A feature-based constraint system for logic programming with entailment. Theoretical Comput

more general than the trees of Herbrand(called constructor trees), and the constraints of arefiner grained and of different ex-pressivity.The basic notion of are functional attributes called features,which provide for record-like descriptions of data avoid-ing the overspecification intrinsic in Herbrand’s constructor-based descriptions.The feature tree structurefixes an algebraic seman-tics for.We will also establish a logical semantics,which is given by three recursive axiom schemesfixing thefirst-order theory.is a constraint system for logic programming,providing a test for unsatisfiability,and a test for entailment between constraints, which is needed for advanced control mechanisms.The two major technical contributions of this paper are(1)an incremental entailment simplification system that is proved to be sound and complete,and(2)a proof showing that satisfies the so-called“independence of negative constraints.”1IntroductionAn important structural property of many logic program-ming systems is the fact that they factorize into a constraint system and an extension facility.Colmerauer’s Prolog II [7]is an early language design making explicit use of this property.CLP(Constraint Logic Programming[9]),ALPS [15],CCP(Concurrent Constraint Programming[20]),and KAP(Kernel Andorra Prolog[8])are recent logic program-ming frameworks that exploit this property to its full extent by being parameterized with respect to an abstract class of constraint systems.The basic operation these frameworks require of a constraint system is a test for unsatisfiability. ALPS,CCP,and KAP in addition require a test for entail-ment between constraints,which is needed for advanced control mechanisms such as delaying,coroutining,synchro-nisation,committed choice,and deep constraint propagation.sorts,and whose edges are labeled with symbols calledfeatures.The labeling with features is deterministic in thatall edges departing from a node must be labeled with distinctfeatures.Thus,every direct subtree of a feature tree can be identified by the feature labeling the edge leading to it.The constructor trees of Herbrand can be represented asfeature trees whose edges are labeled with natural numbersindicating the corresponding argument positions.All but the second and third feature tree in Figure1satisfythe description(1).The constraints of are ordinaryfirst-order formulaetaken over a signature that accommodates sorts as unary andfeatures as binary predicates.Thus the description(1)isactually syntactic sugar for the formulawine(x)∃(()())∃(()())The set of all rational feature trees is made into a corre-sponding logical structure by letting()hold iff the root of is labeled with the sort,and letting()hold iff has as direct subtree via the feature.The feature tree structurefixes an algebraic semantics for.We will also establish a logical semantics,which is givenby three recursive axiom schemesfixing afirst-order theory .We will show that the feature tree structure is a model of.Backofen and Smolka[5]show that is indeed a complete theory,which means that is exactly the theory induced by.However,we will not use this result in the present paper,but show explicitly that entailment with respect to to is the same as entailment with respect to. The two major technical contributions of this paper are (1)an incremental entailment simplification system that is proved to be sound and complete,and(2)a proof showing that satisfies the so-called“independence of negative constraints”[6,13,14].The incremental entailment simpli-fication system is the prerequisite for’s use with either of the constraint programming frameworks ALPS,CCP or KAP mentioned at the beginning of this section.The in-depence property means among other things that constraints can essentially be handled through en simplification.One origin of is A¨ıt-Kaci’s-term calculus[1 is at the heart of the programming language LOGIN further extended in the language LIFE[4]with funct feature structures thanks to a generalization of the of residuation of Le Fun[3].Other precursors offeature descriptions found in so-called unification g [12,11]developed for natural language process also the formalisms of Mukai[16,17].These earl structure formalism were presented in a nonlogic Major steps in the process of their understanding an reformulation are the articles[19,22,10,21].The paper is organized as follows.Section2de basic notions and shows that the feature tree structu