J. C. Shepherdson, Negation in logic programming, in Foundations of Deductive Databases and Logic Programming, J. Minker editor, Morgan Kaufmann, Los Altos, CA (1988), pp. 19-88.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....power of the logical language that subsumes the theory of Logic Programming has long been an active area of research. Examples of such approaches include consideration of negative literals in the body of clauses, interpreted either as negation as failure or as true negation (for surveys see [3, 12]) and answering queries more complex than existentially quantified conjunctions of atoms [7] All these investigations, however, have been in the realm of classical deductive logic. Since the class of Prolog programs determines an acceptable indexing of the class of all partial recursive ....

Shepherdson, J.: Negation in Logic Programming. In Minker, J., ed., Foundations of Deductive databases and Logic Programming. Morgan Kaufmann. (1988) 19--88

....demonstrated by Prolog, the first and yet the most popular logic programming language which implements SLDNF resolution. However, SLDNF resolution suffers from two serious problems. One is that the declarative semantics it relies on, i.e. the completion of programs [8] incurs some anomalies (see [15, 29] for a detailed discussion) and the other is that it may generate infinite loops and a large amount of redundant sub derivations [2, 9, 35] The first problem with SLDNF resolution has been perfectly settled by the discovery of the well founded semantics [33] Two representative methods were ....

J. C. Shepherdson, Negation in logic programming, in: (J. Minker, ed.) Foundations of Deductive Databases and Logic Programming, Morgan Kaufmann, 1988, pp. 19-88.

....is defined in terms of ESLDNF computations only. 1 Introduction Let P be a general logic program and F be the set of goals which have a finitely failed SLDNF tree. We will use the following two definitions. The first definition uses three valued models, the second not. Definition 1 (Shepherdson, [6], p. 58) A program P is negation complete i# for every goal L 1 , L n if comp(P ) 3 #(L 1 # . # L n ) then (L 1 , L n ) # F. Of course, the original definition due to Shepherdson is for classical logic and not for three valued logic. Definition 2. A program ....

J. C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 19--88. Morgan Kaufmann, Los Altos, 1987.

....succeeds, and false if it fails. Chapter 4 Cut elimination and the completeness of ESLDNF resolution The main result of this chapter will be that a program P is negation complete if, and only if, it has the cut property. For this we need the following two definitions. Definition 58 (Shepherdson, [28], p. 58) A program P is negation complete if for every goal L 1 , L n if comp(P ) 3 #(L 1 # . # L n ) then L 1 , L n fails in ESLDNF resolution. The original definition due to Shepherdson is for classical logic and SLDNF resolution and not for three valued ....

....By the previous theorem it follows that the goal L 1 , L n fails in ESLDNF resolution. # Until now we have only considered negative goals. In order to bring the results on negative goals in connection to positive goals we need the following two definitions. Definition 73 (Shepherdson, [28], p. 58) A program P is answer complete for the goal L 1 , L n if for every substitution # if comp(P ) 3 #(L 1 # # . # L n #) then there exists an ESLDNF computed answer # of L 1 , L n and a substitution # such that (L 1 # . # L n )## = L 1 # . ....

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J. C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 19--88. Morgan Kaufmann, Los Altos, 1987.

....the notion A(F, #) v is defined. The quantifiers are interpreted as infinite disjunctions or conjunctions. For example A(#XF,#) t i# there is an element a # A such that A(F, #[X a] t and A(#XF,#) f i# for all a # A A(F, #[X a] f . A survey of this logic can be found in [12] and a survey of 3 valued logic can be found in [6] What is the 3 valued interpretation of a sequent # # # of S Intuitively the sequent # # # is true i#, if all F in # are true then there is a G in # which is true and if all G in # are false then there is a F in # which is false. Let ....

J. C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 19--88. Morgan Kaufmann, Los Altos, 1987.

....This result is interesting, since the first property, negation complete, is defined semantically in terms of threevalued models, and the second property, the cut property, is defined in terms of SLDNF resolution only. We need the following two definitions. Definition 7.1. Shepherdson, [28], p. 58) A program P is negationcomplete i#, for every goal, L 1 , L n if comp(P ) 3 #(L 1 # . # L n ) then (L 1 , L n ) # F. Definition 7.2. A program P has the cut property i#, for every goal #, for every clause A : L 1 , L n of P and every ....

