obtained,
i.e.,
(or goals) to work on next, see the section on control
onethathasno subformula
outside
thefocus.Without
knowledge.
thiscontrol-rule
thesearch
spa~:e
growstoolargewhen
Backcha±n
is chosenunnecessarily.
202
Satisfiability andLogic
From: AIPS 1998 Proceedings. Copyright © 1998, AAAI (www.aaai.org). All rights reserved.
supermethod:UNWRAPHYP
character predictable
prem/ses
O L1
conclusions~LIST
appl-cond.
F <--- formula(L1) &termoccs(focus
submethods
control
(AndE Skolem-f Backchain)
(attack-latest)
As opposed to ’unpredictable’, the ’predictable’ classification means that the main output can be anticipated without actually expanding the supermethod.
For UNNRAPHYP
this holds because the eventually resulting assumption is marked by a focus before applying UNNRAPHYP.
However, this anticipation may not be
fully reliable and the subgoals in LISTthat arise during
the expansion cannot be predicted.
In a multi-strategy planner as described in (Kambhampati, Knoblock, & Yang 1995; Melis 1996), the expansion of supermethods can be a refinement strategy.
This expansion strategy yields the subplan that is introduced into the plan at a hierarchically lower level. The
expansion strategy can be invoked flexibly depending on
theplanning
stateandhistory
as wellas on resources
andproperties
of therespective
supermethod.
For instance,
the supermethod
UNNRAPHYP
doesnot
haveto be expandedimmediately
becauseits characteristic
’predictable’.
However,
an expansion
couldbe
preferred
if enoughresources
areavailable
andif the
userwantsto checktheND-proof
resulting
fromrecursive expansion.
For supermethods
for whichnone of
the resulting
goalsandassumptions
canbe predicted,
however,
theexpansion
hasto takeplacerightaway.
Control
Knowledge
Control knowledge in proof planning is used to reduce
the search and to prefer proof plans with a structure
that is comprehensible for the user. Several experiences (Minton 1989; Weld 1994) indicate the superiority of a separate representation of control knowledge
by control-rules.
This modular representation is well
suited for modifications, for the user’s comprehension,
and for learning control knowledge. Weadopted this
approach.
Currently, we distinguish the following classes (kinds)
of control-rules that correspond to different decisions of
the planner.
¯ strategy rules guide the choice of a refinement strategy,
¯ operator rules restrict
tors,
and rate the choice of opera-
¯ sequent rules,
with the subclasses
goal and
assumption, guide the choice of sequents to work on
next.
Currently, the syntax of control-rules is essentially
(control-rule
name
kind
if (conditions)
then
(prefer
[ select[reject
] iterate
(list))
Themeta-predicates
usedin therule’scondition
return
allsatisfying
binding
alternatives
in casean argument
is notinstantiated;
otherwise
the)’
return
a truth
value.
Examples
for meta-predicates
are goal-matches(x,
F)
andlast-operator(z)
whichyieldinstantiations
for
ifz is a variable.
Otherwise
theyreturn
thetruthvalue,
e.g.,of goal-matches(a,
F).
Inthefollowing,
wepresent
a setof control-rules
that
produced
a satisfying
search
behavior
in planning
limit
theorems
ratherthancompeting
forthemostefficient
control.
Forfigures
of thissearchseetheconcluding
section.
Mostof thervlesaredesigned
in orderto capturethefollowing
global
mathematical
control
storyin
limit
proofs:
1. Linear
ineqnalities
canbe proved
by simpleestimations
or by complex
estimations
thatarebasedon
simpler
inequalities
withauxiliary
variables.
2. The
complex
estimations
typically
comparea subformula
of a proofassumption
withtheinequality
to be estimated.
This mathematical knowledge ’how- to’ can be translated into the following verbally expressed control
knowledge that talks about operators.
1. Linearinequality
goalscan be satisfiedby
Solve<b,
SOLVE*,or by LimHeuristic.
2. Thelatter
requires
somepreparation
by UNNRAPHYP.
In planning
thelimittheorems,
UNNRAPHYP
extracts
thesubformulas thatneedsto be employedby LimHeuristic
(Itis the(0)assumption
of LimHeuristic).
