General Domain Circumscription and Its First-order Reduction General Domain Circumscription and Its First-order Reduction

We rst de ne general domain circumscription GDC and provide it with a semantics GDC subsumes existing domain circumscription proposals in that it allows varying of arbitrary predicates functions or constants to maximize the minimization of the domain of a theory We then show that for the class of semi universal theories without function symbols that the domain circumscription of such theories can be construc tively reduced to logically equivalent rst order theories by using an extension of the DLS algorithm previously proposed by the authors for reducing second order formulas We also show that for a certain class of domain circumscribed theories that any arbitrary second order circumscription policy applied to these theories is guaranteed to be reducible to a logically equivalent rst order theory In the case of semi universal theories with functions and arbitrary theories which are not separated we provide additional results which although not guaranteed to provide reductions in all cases do provide reductions in some cases These results are based on the use of xpoint reductions The rst author has been supported in part by the Swedish Council for the Engineering Sciences TFR The second and third authors are a liated with the Institute of Informatics Warsaw University and have been supported in part by KBN grant P General Domain Circumscription and its First Order Reduction Patrick Doherty and Witold Lukaszewicz and Andrzej Sza las Abstract We rst de ne general domain circumscription GDC and provide it with a semantics GDC subsumes existing domain circumscription proposals in that it allows varying of arbitrary predicates functions or constants to maximize the minimization of the domain of a theory We then show that for the class of semi universal theories without function symbols that the domain circumscription of such theories can be constructively reduced to logically equivalent rst order theories by using an extension of the DLS algorithm previously proposed by the authors for reducing second order formulas We also show that for a certain class of domain circumscribed theories that any arbitrary second order circumscription policy applied to these theories is guaranteed to be reducible to a logically equivalent rst order theory In the case of semi universal theories with functions and arbitrary theories which are not separated we provide additional results which although not guaranteed to provide reductions in all cases do provide reductions in some cases These results are based on the use of xpoint reductionsWe rst de ne general domain circumscription GDC and provide it with a semantics GDC subsumes existing domain circumscription proposals in that it allows varying of arbitrary predicates functions or constants to maximize the minimization of the domain of a theory We then show that for the class of semi universal theories without function symbols that the domain circumscription of such theories can be constructively reduced to logically equivalent rst order theories by using an extension of the DLS algorithm previously proposed by the authors for reducing second order formulas We also show that for a certain class of domain circumscribed theories that any arbitrary second order circumscription policy applied to these theories is guaranteed to be reducible to a logically equivalent rst order theory In the case of semi universal theories with functions and arbitrary theories which are not separated we provide additional results which although not guaranteed to provide reductions in all cases do provide reductions in some cases These results are based on the use of xpoint reductions Introduction In many common sense reasoning scenarios we are given a theory T specifying general laws and domain speci c facts about the set of phenomena under investigation In addition one provides a number of closure axioms circumscribing the domain of individuals and certain properties and rela tions among individuals The closure machinery normally involves the use of non monotonic rules of inference or in the case of circumscription a second order axiom In order for a circumscribed theory to be useful it is necessary to nd a means of computing inferences from the circumscribed theory in an e cient manner Unfortunately the second order nature of circumscription axioms creates an obstacle towards doing this In previous work we proposed the use of an algorithm DLS which when given a second order formula as input would terminate with failure or output a logically equivalent rst order formula Since circumscription axioms are simply second order formulas we showed that the DLS algorithm could be used as a basis for e ciently computing inferences for a broad class of circumscribed theories by rst reducing the circumscription axiom to a logically equivalent rst order formula and then using classical theorem proving techniques to compute inferences from the original theory augmented with the output of the algorithm In the DLS algorithm was generalized using a reduction theorem from It was shown that a broad subset of second order logic can be reduced into xpoint logic Moreover