Gelder, A. V. 1993. The Alternating Fixpoint of Logic Programs with Negation. Journal of computer and system sciences 47:185--221.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....true, then F [d] would be true. Or, the bodies of the grounding correspond to the partial models of the rule body F [d] Define a consistent set S of literals of At I as a set of positive or negative literals based on At I that does not contain a pair of complementary literals p(d) p(d) 4 In (Gelder 1993), an alternative way of defining the wellfounded semantics of predicate rules is proposed; it is based on a different treatment of positive and negative occurrences of predicates in the body of rules. I believe both techniques are equivalent but haven t proven this. 5 Note that Sigma I may be ....

Gelder, A. V. 1993. The Alternating Fixpoint of Logic Programs with Negation. Journal of computer and system sciences 47:185--221.

....obtained. More recently, a paraconsistent extension of this semantics (WFSX p ) has been proposed in [1, via an alternating fixpoint definition that we now recapitulate. We begin by recalling the definition of Gelfond Lifschitz Gamma operator, used in the alternating fixpoint definition of WFS [19],WFSX , and WFSX p . Definition 23. 21] Let P be an extended program, I a two valued interpretation. The GL transformation P I is the program obtained from P by removing all rules containing a default literal not A such that A 2 I, and by then removing all the remaining default literals from ....

A. V. Gelder. The alternating fixpoint of logic programs with negation. In 8th Symposium on Principles of Database Systems. ACM SIGACT-SIGMOD, 1989.

....intent of how they are interpreted. Disjunctive Database Rule (DDR) was proposed which always inteprets disjunctions inclusively [21] The DDR has been shown to be equivalent to the so called Weak Generalized Closed World Assumption in [17] Stable model semantics [7] and well founded semantics [26, 27] are attempts to assign a natural meaning to normal logic programs. Well founded semantics is extended to disjunctive databases by the so called weak well founded semantics and strong well founded semantics [20] Strong well founded semantics generalizes GCWA and infers a subset of perfect model ....

Van Gelder, A. "The Alternating Fixpoint of Logic Programs with Negation," Proceedings of the 8th ACM Symposium on Principles of Database Systems , Philadelphia, 1989, pp. 1-10.

.... y oe (y = z oe x = z) 3: 8x 1 ; xn ; y 1 ; yn ( x 1 = y 1 : xn = yn) oe (f(x 1 ; xn) f(y 1 ; yn) 4: 8x 1 ; xn ; y 1 ; yn ( x 1 = y 1 : xn = yn) oe (p(x 1 ; xn) j p(y 1 ; yn) 2 The notion of alternating fix point was introduced byVan Gelder [8] and later studied by a number of authors. It has been shown that normal alternating fix points characterize almost all the semantics for logic programs with default negation (cf. 21] where f is any n ary function symbol and p is any n ay predicate symbol. We translate A into a priority ....

A. Van Gelder, `The alternating fixpoint of logic programs with negation ', J. Computer and System Sciences, 185--221, (1993). The preliminary version appeared in PODS '89.

....works on a fixed computation strategy, a feature necessary for efficient implementation. Also, the definition of an SLG evaluation reflects the fact that computing threevalued stable models may require transfinite induction, a point first made by Van Gelder for the alternating fixpoint technique [79]. Unlike the alternating fixpoint, SLG is a goal directed resolution strategy, and computes only those atoms of the model which are needed to support an initial query. To reflect this property, statements about its correctness are made with respect to the restriction of a model. Definition 2.2.7 ....

....and, based on that model, the program is rewritten to make clauses definite. Using these definite clauses, a fixpoint is reached with a new model. The process of finding a model, and rewriting program clauses to make them definite continues until a larger fixpoint is reached. In approaches like [79] or [6] this iterated fixpoint is central to the formulation of the evaluation method, while in SLG it is perhaps less obvious. In SLGO , a system of clauses may be evaluated until only suspended and answer clauses remain. At completion, the suspended clauses are either reactivated, or delayed. ....

A. van Gelder. The alternating fixpoint of logic programs with negation. In Proc. of 8th PODS, pages 1--10. ACM, 1989.

....assumptions about programs. In this paper the well founded approach will be extended to a richer range of logics than just classical. Again, bilattices provide the right tools. Rather than extending the original characterization of the well founded model, we will build on the equivalent one of [21] involving alternating fixpoints. We believe that by moving to a more general setting, not only do we gain a wider range of applicability, but proofs become more transparent and unnecessary details fall away. We note that some of the results appearing here have been obtained, in a less general ....

