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). References Bartle, R., and Sherbert, D. 1982. Introdue:tion to Real Analysis. NewYork: John Wiley& Sons. Benzmueller, C.; Cheikhrouhou, L.; Fe.hrer, D.: Fiedler, A.; Huang, X.; Kerber, M.; Kohlhase, M.: Konrad, K.; Meier, A.; Melis, E.; Schaarschmidt, W.: Siekmann, J.; and Sorge, V. 1997. OMEGA:T~>wards a mathematical assistant.. In McCune.W.: ed.. Proceedings 14th International Confervnce on Automated Deduction (CADE-14), 252-255. Townsvillc: Springer. Bledsoe, W., and Hines, L. 1980. Variable elimination and chaining in a resolution-b&sed prover for inequalities. In Proceedings o[ the Fi[t.h Con[erence on Automated Deduction (CADE). Bledsoe, W.; Bo}’er, R.; and Henneman, W. 1972. Computerproofs of limit theorems. Artificial lnteUigence 3(1):27-~}0. 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