....by Lemma 10.3 (4) we obtain that (L 1 , L n ) R yes . # This theorem says nothing on computed answers. In order to handle computed answers we need the following two definitions which are the analoga to the definition of negation complete and cut property. Definition 7.17. Shepherdson, [28], p. 58) A program P is answercomplete for the goal L 1 , L n i#, for every substitution #, if comp(P ) 3 #(L 1 # # . # L n #) then there exists a substitution # such that (L 1 , L n ) R # and (L 1 , L n )# # (L 1 , L n )#. Definition 7.18. A ....

J. C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations 36 of Deductive Databases and Logic Programming, pages 19--88. Morgan Kaufmann, Los Altos, 1987.

....relies on the explicit construction of answers to a goal which may possibly involve negation. This is commonly achieved by explicitly computing the negation of the solutions to the corresponding positive goal. This scheme allows one to overcome the restrictions imposed by Negation as Failure (NAF) [14, 1]; in particular, negative literals in a goal are not restricted to be ground, and answers are not limited to true false. The term Constructive Negation was rst introduced in [3] as a technique for extending the NAF rule in the context of Logic Programming. The semantic properties of CN in logic ....

....constraints are equations between terms. 1 A sample CLP program is p(X; Z) X = f(a; Y) X = f(b; Y) q(X; Y) X = c Y = c q(X; Y) X 6= c p(X; Y) A sample goal is :p(X 1 ; X 2 ) Given a CLP program P we use P to denote the theory obtained by the following transformation of P [14]: assume that p( X) A 1 ; p( X) An are all the clauses in P de ning the predicate symbol p, and let vars( A i ) n f Xg = f Y i g, then such clauses are rewritten as: 8 X p( X) W n i=1 9 Y i A i . The formula 8 X:p( X) is introduced in ....

J. C. Shepherdson. Negation in Logic Programming. In J. Minker, ed., Foundations of Deductive Databases and Logic Programming, pages 19-88. Morgan Kaufmann, 1988.

....atoms Q such that P ; EFT j= Q. The ground finite failure set GFF (P ) is the complement (in the set of all ground atoms) of gm(P ; FT ) A program P is canonical over A [2] if, for every ground atom Q, if P ; EFT 6j= Q then gm(P ; A) j= Q. The model gm(P ; A) is said to be n generic [25]. We say P is canonical if P is canonical over FT . In this case FF (P ) GFF (P ) Canonical programs were introduced in [12] where it was shown that every program is equivalent (with regard to successful and finitely failed executions) to a canonical program. See also the work of Wallace ....

....of AP;Q in F . Thus every program is canonical over F , and gm(P ; F) is n generic, for each program P . If we take the view that the language of function symbols is determined by the program P then it is only possible to discuss canonicality one program at a time. In this context, Shepherdson [25] gave a non constructive argument that every program has a n generic model. Blair and Brown [2] gave a complex construction of an algebra over which P is canonical. In [13] Kunen uses an ultrapower construction to build a three valued model of P ; EFT . Although the original aim of this ....

J.C. Shepherdson, Negation in Logic Programming, in: Foundations of Deductive Databases and Logic Programming, J. Minker (Ed), Morgan Kaufmann, 19--88, 1988.

....the function f and the query language L. In some cases the question may be effectively undecidable. Note that in logic programming, queries involving negation and a function symbol (as is the case with our query languages) posed in the presence of a closed world assumption are also undecidable [40]. The issue of how much (and what type of) computational resources are needed to answer the questions posed by a QIM are not investigated in this work. Formally, a QIM is a total algorithmic device which, if the input is a string of bits b, corresponding to the answers to previous queries, ....

SHEPHERDSON, J. Negation in logic programming. In Foundations of Deductive Databases and Logic Programming, J. Minker, Ed., Morgan Kaufmann Piblishers, Los Altos, CA, 1988.