a class of xpoint formulas was characterized which can be reduced into their rst order equivalents In this paper we extend the previous work in three ways We de ne a general form of domain circumscription which subsumes existing domain cir cumscription proposals in the literature and We call the generalization general domain circumscription GDC GDC distinguishes itself from other proposals in the following manner When circumscribing the domain of a theory T it is permitted to vary arbitrary predicates functions or constants to maximize the minimization of the domain of individuals We characterize a class of theories which when circumscribed using GDC are guaranteed to be reducible to equivalent rst order theories which are constructively generated as output from extended versions of the original DLS algorithm Included in this class are theories for which both McCarthy s original domain circumscription and Hintikka s mini consequence are always reducible to rst order logic We characterize a class of theories which when rst circumscribed using GDC and then circumscribed using an arbitrary circumscription policy are guaranteed to be reducible to equivalent rst order theories which are constructively generated as output from the extend ed versions of the original DLS algorithm mentioned in the previous item We approach the characterization and reduction problems in the following manner Given a theory T we show that if the domain closure axiom is entailed by the domain circumscribed theory CircD T then CircD T is always reducible to a logically equivalent rst order theory We then characterize a class of theories where the domain closure axiom is not only en tailed by the domain circumscribed theory but can be automatically generated and used in the extended algorithm to reduce theories from this class to their corresponding rst order equivalents Given a theory in the class characterized above and an arbitrary circumscription policy applied to that theory we show that the extended version of the DLS algorithm will always generate a rst order theory logically equivalent to the second order circumscribed theory The key to the approach is determining when a domain circumscribed theory CircD T entails it s domain closure axiom Semantically a possible answer is when the cardinalities of all minimal models of the domain circumscribed theory have the same nite upper bound Syntactically we can characterize two classes of theories that provide such constraints when minimized Universal theories without function symbols where the general domain circumscription pol icy can include arbitrary constants and predicates that vary Semi universal theories without function symbols where the general domain circumscription policy can include arbitrary constants and predicates that vary The class of semi universal theories is a broad class of theories much more expressive than uni versal theories which have previously been studied in the context of restricted forms of domain circumscription In the case of universal and semi universal theories with function symbols where the general domain circumscription policy can include arbitrary constants predicates and func tions that vary reducible classes of theories are di cult to characterize In this case we provide additional results which guarantee reduction non constructively and additional methods which although not guaranteed to provide rst order reductions in all cases do provide reductions in some cases The paper is organized as follows Section consists of preliminary de nitions and notation In Section general domain circumscription is introduced together with it model preferential seman tics In Section the original DLS algorithm is brie y described together with two limitations associated with the basic algorithm In Section two generalizations of the basic DLS algorithm are described which deal with the limitations previously described In Section reducibility re sults concerning di erent specializations of general domain circumscription are presented together with a number of concrete examples In Section we consider the potential for reducing a larg er class of arbitrarily circumscribed theories which are rst circumscribed using general domain circumscription Preliminaries In this paper the term theory always refers to a nite set of sentences of rst order logic Since each set is equivalent to the conjunction of its members a theory may be always viewed as a single rst order sentence In the sequel we shall never distinguish between a theory T and the sentence being the conjunction of all members of T Unless stated otherwise the term function symbol refers to a function symbol of arity n where n Notation An n ary predicate expression is any expression of the form x A x where x is a tuple of n individual variables and A x is any formula of rst order classical logic If U is an n ary predicate expression of the form x A x and is a tuple of n terms then U stands for A As usual a predicate constant P is identi ed with the predicate expression x P x Similarly a predicate variable is identi ed with the predicate expression x x An n ary function expression is any expression of the form x x where x is a tuple of n individual variables and x is any term of