Theverbally
expressed
control
knowledge
canbe capturedin control-rules
suchas
(control-rule
prove-in¯quality
(kind operator)
(if (goal-matches("goal"(less "x
y"))))
(then
(prefer((Solve<b
"goal())
(SOLVE* "goal" ())))
(side-effectmark (solve-failed))
The intention
of prove-inequality
is an attemptto
apply Solve<b or alternatively SOLVE*,if a goal is of
the form x < V- If these two fail to be applicable, the
application of LimHeuristic is prepared (in the sideeffect).
The side-effect
has been introduced in order to
avoid an unnecessary evaluation of the meta-predicate
similar-subterm
in case Solve<or Solve*are applicable.
The evaluation
of thismeta-predicate
is
expensivesincefor each instantiation
of "goal",
similar-subterm
returnsinstantiations
of "ass"and
"pos"..Themeta-predicate
computes
thoseformulae
s
at positions
posin someassumptions
asssuchthats is
mostsimilar
to thegoal.Thissimilarity
is measured
by
Melis
203
From: AIPS 1998 Proceedings. Copyright © 1998, AAAI (www.aaai.org). All rights reserved.
instantiations of the variable ~ mayexist. A way to delay the instantiation (until the plan is completed) is the
incremental restriction of the range of the variable by
a c~nstraint solver that is combinedwith the planner.
Constraint solvers represent, objects by specialized
data types and handle them efficiently. For manykinds
of constraints , e.g., for finite domains (He~tenryck
1989) there exist very efficient specialized procedures
( c~x~rol -r ~le ~x~ -LH
for constraint solving (consistency check, entailment
(ki~l operator)
check, and simplification}.
(if"(and (smlve-failed)
Actually, we could have taken off the shelf a con(and(l~st-goal
"goal")
straint
solver far linear arithmetic over tile real num(and.. (similLr-subt
erm("goal"
leUS"
beta, e.g., CLP(T~)(Jaffar et al. 1992). For the first ex"poe")
)) )
.periments, however, we used our ownconstraint solver,
(then
LINEQ,capable of handling value constraints that ave
(iterate
((Focus() ("us")"("poe"))
expressed by linear equalities and inequalities over IR
(t,~IIYPO ("ass"))
that may contain the absolute value function. It can
(ItemovaYocus
()~"ass")
) )))
restrict the range of variables.
The function tell interfaces certain operators with
The rulecan be readas: afterSolve<band SOLVE*
the constraint solw:r. Our most important operators ,.
failed
to be chosen,
thenextoperators
to be chosenare
Focus,UNWARPHYP,
and thenRemoveF.ocus.
Theinstanat the interface between the proof planning and the
constraint-handling
component
are Solve<b,
Solvefb,
tiation
of tileparameters
"ass"and"poe"of theFocus
operator
is obtained
by evaluating
the meta-l~edicate and Solve*<b.Theirpurposeis to removean equasimilax-subterm.
The operatorFocusintroducesa
tionalor aa inequality
goalby addingit to theconloc’,dfocuson s as explained
abo~e.Then~II~APHYP straint
store.
Eachof theseoperators
is applicable
only
workson theassumption
theftcontains
thelocalfocus.
¯ if itresults
ina consiste,t
constraint
store.
ThefuncThe unwrapping
business.is-completed
by re~novin~
the
tiontellinvoked
by theseoperators
a~cesses
theconlec~a]
.focus
because
itis nolongernecessary.
strain¢
solver
midreturns
trueif thecurrent
constraint
When an appropriateassumptionis extracted,
storeis consistent
withthetoldconstraint
and_l.othSOLVE*or alternatively
LimHeuristic
is triedto be
erwise.
applied
as formulated
in thenextcontrol-rule.
SDLVE*
Ttle constraint solver serves two .main purposes:
is triedfirstbecause
itproduces
simple
~)rno subgoals Firstly, it is used during the process..of proof planv.ing .’ "’
as opposedto LimNeuristic.
to determine whether a Solve operator can belegally
(~on~rol-rule.
~t ack~=pped
applied. Secondly, after the ~cnnptetien of the proof "
(kind operator)
plan, the constraint state is condensed into an answ~er
(if (and(lasz-Dperator
RemovaFocus)
assert ion about the values of somevariables that .is in-:
(and~-l~tes~-aasmnFtion
"ass")
ctuded into a proof and that justifies.inequalities
and
(go~l-ma~ches
("’goa.q~’(less
"x""y")
))
equalities
that
follow
from
the
final
cxmstraint
store.