....data. Since such data is too voluminous to be represented in the database explicity, it must be inferred via some metarule. In the case of deductive databases that are restricted to Horn rules (in which k = 1) Reiter s closed world assumption (CWA) Re78] known as negation as failure in PROLOG [Sh88]) allows the inference of a negative atom if the corresponding positive atom cannot be inferred from the database. Alternatively the CWA can be rephrased in model theoretic terms: Horn databases are well known to have a unique minimal model, and the CWA thus reduces to truth evaluation with ....

J. C. Shepherdson, Negation in logic programming, in J. Minker, ed., Foundations of Deductive Databases, pp 19-88, (Morgan Kaufmann, Washington, 1988).

....13 By virtue of the above equivalence result, the three valued xpoint semantics of general program expressions can be related to the operational semantics for general programs proposed in the literature. SLDNF resolution has been proved to be sound w.r.t. three valued semantics (e.g. see [12, 28]) By virtue of Lemma 4, the soundness result for general logic programs can be formulated as follows. Proposition 9 Let P be a general program and G a query. Then: 1. If # is a computed answer substitution (c.a.s. for G in P then there exists a nite n s.t. P ) n j= 3 8G#. 2. If there ....

J.C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 19-88. Morgan Kaufmann, Los Alto, CA, 1988.

....by a fixpoint construction to the system of autoepistemic logic introduced by Moore [1985] The monographs [ Lloyd, 1987 ] and [ Lobo et al. 1992 ] use the approaches to the semantics of logic programming different from ours. A survey of early work on negation as failure can be found in [ Shepherdson, 1988 ] The 1994 special issue of the Journal of Logic Programming and its continuation, celebrating the tenth anniversary of the journal, contain, among others, three surveys closely related to this one by Apt and Bol [1994] by Baral and Gelfond [1994] and by Ramakrishnan and Ullman [1995] They ....

John Shepherdson. Negation in logic programming. In Jack Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 19--88. Morgan Kaufmann, San Mateo, CA, 1988.

....initial model, 7 which for the sake of competeness we define here as follows: An initial model J of F is a reachable model such that for any relation r defined in F , ground instances r(t) are true in J iff they are true in all models of F . 6 Thus an isoinitial model is a generic model (see [13]) 7 This holds only for reachable models. For non reachable models initiality and isoinitiality are independent properties (see [1] 7 In general, the existence of an isoinitial model is of course not guaranteed. However, if a framework F has a reachable model I, then it can be shown that I ....

....an obvious isoinitial model I 0 , and then successively expand F i into F i 1 by adding new function and relation symbols, together with their axioms, in such a way that I i can be expanded into an isoinitial model I i 1 of F i 1 . Let us consider the simple case, where the freeness axioms (see [13]) hold in the data type we want to axiomatise. We use as F 0 the framework consisting of just the type constructor symbols in Fn together with their freeness axioms. The term model generated by the type constructors is of course just the set of ground terms of T . It is an isoinitial model of ....

J.C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming , pages 19--88. Morgan Kaufmann, 1988.

....all ground atoms free for negation in T . Then GCWA(T ) T : 40 Minker [Min82] proves that for T with finite number of constants and no function symbols, T [ GCWA(T ) classically entails a literal q iff q is true in all minimal Herbrand models of T . This result was extended to arbitrary T in [GP86, She88]. The following proposition establishes the connection between Minker s semantics and answer set semantics of disjunctive logic programs. Proposition 4.4 [Gel92b] Let Pi be a program consisting of rules of the form A 0 or : or A k A k 1 : Am (where A s are atoms) and the closed ....

J. Shepherdson. Negation in Logic Programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 19--88. Morgan Kaufman Pub., 1988.

....proof of such kind of properties. Moreover, ranked structures can be seen as a special case of (a three valued version of) Beth structures, used to provide semantics to intuitionistic logic (e.g. see [33] In this sense, our semantics suggests a link, already mentioned by other authors (e.g. see [31]) between logic programming negation and intuitionistic logic that may be worthwhile to study. In particular, it could serve as a basis for extending with negation those approaches to modularity based on the use of an intuitionistic implication (e.g. see [28] In this line, the only work we know ....