....of truth. Stratified semantics, 2] is closely related to Tarski s well known heirarchy of languages. The Kripke Kleene approach of [5] was motivated by Kripke s work, 13] Its bilattice generalization, 10] has its philosophical analog, 6] So too with the alternating fixpoint approach of [21], which was anticipated in [24] This convergence of two disciplines makes me want to believe we are doing something right. 2 Syntax We assume we have a fixed collection of constant, function and relation symbols. Terms and atoms are built up as usual (we allow false as an atom) A literal is an ....

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Allen Van Gelder. The alternating fixpoint of logic programs with negation. In Proc. 8th ACM Symp. on Principles of Database Systems, pages 1--10, Philadelphia, 1989. ACM.

....formula of first order logic. Arbitrary firstorder formulas in rule bodies allow natural descriptions of queries and rules. For instance, the fixpoint logic (FP) namely first order formulas augmented with a least fixpoint operator, has been studied extensively as a recursive query language (see [19] and its references) In active databases, conditions in event condition action rules can be arbitrary relational queries [1] General logic programs were studied by Lloyd and Topor as a basis for deductive databases [9, 10, 11] The semantics of a general logic program is defined using Clark s ....

....programs. Efficient techniques of query evaluation with respect to the well founded semantics have been developed [4, 13, 15, 18] Van Gelder developed a constructive characterization of the well founded semantics of normal logic programs, called the alternating fixpoint partial model [19]. It extends naturally to general logic programs; such an extension is called alternating fixpoint logic [19] One may wonder if the alternating fixpoint logic can be implemented through the standard translation of general logic programs into normal logic programs. As pointed out by Van Gelder ....

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Van Gelder, A. The alternating fixpoint of logic programs with negation. Journal of Computer and System Sciences, 47:185--221, 1993.

....9 , the existential fragment of second order logic. Schlipf proves in [Schlipf1990] an analogous result for DATALOG : with stable model semantics (DATALOG : stable henceforth) Sacc a gives in [Sacc a1993] further insight on the expressive power of DATALOG : stable . Van Gel der analyzes in [Van Gelder1989] the expressive power of DATALOG : with well founded semantics. In all these papers, databases are modeled as finite structures, i.e. finite interpretations of theories. In this work we are concerned with default logic as a query language. Default logic [Reiter1980] is one of the most popular ....

A. Van Gelder. The Alternating Fixpoint of Logic Programs With Negation. In Proceedings PODS-89, pages 1--10, 1989.

....Well founded semantics is closely related to the stable model semantics [11] another major approach to logic programs with negation. The well founded semantics approximates the stable model semantics [17, 10] Moreover, computing the well founded model of propositional programs is polynomial [16] while computing stable models is NP hard [12] Consequently, evaluating the well founded semantics can be used as an effective preprocessing technique in algorithms to compute stable models [15] In addition, as demonstrated by smodels [13] at present the most advanced and most efficient system ....

....most efficient system to compute stable models of DATALOG : programs, the well founded semantics can be used as a powerful lookahead mechanism. Despite the importance of the well founded semantics, the question of how fast it can be computed has not attracted significant attention. Van Gelder [16] described the so called alternating fixpoint algorithm. Van Gelder s algorithm runs in time O(jAt(P )j Thetasize(P ) where 1 A preliminary version of this paper appeared in the Proceedings of CL 2000. 2 On leave from Warsaw University of Technology. 1 At(P ) is the set of atoms occurring ....

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A. Van Gelder. The alternating fixpoints of logic programs with negation. In ACM symposium on principles of database systems, pages 1--10, 1989.

....semantics for logic programs [5] stable models are called default models in [6] 7] in the following, we refer to this variant of DATALOG as DATALOG : stable . Sacc a gives in [8] further insight on the expressive power of interesting variants of DATALOG : stable . Van Gelder analyzes in [9] the expressive power of DATALOG : with well founded semantics [10] In all these papers, databases are modeled as nite structures, i.e. nite interpretations of theories. In this work we are concerned with default logic as a query language. Default logic [11] is one of the most popular NMR ....