rst order classical logic If u is an n ary function expression of the form x x and t is a tuple of n terms then u t stands for t An n ary n function constant f is identi ed with the function expression x f x An n ary n function variable is identi ed with the function expression x x Note that ary function variables are simply individual variables Let U U Un and V V Vn resp u u un and v v vn be tuples of predicate resp function expressions U and V resp u and v are said to be similar i for each i i n Ui and Vi resp ui and vi are predicate resp function expressions of the same arity Truth values true and false are denoted by and respectively If U and V are predicate expressions of the same arity then U V stands for x U x V x If U U Un and V V Vn are similar tuples of predicate expressions i e Ui and Vi are of the same arity i n then U V is an abbreviation for Vn i Ui Vi IfA is a formula n and n are tuples of any expressions then A stands for the formula obtained from A by simultaneously replacing each occurrence of i by i i n For any tuple x x xn of individual variables and any tuple t t tn of terms we write x t to denote the formula x t xn tn We write x t as an abbreviation for x t De nitions De nition A theory T is said to be existential universal i all of its axioms are of the form x T resp x T where T is quanti er free De nition A theory is called semi universal if its axioms do not contain existential quanti ers in the scope of universal quanti ers De nition Let T be a theory without function symbols of positive arity and suppose that for n c cn are all the individual constants occurring in T The domain closure axiom for T written DCA T is the sentence x x c x cn Let c be a tuple of individual constants By DCA c T we shall denote the sentence x x c x cn where c cn are all the individual constants of T excluding constants from c For k by DCA k T we shall denote the sentence z zk x x z x zk x c x cn where c cn are all the individual constants of T We also use notation DCA c k T as a combination of the above De nition A predicate variable occurs positively resp negatively in a formula A if the conjunctive normal form of A contains a subformula of the form t resp t A formula A is said to be positive resp negative w r t i all occurrences of in A are positive resp negative De nition Let be either a predicate constant or a predicate variable and be a tuple of predicate constants or a tuple of predicate variables Then a formula T is said to be separated w r t i it is of the form T T where T is positive w r t and T is negative w r t General Domain Circumscription In this section we provide a de nition of general domain circumscription GDC and its model preferential semantics GDC subsumes both McCarthy s original domain circumscription intro duced in studied in and substantially improved in and Hintikka s mini consequence formulated in and studied in De nition Let P P Pn be a tuple of di erent predicate constants f f fk be a tuple of di erent function constants including perhaps individual constants T P f be a theory and let be a one place predicate variable be a tuple of predicate variables similar to P and be a tuple of function variables similar to f By Axiom P f sometimes abbreviated by Axiom we shall mean the conjunction of a for each individual constant a in T not occurring in f i for each individual constant a in T such that a is fi x xn x xn f x xn for each n ary n function constant f in T not occurring in f and x xn x xn i x xn for each n ary n function constant f in T such that f is fi T stands for the result of rewriting T replacing each occurrence of x and x in T with x x and x x respectively De nition Let P P Pn f f fk and T P f be as in De nition The general domain circumscription for T P f with variable P and f written CIRCD T P f is the following sentence of second order logic T P f x x Axiom P f T x x A formula is said to be a consequence of CIRCD T P f i CIRCD T P f j where j denotes the entailment relation of classical second order logic The second conjunct of the sentence is called the domain circumscription axiom It is not di cult to see that asserts that the domain of discourse represented by is minimal with respect to T where P and f are allowed to vary during the minimization We shall write CIRCD T as an abbreviation for CIRCD T i e if neither predicate nor function constants are allowed to vary This simplest form of domain minimization corresponds closely to McCarthy s original domain circumscription with the augmentation described in We shall write CIRCD T P as an abbreviation for CIRCD T P i e if some predicate constants but not function constants are allowed to vary If P includes all predicate constants occurring in a theory T then CIRCD T P is exactly mini consequence introduced in and improved in Following this form of minimization will be referred to as variable domain circumscription Example Consider a theory T consisting of x P x Q x P a Q b We shall minimize the domain of T without varying predicate or function constants CIRCD T is given by T x x a b x x P x Q x P a Q b x x Substituting x x a x b for we get T x x a x b a a a b b a b b x x a x b P x Q x P a Q b x x a x b Since is equivalent to T x x a x b we conclude that the domain closure axiom