Shepherdson, J. C., Negation in Logic Programming, in: Minker, J. (ed), Foundations of Deductive Databases and Logic Programs, Morgan Kaufmann, Los Altos, CA, 1988.

.... We have also developed a fixpoint theoretic and proof theoretic semantics of SchemaLog (for definite clause programs) In fact, the framework can be easily extended to incorporate the various forms of negation extensively studied in the literature of deductive databases and logic programming (see [She88] for a survey) notably stratified negation, with minor modifications. Even though SchemaLog is quite simple, our study (and our experience) indicates that it has a rich expressive power making it applicable to a variety of problems including database programming (with schema browsing) schema ....

Shepherdson, J.C. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming. Morgan-Koffmann, 1988.

....individual constant, is c (x) means x=c) Notice that CET L can be defined only for languages containing finitely many function symbols, otherwise the conjunction V f2Func :is f (x i ) would not be finite. This is on of the differences between our approach and that of Kunen [11] also cf. [19]) Notice that CET L does not have any finite models. We have U L j= CET L and this shows that CET L is a correct description of U L . CET L is also a more accurate description of U L than CETL for instance CETL does not exclude the conventional Herbrand universe ....

J. C. Shepherdson, Negation in Logic Programming, in: J. Minker (ed.), Foundations of Deductive Databases and Logic Programming, Morgan Kaufmann, 1988, pp.19-88.

....as lfp(T P ) The theorem now follows from Lemma 5.1. The details are identical to those for classical logic programming ( 56] 2 4 X is directed if every finite subset of X has an upper bound in X. 17 Incorporating any of the various forms of negation studied in logic programming (e.g. see [47]) in SchemaLog is not very difficult. We do not discuss this issue further in this paper. 5.2. Proof Theory of SchemaLog In this section, we develop a sound and complete proof theory for SchemaLog as a full fledged logic. We consider arbitrary SchemaLog theories, not just definite clauses. ....

....than the extensions they stand for. We have also developed a fixpoint theoretic and proof theoretic semantics of SchemaLog. In fact, the framework can be extended to incorporate the various forms of negation extensively studied in the literature of deductive databases and logic programming (see [47] for a survey) notably stratified negation, without much difficulty. We have studied an extension of classical relational algebra that is capable of manipulating both schema and data of component databases in a federation, and established its equivalence to a form of relational calculus inspired ....

Shepherdson, J.C. Negation in Logic Programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming. Morgan Kaufmann, 1988.

....set x one can choose a member z which belongs to x and does not intersect x. This is a well known expedient to state that membership does not form cycles. The theory Set is completed by the standard equality axioms (not reported here; cf. e.g. 23] and by the following freeness axioms (cf. [46]) 8x 1 Delta Delta Delta 8xn 8 y 1 Delta Delta Delta 8 yn (f(x 1 ; xn ) 6= g(y 1 ; ym ) 8x 1 Delta Delta Delta 8xn 8 y 1 Delta Delta Delta 8 yn (f(x 1 ; xn ) f(y 1 ; yn ) x 1 = y 1 Delta Delta Delta xn = yn ) 8x ( t[x] 6= x ) ....

....of partial p shown above. Thus, what we really need is just some form of negation in clause bodies. The use of different forms of negation in flogg has been investigated in previous works. In particular: flogg: A LOGIC LANGUAGE WITH SETS 31 Negation As Failure: the use of negation as failure [12, 14, 46] in flogg has been considered in [16, 19] The extension of SLDNF resolution to include the new features of our language is quite straightforward and appears to be easily implementable. On the other hand the syntactic restrictions imposed by negation as failure on the source program, in order to ....

Shepherdson, J. C., Negation in Logic Programming. In J. Minker (ed.), Foundations of deductive databases and Logic Programming . Morgan Kaufmann, Los Altos, CA, 1987.