A. Van Gelder, \The Alternating Fixpoint of Logic Programs With Negation", in Proceedings PODS-89, 1989, pp. 1-10.

....As an immediate example, it becomes possible to define modal operators that combine the features of Kripke s and of Moore s. We have also implemented an automated theorem prover that handles modal operators of the sort we have defined. The implementation extends existing work on stratification, [1, 23, 74, 75] which can be used to compute the consequences of some autoepistemic theories, to our more general setting. This truth value mapping approach to modality allows modality to be interpreted very generally. For example, causality, temporal notions, and even negation can be viewed as modal ....

A. Van Gelder. The alternating fixpoint of logic programs with negation: Extended abstract. In Proceedings of the ACM SIGACT-SIGMOD Symposium on Principles of Database Systems, 1989.

....of U P (L) has the form of a coinduction rather than the form of an induction (see [9] Secondly, note that the authors of [105] did not explicitly tell us how to obtain U P (L) constructively. Subsequently, some equivalent constructions of well founded partial models were proposed. In [104], van Gelder defined the alternating fixpoint construction. In [16] Bidoit and Froidevaux used the notion of a potentially founded set, the dual of the notion of unfounded sets. However, it is worth noting a construction is in fact implicitly there in [105] Since Gamma P;L is a monotonic ....

A. van Gelder. The alternating fixpoint of logic programs with negation. Journal of Computer and System Sciences, 47:185--221, 1993.

....of the well founded semantics can be made sufficiently fast, that calculation may also prove a highly useful subroutine in computing under the stable semantics. The standard calculation methods for the well founded semantics are based upon the alternating fixed point algorithm of Van Gelder [13]. Several researchers have observed that, for propositional logic programs, the (worst case) time taken by that algorithm to compute the well founded partial model is quadratic in the size of the program. More specifically, for a propositional logic program P with atom set A, the complexity is ....

....of every disjunct is false in , and undefined otherwise. Moreover, this can be computed by substituting T literals true in and F for literals false in and doing standard simplifications. The algorithm embodied in Van Gelder s alternating fixed point definition of the well founded semantics [13] translates almost immediately into the following algorithm on their corresponding hypergraphs: Algorithm1. Van Gelder) We are given a program P and its hypergraph interpretation H. We maintain a partial truth assignment , consisting of the truth values already inferred, and two approximations ....

A. Van Gelder. The alternating fixpoint of logic programs with negation. In Eighth ACM Symposium on Principles of Database Systems, pages 1--10, 1989. Available from UC Santa Cruz as UCSC-CRL-88-17.

....models, since there is one that is, in a certain sense, minimal. This minimal stable model was characterized in more than one way van Gelder, Ross and Shlipf [40,41] gave a construction that led to its standard name, the well founded model. Van Gelder gave an alternating fixpoint construction [39]. And Pryzmusinski gave yet another construction that established its minimality [30] The investigation of stable models fits well with the four valued approach presented earlier. In addition, extracting the algebraic features behind the constructions makes it clear that they are really quite ....

Van Gelder, A. The alternating fixpoint of logic programs with negation. In Proc. 8th ACM Symp. on Principles of Database Systems (Philadelphia, 1989), ACM, pp. 1--10.

....ed interpretation of a theory T i it is a justi ed interpretation of all its de nitions and a (3 valued) model of the classical logic sentences of T . An interpretation I is a model of a de nition D, resp. theory T , i it is a total (i.e. 2 valued) justi ed interpretation of D, resp. T . 7 In [16], an alternative way of de ning the well founded semantics of predicate rules is proposed; it is based on a di erent treatment of positive and negative occurrences of predicates in the body of rules. I believe both techniques are equivalent but haven t proven this. 7 The above model theory is ....

A. Van Gelder. The Alternating Fixpoint of Logic Programs with Negation. Journal of computer and system sciences, 47:185-221, 1993.

....in the domain, the empty argument. Definition 2.5 Let Pi = hR; OEi be any priority program. A fixpoint I of R 2 is called an alternating argument of Pi. It is said to be normal (also called self extensible) if I R(I) 2 8 The notion of an alternating fixpoint was introduced by Van Gelder [48] and later studied by a number of authors (e.g. 4, 55] Monotonic augmentation of an argument provides an important intuition of argumentation, especially for skeptical reasoning; one should be able to form a unique, conservative skeptical argument to support one s assertion incrementally, ....