model of.Moreover,Section2discusses the di in expressivity between Herbrand and.Sectiona basic simplification system that decides satisfiapositive constraints.Section4is not committed to discusses the notion of incremental entailment chec its connection with the indepence property and n Section5gives the entailment systems,provescomplete and terminating,and also proves the in property.2Feature Trees and Constraints To give a rigorous formalization of feature trees,w two disjoint alphabets and,whose symbols a sorts and features,respectively.The letters,always denote sorts,and the letters,,will alway features.Words over are called paths.The conc of two paths and results the path.The s denotes the empty path,==,and denot of all paths.A tree domain is a nonempty set that 21=∈.Thus,is given as the mapping :1defined on its domain by()=().A feature tree is called a subtree of a feature tree if it is a subtree=at one of its nodes,and a direct subtree if ∈.A feature tree with domain is called rational if(1) has onlyfinitely many subtrees and(2)isfinitely branching, which is:for every∈,=∈∈isfinite.Assuming(1),(2)is equivalent to saying that there existfinitely many features1...such that 1....Constraints over feature trees will be defined asfirst-order formulae.Wefirstfix afirst-order signatureby taking sorts as unary and features as binary relation symbols.Moreover,wefix an infinite alphabet of variables and adopt the convention that,,always denote variables. Under this signature,every term is a variable and an atomic formula is either a feature constraint(()in standard notation),a sort constraints(()in standard notation), an equation=,(“false”),or(“true”).Compound formulae are obtained as usual by the connectives,,, ,¬and the quantifiers∃and∀.We use˜∃and˜∀to denote the existential and universal closure of a formula, respectively.Moreover,()is taken to denote the set of all variables that occur free in a formula.The letters and will always denote formulae.In the following we won’t make a distinction between formulae and constraints,that is, a constraint is a formula as defined above.-structures and validity of formulae in-structures are defined as usual.Since we consider only -structures in the following,we will simply speak of structures.A theory is a set of closed formulae.A model of a theory is a structure that satisfies every formulae of the theory.A formula is a consequence of a theory(=) if˜∀is valid in every model of.A formula is satisfiable in a structure if˜∃is valid in.Two formulae,are equivalent in a structure if˜∀()is valid in.We say that a formula entails a formula in a structure [theory]and write=[=]if˜∀()is valid in[is a consequence of].A theory is complete ifsays that has no subtree at,that is,that there isdeparting from’s root that is labeled with.To shas subtree at path1,we can use the const ∃1∃1(11122 (1)Now let’s look at statements we cannot expres precisely,statements of whom the authors belicannot be expressed).One simple unexpressible sis“is a subtree of”(that is,“∃:=”).Mwe cannot express that is smaller than.Finalassume that the alphabet of features is infinite,w say that has subtrees at features1...but no at any other feature.In particular,we then cannot sis a primitive feature tree,that is,has no proper subThe theory0is given by the following tw schemes:∀∀∀(=)(for every feature)∀()(for every two distinct sorts and).Thefirst axiom scheme says that features are fuand the second scheme says that sorts are mutuallyClearly,is a model of0.Moreover,0is in(for instance,∃()is valid in but invalid in otheof0).We will see in the next section that0important role with respect to basic constraint simplNext we introduce some additional notation needrest of the paper.This notation will also allow usa third axiom scheme that,as shown elsewhere[5]to a complete recursive axiomatization of.Throughout the paper we assume that the conjuformulae is an associative and commutative operhas as neutral element.This means that we ()with(),and with(bu example,with).A conjunction o formulae can thus be seen as thefinite multiset formulae,where conjunction is multiset union,an3Given a solved-clause,we say that a variable is dependent in if contains a constraint of the form,or, and use()to denote the set of all variables that are dependent in.The theory is obtained from0by adding the axiom scheme:˜∀∃(for every solved-clause and=()). Theorem2.1The feature tree structure is a model of the theory.For a proof of this theorem see[5].This paper also proves that is a complete theory if the alphabets of sorts and features are infinite.3Basic SimplificationA basic constraint is either or a possibly empty conjunction of atomic formulae of the form,,and=. The followingfive basic simplification rules constitute a simplification system for basic constraints,which,as we will see,decides whether a basic constraint is satisfiable in .1.