....intuitive meaning to negation incorporating some form of default reasoning. The first approach, due to Clark [6] was to define the completion of a program. The semantics of the program is then given by the logical consequences of the completion. For a detailed description of this approach see [27, 28, 13]. An alternative approach was taken by Fitting [7] and Kunen [11] who interpreted the completion in terms of 3 valued logic in order to overcome some anomalies with the completion when interpreted in a 2 valued sense. Based on the completion, Clark proposed a top down procedural semantics known ....

J. C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 19--88, Los Altos, CA, 1988. Morgan Kaufmann.

....symbols by total computable functions. Indeed, we only ever construct such frameworks. To construct such a framework F n for axiomatizing a type T , we proceed incrementally. We start with a small framework F 0 with an obvious isoinitial model I 0 . For instance, if the freeness axioms (see [16]) hold in T , then F 0 consists of just the constructor symbols in F n together with their freeness axioms. The term model generated by the constructors is of course just the set of ground terms of T . It is an isoinitial model of their freeness axioms, i.e. it is an isoinitial model of F 0 . Then ....

J.C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming , pages 19--88. Morgan Kaufmann, 1988.

....symbols by total computable functions. Indeed, we only ever construct such frameworks. To construct such a framework F n for axiomatising a type T , we proceed incrementally. We start with a small framework F 0 with an obvious isoinitial model I 0 . For instance, if the freeness axioms (see [15]) hold in T , then F 0 consists of just the constructor symbols in F n together with their freeness axioms. The term model generated by the constructors is of course just the set of ground terms of T . It is an isoinitial model of their freeness axioms, i.e. it is an isoinitial model of F 0 . ....

....expanded into an isoinitial model I i 1 of F i 1 . For example, the following framework NAT axiomatises the abstract data type of Peano arithmetic: Framework NAT ; sorts: Nat ; functions: 0 : Nat ; s : Nat Nat ; Nat ; Nat) Nat ; 9 Thus an isoinitial model is a generic model (see [15]) 10 This holds only for reachable models. For non reachable models initiality and isoinitiality are independent properties (see [1] axioms: 8x ( 0 = s(x) 8x; y (s(x) s(y) x = y) 8x (x 0 = x) 8x; y (x s(y) s(x y) 8x (x 0 = 0) 8x; y (x s(y) x x y) 8(H(0) 8i ....

J.C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming , pages 19-- 88. Morgan Kaufmann, 1988.

....of all ground atoms free for negation in T . Then GCWA(T ) T : Minker [Min82] proves that for T with finite number of constants and no function symbols, T [ GCWA(T ) classically entails a literal q iff q is true in all minimal Herbrand models of T . This result was extended to arbitrary T in [GP86, She88]. The following proposition establishes the connection between Minker s semantics and answer set semantics of disjunctive logic programs. Proposition 4.4 [Gel92b] Let 5 be a program consisting of rules of the form A 0 or : or A k A k 1 : Am (where A s are atoms) and the closed world ....

J. Shepherdson. Negation in Logic Programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 19--88. Morgan Kaufman Pub., 1988.

....logic program has been an attractive alternative to the explicit representation approach. For Horn programs, Reiter (1978) introduced a rule of negation called the Closed World Assumption (CWA) There is, however, a difficulty associated with the CWA. It is, in general, as shown by Apt (1989) and Shepherdson (1988) , computationally intractable. Hence, a weaker definition, called Negation As Failure (NAF) rule defined by Clark (1978) is used in practice. SLDNF resolution (1978) incorporates a procedural interpretation of the NAF rule. In order to capture more situations with logic programming, there has ....

Shepherdson J.C. (1988). Negation in Logic Programming. In Minker J., editor, Foundations of Deductive Databases and Logic Programming, pages 19--88. Morgan Kaufman Pub.

....to T Sigma . Normally, Sigma will be understood and we will omit it. A closed framework F is a (first order) theory with the following key properties: ffl Freeness . A subset of the constant and function symbols of F , which we shall call the construction symbols , satisfy the freeness axioms [14]. ffl Reachability . There is at least one model of F reachable by the construction symbols. This means that every element of the domain of the model is represented by a ground term using the construction symbols. We will call such a term a construction. ffl Atomic completeness . For every ....