for T i e the sentence x x a x b is a consequence of CIRCD T Example Let T consist of P a P b We minimize the domain of T with a allowed to vary during the minimization CIRCD T a is given by T xa x x xa b P xa P b x x Substituting x x b for and b for xa one easily calculates that implies T x x b Accordingly we conclude that the domain of T consists of one object referred to by both a and b In fact CIRCD T is slightly stronger in that it is based on a second order axiom rather than on a rst order schema Note that in variable domain circumscription all predicate constants but no function constants are allowed to vary during the minimization process Since a is an individual constant the variable corresponding to a is in fact an individual variable Accordingly we denote it by xa rather than by We now proceed to give a semantics for general domain circumscription We start with some notational conventions Given a model M we shall write j M j to denote the domain of M By V v A we denote the truth value of formula A in M with respect to a valuation v Note that if A is a formula of second order logic then v provides an interpretation not only for individual variables but also for predicate and possibly function variables If v is a valuation for a model M d is an element of jM j and x is an individual variable then v x d denotes an assignment that is identical to v except for the variable x which is assigned the value d De nition Let P f and T P f be as in De nition Let M and M be models of T We say that M is a P f submodel of M written M U u M i jM j jM j and for each predicate function or individual constant C occurring neither in P nor in f the interpretation of C in M is the restriction of the corresponding interpretation in M to jM j A model is said to be P f minimal i it has no proper P f submodels Theorem Let P f and T P f be as in De nition A formula A is a consequence of CIRCD T P f i A is true in all P f minimal models of T Proof It su ces to show the following a model M of T is a model of CIRCD T P f i M is a P f minimal model of T Assume to the contrary that M is a model of CIRCD T P f andM is not P f minimal Thus there is a model N of T which is a proper P f submodel of M Clearly j N j j M j Since M is a model of CIRCD T P f V v x x Axiom T x x for each assignment v Suppose that v is j N j and v i resp v i is the relation resp function corresponding to the interpretation of Pi resp fi in N Since N is a P f submodel of T this assignment is well de ned It is easily veri ed that V M v x x Axiom T Thus in view of we infer that V M v x x This means that j M j j N j A contradiction Assume to the contrary that M is a P f minimal model of T and M is not a model of CIRCD T P f Thus there is an assignment v such that V v x x Axiom T V v x x Consider the set S given by d S i V v x d x In view of S is a proper subset of j M j Moreover in view of S is non empty We de ne a model N such that j N j is S and the interpretation of each predicate and function constant not in P f in N is the restriction of the corresponding interpretation in M to j N j Using one easily checks that N is a model of T Thus since j N j jM j N is a proper P f submodel of M A contradiction An Optimization Technique In this section we propose a technique that allows one to reduce the size of a domain circumscrip tion axiom The technique allows one to sometimes remove counterparts of universal formulas from the axiom More precisely let theory T consist of axioms including universal axioms of the form x xn A x xn and suppose that all predicate and!or function constants occurring in A x xn are not allowed to vary during the minimization Then each such axiom reappears in as a part of T equivalent to the formula x xn x xn A x xn Since together with the corresponding axiom of T reduces to it can be removed from T We thus have the following principle Remove a counterpart of every universal axiom A of T from T in the domain cir cumscription axiom provided that none of the predicate and!or function constants occurring in A are allowed to vary If B is the resulting formula then T B is equiv alent to CIRCD T P f Observe also that one can remove formula x x from whenever T contains a constant symbol This follows from the fact that for each constant symbol say a one has a as a conjunct of Axiom Thus x x follows from Axiom and can be removed It should be emphasized that the DLS algorithm works successfully without the above mentioned optimizations However as shall be seen in the examples in Section they usually considerably decrease the complexity of the reduced formula DLS Algorithm The Basic DLS Algorithm In this section we brie y describe the DLS algorithmmentioned in the introduction Its complete formulation can be found in The algorithm was originally formulated in a weaker form in in the context of modal logics It is based on Ackermann s techniques developed in connection with the elimination problem see The DLS algorithm is based on the following lemma proved by Ackermann in see The proof can also be found in Lemma Ackermann Lemma Let be a predicate variable and A x z B be for mulas without second order quanti cation Let B be positive w r t and let A contain no occurrences of at all Then the following equivalences hold