A. Van Gelder. The alternating fixpoint of logic programs with negation. JCSS, pages 185--221, 1993.

....space of valuations the structure of a complete semi lattice, monotone mappings have least fixed points; the least fixed point of # # P is the well founded model for P; 3. # # P is anti monotonic with respect to # t (this leads to the alternating fixpoint approach to the well founded model [15], something we do not need here) 5 Results, Proofs, and Constructions In this section we prove the existence of a largest intrinsic stable model, give a sort of construction for it, and prove some related results of interest. As a matter of fact, essentially nothing in this section depends on ....

Van Gelder, A. The alternating fixpoint of logic programs with negation. In Proc. 8th ACM Symp. on Principles of Database Systems (Philadelphia, 1989), ACM, pp. 1--10.

....Fit85, Fit86] Kunen [Kun87, Kun89] and others [LM85, Myc83] used Kleene s strong three valued logic while Van Gelder, Ross and Schlifp [VGRS91] used a different three valued logic to give the well founded semantics of a logic program. Przymusinski [Prz89a, Prz89d] Dung [Dun93a] Van Gelder [VG89], and many others gave alternative formalizations of the well founded semantics. The semantics of Fitting and Jacob differ from the well founded semantics. For the program consisting of the rule p p, Fitting and Jacob assign the truth value unknown to p while the well founded semantics (also ....

....in classical two valued or three valued theories. Others moved closer to non traditional non monotonic logics. To give the reader a flavor of these developments, we introduce the well founded semantics, and compare it with the stable models semantics. We will follow the ideas from [BS91] and [VG89]. Definition 2.3 [BS91] For any general logic program Pi and a set of atoms S, consider F Pi (S) a( Pi S ) where a and Pi S are as in Definition 2.1 of stable models. C, a set of 16 interpretations 8 is said to be a stable class of Pi iff C = fF Pi (S) S 2 Cg. A stable class is ....

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A. Van Gelder. The alternating fixpoint of logic programs with negation. In Proc. of PODS-89, pages 1--10, 1989.

....also a dual largest stable model, S k P , which has not been considered before. There is another ordering based on degree of truth. Taking the meet and the join, in the truth ordering, of the two extreme stable models s k P and S k P just mentioned, yields the alternating fixed points of [29], denoted s t P and S t P here. From s t P and S t P in turn, s k P and S k P can be produced again, using the meet and join of the knowledge ordering. All stable models are bounded by these four valuations. Further, the methods of proof apply not just to logic programs ....

....to the setting of Belnap s logic. Also the family of stable models is bounded from above and below in the truth ordering by two valuations that are not necessarily stable models themselves, but which are extremal in a certain sense. This, essentially, is the alternating fixpoint construction of [29]. Finally, we will show that the smallest and biggest stable models in the knowledge ordering, and the extremal valuations in the truth ordering, are closely connected, and can be calculated from each other. This is new. The results just sketched actually hold in a much broader setting than was ....

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Van Gelder, A. The alternating fixpoint of logic programs with negation. In Proc. 8th ACM Symp. on Principles of Database Systems (Philadelphia, 1989), ACM, pp. 1--10.

....body of each rule is a conjunction of literals, its well founded semantics is characterized by a unique three valued model, called the well founded partial model. It is well defined for all normal logic programs. The well founded semantics of normal logic programs has been extended by Van Gelder [40] to general logic programs, where the body of each rule may be an arbitrary first order formula. The resulting semantics is called the alternating fixpoint logic [40] The well founded semantics captures the intuitive meaning of the program in Example 1.1, including that a is neither a winning ....

....is well defined for all normal logic programs. The well founded semantics of normal logic programs has been extended by Van Gelder [40] to general logic programs, where the body of each rule may be an arbitrary first order formula. The resulting semantics is called the alternating fixpoint logic [40]. The well founded semantics captures the intuitive meaning of the program in Example 1.1, including that a is neither a winning nor a losing position. However, it is inadequate in dealing with reasoning by cases or multiple alternative situations. For the program in Example 1.2, every ground ....

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A. Van Gelder. The alternating fixpoint of logic programs with negation. Journal of Computer and System Sciences, 47:185--221, 1993.

....Issue: How to handle unknown undefined literals ffl One issue involves dynamically changing the computation rule ffl A second issue involves representing atoms that are neither true nor false. ffl XSB implements delay and simplification [95] ffl WFOS [101] uses the Alternating Fixpoint of [113] 81 Implementation of Tabling: Optimizations ffl Tabling is weak for acyclic right recursive queries Left: ancestor(X,Y) parent(X,Y) ancestor(X,Y) ancestor(X,Z) parent(Z,Y) Right: ancestor(X,Y) parent(X,Y) ancestor(X,Y) parent(X,Z) ancestor(Z,Y) What if parent is a chain of ....