≠3.=[]∈()and≠5.====Using the same steps up to the last one,the c==simplifiesthe last step,Rule2instead of Rule3is applied).Proposition3.2If the basic constraint simplifithen0=.Proof.The rules3,4and5perform equivalencemations with respect to every structure.The rulescorrespond exactly to the two axiom schemes ofperform equivalence transformations with respectmodel of0.We say that a basic constraint binds a variable=∈and occurs only once in.At this pimportant to note that we consider equations as ordeis,assume that=is different from=ifsay that a variable is eliminated,or bound by,ito some variable.Proposition3.3The basic simplification rules are ting.Proof.First observe that the simplification rules dnew variables and preserve eliminated variables.more,rule4increases the number of eliminated varone.Hence we know that if an infinite simplificatiexists,we can assume without loss of generality themploys the rules1,3and5.Since rule1decrenumber of feature constraints“”,which is not iby rules3and5,we know that if an infinite simpchain exists,we can assume without loss of generit only employs the rules3and5.Since this iimpossible,an infinite simplification chain cannot eA basic constraint is called normal if none ofsimplification rules applies to it.A constraint isnormal form of a basic constraint if can be sto and is normal.A solved constraint is aconstraint that is different from.42.the graph of is a solved clause;3.no primitive constraint appears more than once in. Proposition3.5Every solved constraint is satisfiable in every model of.Proof.Let be a solved constraint and be a model of .Then we know by axiom scheme that the graph of a solved constraint is satisfiable in an-model.A variable valuation into such that=can be extended on all eliminated variables simply by()=() if=∈,such that=.Theorem3.6Let be a normal form of a basic constraint .Then is satisfiable in if and only if≠. Proof.Since and are equivalent in every model of 0and is a model of0,it suffices to show that is satisfiable in if and only if≠.To show the nontrivial direction,suppose≠.Then is solved and we know by the preceding proposition that is satisfiable in every model of.Since is a model of,we know that is satisfiable in.Theorem3.7For every basic constraint the following statements are equivalent:=˜∃∃:=˜∃=˜∃Proof.The implication12holds since is a model of 0.The implication31follows from the fact that is a model of.It remains to show that2 3.Let be satisfiable in some model of0.Then we can apply the simplification rules to and compute a normal form such that and are equivalent in every model of0.Hence is satisfiable in some model of0.Thus≠,which means that is solved.Hence we know by the preceding proposition that is satisfiable in every model of.Since and are equivalent in every model of0,we have that is satisfiable in every model of .will not determine∃when∃is considered this may change when is strengthened tobasic idea leading to an incremental entailment ch to simplify with respect to the context and variables.Given,and,simplification mu formula such that=∃∃The following facts provide some evidence that th right invariant for entailment simplification.Proposition4.1Let=∃∃.Then:1.=∃iff=∃;2.=¬∃iff=¬∃;3.if=,then=¬∃;4.if∃is valid in,then=∃.Statements1and2say that it doesn’t matter wh tailment and disentailment are decided for or.S 3gives a local condition for disentailment,and S 4gives a local condition for entailment.The en simplification system for feature trees given in section will in fact decide entailment and disentai simplifying such that the condition of Statement4 the case of entailment,and that the condition of Sta is met in the case of disentailment.In practice,one can ensure by variable renam no variable of occurs in.The next fact s then it suffices if entailment simplification respects convenient invariant=This is the invariant respected by our system(cf.tion5.4).Proposition4.2Let()=∅.Then:5system satisfies the independence property if=∃11...∃∃:=∃for all basic constraints,1...and allfinite sets of variables1....Proposition4.3If a constraint system satisfies the indepen-dence property,then the following statements hold(,and 1...are basic constraints):1.