(then
In LINEQthe constraint store is represented as a
(prefer((~gLVE*4’Eoal"("us"))
(disjunctive.) list of branches, e’~h of which cc~ta£ns,
~LimHeuristic
"goal"
("ass"))
))
for every knownequality class, a list of upper and lower --..
As mentioned ezalier, we have contro|-rutes ~h~t bebounds. These bounds are terms. The explicit reprelong to the ~ontrol of particular supermethods. For
sentat.ion of the constraint’st~’re facilitates backtracking. :
instazice,
attAr-~-latest
governs the planning for
toearlier
planning
states
bystoring
complete
constraint
UNWRAPHYP.
starein plannodes.
(control-rule
Et~ack-latest
WheneverLLNEQdeliversa new inequality
~o the
(kind sequent)
constraint
store,
it
is
solved
for
every
variable
it
con-,
(if (in-latest-assumption(colored
"ass")))
rains.
In
every
branch,
the
resulting
bound
is
added
(then
totheappropriate
boundlist.forthevariable
and~om(select-sequen~s (() ("ass") )
paredto everytermin the otherboundlist.For inTiffs mcans that only the latest assumption that
stm~ce,
forthelistsof upperandlowerbounds
u andl
carries a focus is admitted as an input of the next
the newboundis introduced
intou andcompared
with
UNWRAPHYP
submethod.
eachmemberof I. Theintroduction
andthecomparison
yieldnewinequalities
thatarerccursively
addedto the
Introducing
a Constraint Solver
store.Therecursion
endswhensucha compm’ison
fails,
thenewinequalit.y
isalready
entailed
bythestore,,
oran
In many proofs, mathematical objects with certain
inequality
contains
no morevariables.
A failedbranch
properties need to be constructed, for example, the existentially quantified 6 in LIM+. Pure planning can be
is removed
fromthebranch
list;a constraint
storewith
no branches
is considered
inconsistent.
difficult in this case because infinitely manypotential
comparing
thenumberof fmlction
symboloccurrences,
exceptthoseof + and*, in s "and"goal".
Thischaracterizes
thelikelihood
of L~ml~Ieuristic’s
applicability
to thegoal"goal".andassumption
s.
The rule before-LHprefersthe sequenceFocus,
]~WRAFHYP,
RemoveFocus
of operators
to be. applied
in
thepreparation,
of LimHeurietic.
204
Satisfiability and Logic
From: AIPS 1998 Proceedings. Copyright © 1998, AAAI (www.aaai.org). All rights reserved.
Let us consider an example. In planning the proof of
the LIM+theorem, the following constraints are told
to the constraint solver in a row: 0 < D, 1 < M, E1 <
div(e, 2 * M),X1 = x, E2 < div(e, 2),X2 = z,D <
62,D < 61. The final constraint
store of the LIM+
looks as follows:
troduction of a lemmaor for the introduction of a case
split (H V --H ~ G) in backward planning there are
infinitely manylegal instantiations of H. Furthermore,
in proof planning there is no goal interaction in the
original object-level sense.
Typically, the knowledge acqusition, for mathematicai domains is difficult.
Wepresented a prototypical
mathematical domain for proof planning and concen0 < E2< div(~, 2);
trated on operators and control knowledge. For this
0 < D < 62,61;
domain we have demonstrated a general way to design
0 < E1 < div(e, (2 * Af)),div(c,
operators from existing special-purpose heuristics. For
1 < M < div(e, (2 * Et))
the design of the operator LimHeuristic we rationally
-ov < Xl = x = X2 < +oo
That is, a lower bound for E2 is 0 and an upper bound
reconstructed Bledsoe’s limit heuristic that was devised
for E2 is 1/2 ¯ e; a lower bound for D is 0 and upper
for a special-purpose program in the ?0th. Compared
bounds are 61,62, etc. Fromthis final constraint store
with this program, proof planning with operators has
an assertion can be extracted that corresponds to mathseveral advantages: It relies on a general-purpose probematicians’ proof assumption: "Let el, e2 < div(e, 2)
lem solver; it provides high-level, hierarchical representations of proofs that can be expanded to chec~ble
and let. 6 = min(6t, 62) ....
calculus-level proofs; and it employsdeclarative control
Evaluation By proof planning ill the presented doknowledge that is modularly organized.
main we have automatically planned proofs of many
We have devised mathematical control knowledge
limit theorems, e.g., of LIM+and LIM*. In the techni(which is an ambitious enterprise that will be contincal report (Melis 1997), we discussed experiments that
ued) andconstraint
solvingmechanisms.