A. van Gelder. The alternating fixpoint of logic programs with negation. In Proc. of 8th PODS, pages 1--10. ACM, 1989.

....multiplicity of stable models leads to several useful forms of nonmonotonic reasoning, such as abduction and skeptical reasoning. Syntactically, several extensions have been proposed to enhance the expressive power of specifications of nonmonotonic reasoning. They include general logic programs [24] that allow arbitrary first order formulas in rule bodies, abductive logic programs with integrity constraints [9] Datalog programs with choice [22] extended logic programs with classical negation and disjunction [12] and epistemic specifications [10] Pragmatically, different operational ....

....head atoms, namely r(f(X,b) and r(f(a,Y) They are treated separately by the current implementation. 2 3. 4 Handling First Order Formulas General logic programs allow arbitrary first order formulas in rule bodies, whose semantics is characterized by Van Gelder s alternating fixpoint logic [24]. It has been shown [4] that a straightforward translation of general logic programs into normal logic programs does not preserve the semantics of alternating fixpoint logic. However, every general logic program can be translated into an equivalent one that contains two kinds of rules, normal ....

A. Van Gelder. The alternating fixpoint of logic programs with negation. Journal of Computer and System Sciences, 47:185--221, 1993.

....by a quantifier free first order formula. This yields a nice new normal form for well founded Datalog and implies that it is sufficient to consider draw free games in order to evaluate arbitrary Datalog programs under the well founded semantics. 1 Introduction The well founded semantics (WFS) [VGRS88, VG93] has become popular as an intuitive and well behaved 1 semantics for the language of logic programs containing negative cyclic dependencies, like the famous program P game : win(X) move(X; Y ) win(Y ) A position X in a game is won, if there is a move to some position Y which is not won ....

....definable in a language. The query associated with a logic program P and database D is defined in terms of the true atoms of WFS(P;D) hence undefined and false atoms belong to the complement of the query) Van Gelder showed that Datalog evaluated under WFS is equivalent to (least) fixpoint logic [VG89, VG93]. A natural question arising is: What is the expressive power of programs which never yield the truth value undefined, i.e. which are total for all databases D For logic programs over infinite Herbrand structures, the restriction to total programs results in loss of expressive power from Pi 1 ....

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A. Van Gelder. The Alternating Fixpoint of Logic Programs with Negation. Journal of Computer and System Sciences, 47(1):185--221, 1993.

....definable in a language. The query associated with a logic program P and database D is defined in terms of the true atoms of WFS(P;D) hence undefined and false atoms belong to the complement of the query) Van Gelder showed that Datalog evaluated under WFS is equivalent to (least) fixpoint logic [VG89, VG93]. A natural question arising is: What is the expressive power of programs which never yield the truth value undefined, i.e. which are total for all databases D For logic programs over infinite Herbrand structures, the restriction to total programs results in loss of expressive power from Pi 1 ....

....the case, since S Datalog (stratified Datalog) is equivalent to LFP (and thus captures PTIME) on ordered databases, and WFS is 2 valued for S Datalog (see e.g. AHV95] As we will show, the question can also be answered affirmatively in the absence of order. First, using results of van Gelder [VG89] and Grohe [Gro94] we show that every W Datalog program can be transformed into a normal form which corresponds to a certain game. The main result is that one can reduce such games to draw free games, which is equivalent to the fact that the corresponding W Datalog program is total. 3.1 Games A ....

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A. Van Gelder. The Alternating Fixpoint of Logic Programs with Negation. In Proc. ACM Symposium on Principles of Database Systems, pages 1--10, 1989.

.... Obj:Class1 Class1: Class Class1[A W] Class1 = Class definedSet(Obj,A) Obj[A V] conflictSet(Obj,Class,A) definedSet(Super,A) Obj:Super not Super: Class not Class: Super. Negation here is implemented using the well founded semantics for negation [5, 6] (as indicated by the tnot operator) One problem with the current implementation of behavioral inheritance is that the well founded semantics for negation in the presence of equality is not yet sufficiently developed. Since Flora s 5 SYNTAX OF FLORA 18 treatment of inheritance relies on ....

A. Van Gelder. The alternating fixpoint of logic programs with negation. In ACM-SIGACTSIGMOD Conference on Principles of Database Systems, pages 1--10, New York, 1989. ACM.

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