¬∃11...¬∃unsatisfiable in iff∃:=∃;2.if¬∃11...¬∃is satisfiable in,then¬∃11...¬∃=∃iff=∃. 5Entailment SimplificationWe now return to the feature tree constraint system. Throughout this section we assume that is a solved con-straint and is afinite set of variables not occurring in. We will call the context,the variables in local,and all other variables global.If is a theory and and are possibly open formulae, we write=(read:entails in)if˜∀()is valid in.Theorem5.1For every basic constraint,the following equivalences hold:=¬∃=¬∃=¬∃Proof.Implication“23”holds since0.Impli-cation“31”holds since is a model of.To show implication“12”,suppose=¬∃.Then we know by Proposition4.2that is unsatisfiable in.Thus we know by Theorem3.7that is unsatisfiable in every model of0.Hence we know by Proposition4.2that=0¬∃.==≠-oriented and pivoted3.∈=5.==∉∈7.We say that a basic constraint simplifies to a cwith respect to and if=or simpliinfinitely many steps each licensed by one ofsimplification rules given above.The notions of nonormal form with respect to are defined accordinExample5.2Let=andThen we have the following simplification chain witto and:====by Rule==by Rule===by Ruleby RuleLet us now take as context˜=˜====is normal with respand.We shall see that this normal form tell˜does not determine˜.If˜gets strengthened6We observe that1is equal to1,modulo renaming by1=and by1=,and modulo the repetition of .Lemma5.3Let simplify to1with respect to and,not using Rule E6(in an entailment simplification step).Then simplifies to some1which is equal to1up to variable renaming and repetition of conjuncts.Proof.Clearly,each entailment simplification rule,except for E6,corresponds directly to a basic simplification rule (namely,E1and E2to B1,E3to B2,E4to B3,E5and E7 to B4,and E8to B5).If the application of the entailment simplification rule to relies on a condition of the form=or=where ≠or≠,then=∈or=∈,and Rule B4isfirst applied to,eliminating by(by).When comparing1and1,renamings takeaccount of these variable eliminations.Note that,if the rule applied to is E2,then has one feature constraint less than—which,after renaming,has a repetition of exactly this constraint.Proposition5.4If simplifies to with respect to and ,then and are equivalent in every model of 0.Proof.follows from Lemma5.3and Proposition3.2. Proposition5.5The entailment simplification rules are ter-minating,provided and arefixed.Proof.First we strengthen the statement by weakening the applicability conditions=in Rules E1and E4 to=.Then from Lemma5.3follows:(*)Each entailment simplification rule applies to1with respect to and if and only if it applies to1with respect to and—except possibly for E5,when the corresponding variable has already been eliminated in an“extra”basic simplification step.of0.Proposition5.6For every basic constraint one c pute a normal form with respect to and=∃iff=∃,and=∃=∃.Proof.Follows from Propositions5.4,5.5,4.2and In the following we will show that from the en normal form of with respect to it is easy to tell we have entailment,disentailment or neither.More basic normal form of is exactly in the (and in the second,where=),and“almos third case(cf.Lemma5.3).Proposition5.7A basic constraint≠is nor respect to and if and only if the following co are satisfied:1.is solved,-oriented,and contains no variis bound by;2.if=and∈,then∈for ever3.if=and∈and∈,then4.if=and∈,then∈for every5.if=and∈and∈,then=Lemma5.8If≠is normal with respect tothen is satisfiable in every model of.Proof.Let≠be normal with respect toFurthermore,let=and=be th decompositions in normalizer and graph.Since the bound by occur neither in nor in,it suffices that is satisfiable in every model ofLet()be the basic constraint that is obtainby applying all bindings of.Then equivalent to()and no variable b occurs in().Hence it suffices to s7We say that a variable is dependent in a solved constraint if contains a constraint of the form,or=. (Recall that equations are ordered;thus is not dependent in the constraint=.)We use()to denote the set of all variables that are dependent in a solved constraint.In the following we will assume that the underlying signature has at least one sort and at least one feature that does not occur in the constraints under consideration. This assumption is certainly satisfied if the signature has infinitely many sorts and infinitely many features. Lemma5.10Let1...be basic constraints different from,and1...befinite sets of variables disjoint from().Moreover,for every=1...,let be normal with respect to and,and let have a dependent variable that is not in.Then¬∃11...