Thelevelat
have shown that the planning would not have succeeded
whichour controlknowledge
is represented
makcsit
without control-rules. With the control-rules that are
possible
to communicate
it to theuserandto learnconused for all problems from the limit domain, the plantrolknowledge.
Theintegration
of constraint
solvers
ning effort is infinitesimal comparedto classical theothathelpinsearching
forvariable
instantiations
opens
rem proving. LIM+that is at the edge of what classiup newopportunities
for proof planning as well as for
cal automated theorem provers can prove today is relgeneral AI-planning.
atively simple in our proof plan setting; its whole planWhycan all this be interesting for the ALplanning
ning takes 214 matchings while planning for LIM*takes
community? Well, on the one hand, the domain defi347 matching attempts and yields a larger proof plan.
nitions that include operators, supermethods, controlHence, LIM* that could not be proved by other currules, and constraint solvers are interesting in itself. On
rent proof planners and general theorem provers can be
the other hand, they can provide ideas for modeling
proved with the OMEGA
planner. The resulting
plan
other realistic domains and for means of search reducfor LIM*
tion in planning. In particular, several ideas can be
NORMAL
SOLVE<-F UNWRAPHYP
generalized beyond proof planning, e.g.,
ILEMOVEFOCUSLIMHE~ISTIC
UNNRAPHYP
REMOVEFOCUS ¯ the handling of potentially infinite branching for
LIMHEURISTICSOLVE<BSOLVE<BSOLVE*UNWRAPHYP
the instantiation of variables by constraint solvREMOVEFOCUS
LIMHEURISTIC
SOLVE<BSOLVE<B
ing. Even if the instantiation of variables in AISOLVE*SOLVE<BSOLVE<BSOLVE*SOLVE<BSOLVE<B
planning has a large search space rather than an
SOLVE<B
SOLVE*SOLVE<B
SOLVE*is still tiny cominfinte one, this technique can be useful. These
pared with the caiculus-level proof of some 300 steps
constraint solvers do not have to be implemented
that results from expanding the plan.
from scratch but can be reused, e.g., from constraint logic programming.
Conclusion
¯ useful extensions of control-rules such as iterate
rules,
Proof planning is an application of AI-planning and
¯ other tools that can be employed to focus the
can make use of experiences in AI-planning, e.g., by
search such as the Focus operator, and
using control-rules.
Comparedto the planning in con¯ supermethods that plan for a subproblem with a
ventionai domains, the proof planning domains are departicular set of operators and control-rules.
scribed as mathematical theories, such as group theory or limit, that contain axioms, theorems, operators,
Previous
Work
control-rules, and domain-specific constraint solvers.
The operators represent complex inference actions and
As mentioned above, CLAMhas been the first proof
the control-rules represent mathematical knowledge on
planner. It works with a uniform difference-reduction
how to proceed in proofs. The objects are mathematsearch heuristic which is appropriate in a class of proofs,
ical objects such as numbers, lists, or trees. A characin particular in inductive proofs, where the differences
teristic particular for proof planning is the potentially
between induction conclusion and induction hypotheinfinite branching in search. For instance, for the insis have to be reduced by rewriting the induction conMelis
205
From: AIPS 1998 Proceedings. Copyright © 1998, AAAI (www.aaai.org). All rights reserved.
clusion in order to apply the induction hypotheses.
CIfiM knows operators such as induction, fertilize~
symbolic-evaluation that are important for a class of
typical inductive proofs.
Related from the knowledgeacquisition perspectiw, is
the work of Bledsoe and Hines on special-purpose theorem pro~ers (Bledsoe & Hines 1980). Related with respect to planningwith control-rules is Prodigy. (Mint, on
et al. 1989).
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