¬∃is satisfiable in every model of.Proof.Let=be the unique decomposition of into normalizer and graph.Since the variables bound by occur neither in nor in any,it suffices to show that ¬∃11...¬∃is satisfiable in every model of.Thus it suffices to exhibit a solved clause such that and,for every=1...,()is disjoint with and is unsatisfiable in every model of.Without loss of generality we can assume that every is disjoint with()and()for all.Hence we can pick in every a dependent variable such that∉for any.Let1...be all variables that occur on either side of equation=∈,=1...(recall that isfixed for ).None of these variables occurs in any since every is-oriented.Next wefix a feature and a sort such that neither occurs in or any.Now is obtained from by adding constraints as follows: if∈,then add;if∈,then add; to enforce that the variables1...are pairwise distinct, add1 (211)It is straightforward to verify that these additions to yield a solved clause as required.hence that∃is satisfiable in.Thus≠¬∃is unsatisfiable in and is a model o know by Lemma5.10that().To show the other direction,suppose≠and.Then=˜∀∃by Proposition5.11,an=˜∀∃.Thus=∃,and hence= Proposition5.6.Theorem5.13Let be a basic constraint.Theniff=∃.Proof.One direction holds since is a model o show the other direction,suppose=∃.Wit of generality we can assume that is normal with rand.Hence we know by Theorem5.12that().Thus=˜∀∃by Proposition hence=∃.Theorem5.14(Independence)Let1...b constraints,and1...befinite sets of variabl =∃11...∃∃:=∃Proof.To show the nontrivial direction,suppo ∃11...∃.Without loss of generality assume that,for all=1...,is disjoint fro is normal with respect to and1,and≠¬∃11...¬∃is unsatisfiable in a model of,we know by Lemma5.10that( for some.Hence=∃by Theorem5.12.6ConclusionThe constraint system is mathematicallly well and possesses the properties which make it practi for advanced logic programming systems.The pre above admits the possibility of extending with order(and a greatest-lower-bound operation)on tha rather straightforward way,thus accounting for thbasis of another part of LIFE[4].8on Programming Language Implementation and Logic Programming(Passau,Germany),J.Maluszy´n ski and M.Wirsing,editors.LNCS528,pages255–274, Springer-Verlag,1991.[5]R.Backofen and G.Smolka.A complete and decid-able feature theory.Research report,German Research Center for Artificial Intelligence(DFKI),Stuhlsatzen-hausweg3,6600Saarbr¨u cken11,Germany,1992.To appear.[6]A.Colmerauer.Equations and inequations onfinite andinfinite trees.In Proceedings of the2nd International Conference on Fifth Generation Computer Systems, pages85–99,1984.[7]A.Colmerauer,H.Kanoui,and M.V.Caneghem.Pro-log,theoretical principles and current trends.Technol-ogy and Science of Informatics,2(4):255–292,1983.[8]S.Haridi and S.Janson.Kernel Andorra Prolog andits computation model.In D.Warren and P.Szeredi, editors,Logic Programming,Proceedings of the7th International Conference,pages31–48,Cambridge, MA,June1990.The MIT Press.[9]J.Jaffar and ssez.Constraint logic program-ming.In Proceedings of the14th ACM Symposium on Principles of Programming Languages,pages111–119,Munich,Germany,Jan.1987.[10]M.Johnson.Attribute-V alue Logic and the Theory ofGrammar.CSLI Lecture Notes16.Center for the Study of Language and Information,Stanford University,CA, 1988.[11]R.M.Kaplan and J.Bresnan.Lexical-FunctionalGrammar:A formal system for grammatical represen-tation.In J.Bresnan,editor,The Mental Representation of Grammatical Relations,pages173–381.The MIT Press,Cambridge,MA,1982.The MIT Press.[16]K.Mukai.Partially specified terms in logic pming for linguistic analysis.In Proceedings oInternational Conference on Fifth Generatiputer Systems,1988.[17]K.Mukai.Constraint Logic ProgrammingUnification of Information.PhD thesis,Tokyoof Technology,Tokyo,Japan,1991.[18]M.Nivat.Elements of a theory of tree codeNivat,A.Podelski,editors,Tree Automata(Aand Open Problems),Amsterdam,NE,1992.Publishers.[19]W.C.Rounds and R.T.Kasper.A completcalculus for record structures representing linformation.In Proceedings of the1st IEEE Syon Logic in Computer Science,pages38–43MA,1986.[20]V.Saraswat and M.Rinard.Concurrent cprogramming.In Proceedings of the7th AnnSymposium on Principles of Programming Lapages232–245,San Francisco,CA,January1[21]G.Smolka.Feature constraint logics for ungrammars.The Journal of Logic Programming87,1992.[22]G.Smolka and H.A¨ıt-Kaci.Inheritance hieSemantics and unification.Journal of Symboputation,7:343–370,1989.9。