J.C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming , pages 19-88. Morgan Kaufmann, 1988.

....functions. Indeed, we only ever use such frameworks, with (computable) isoinitial models. To construct such a framework F n for axiomatising a type T , we proceed incrementally. We start with a small framework F 0 with an obvious isoinitial model I 0 . For instance, if the freeness axioms ([14]) hold in T , then we use as F 0 the framework consisting of just the constructor symbols in F n together with their freeness axioms. The term model generated by the constructors is of course just the set of ground terms of T . It is an isoinitial model of their freeness axioms, i.e. it is an ....

J.C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming , pages 1988. Morgan Kaufmann, 1988.

....ones. In the third section we treat disjunctive logic programming and in the fourth section we give an overview of implemented systems and where they can be obtained. 3 Normal logic programs Several recent overviews of normal logic programming semantics can be found in the literature (e.g. [She88,She90,PP90,Mon92,AB94,Dix95c,BD96b]) Here, for the sake of this text s self sufficiency and to introduce some motivation, we distill a brief overview of the subject. In some parts we follow closely the overview of [PP90] The structure of this section is as follows: first we present the language of normal logic programs and give ....

....that this semantics is not recursively enumerable, and proposed a modification. Unfortunately, Fitting s semantics inherits several problems of Clark s completion, and in many cases leads to a semantics that appears to be too weak. This issue has been extensively discussed in the literature (see [She88,Prz89c,GRS91], Dix95c,BD96b] Forthwith we illustrate some of these problems with the help of examples: Example 10. Consider 5 program P : edge(a; b) edge(c; d) edge(d; c) reachable(a) reachable(X) reachable(Y ) edge(X; Y ) 4 Here we adopt the designation of negation by default . Recently, this ....

John C. Shepherdson. Negation in Logic Programming. In Jack Minker, editor, Foundations of Deductive Databases, chapter 1, pages 19--88. Morgan Kaufmann, 1988.

....function f; CET8. t[X] 6= X; for any term t[X] different from X, but containing X. The first five axioms describe the usual equality axioms and the remaining three axioms are called unique names axioms or freeness axioms. The significance of these axioms to logic programming is widely recognized [13, 23, 24, 14, 9]. Consequently, instead of talking about the theory P we will talk about the theory CET (P ) P CET . The equality axioms (CET1) CET5) ensure that we can always assume that the equality predicate = is interpreted as identity in all models of CET(P) Consequently, models of CET(P) can be ....

Shepherdson, J., "Negation in Logic Programming", in: Foundations of Deductive Databases and Logic Programming (ed. J.Minker), Morgan Kaufmann 1988, 19-88. paper.

....properties: Reachability . There is at least one model of F reachable by a subset of the constant and function symbols of LF , called the construction symbols . The ground terms containing only construction symbols will be called constructions . Freeness . F proves the freeness axioms [13] for the construction symbols. Atomic completeness . F is atomically complete. Thus, by Theorem 4, a closed framework is a theory admitting an isoinitial model. The existence of a set of construction symbols satisfying the freeness axioms is not strictly necessary (reachability sufficies) It ....

J.C. Shepherdson. Negation in Logic Programming. in J. Minker, editor, Foundations of Deductive Databases and Logic Programming , pages 19-88. Morgan Kaufmann, 1988.

....programs. This research has proceeded in two general directions, which may be summarized as the program completion approach and the canonical model approach. 1. 1 Program Completion Semantics The original program completion approach, due to Clark [6] and discussed in detail by Shepherdson [37, 38], Kunen [17] and Lloyd [20] has been to define a new program, called the completed program (sometimes called the completed database) The completed program is treated simply as a first order formula (see Section 4) Then the negative literals that are logical consequences of the completed ....

.... Lassez and Lloyd showed that SLDNF was complete (in the same sense) for Horn programs with non floundering queries consisting of a conjunction of positive and or negative literals [15] SLDNF was further investigated for general logic programs by Lloyd [20] who coined the term SLDNF) Shepherdson [37, 38] (q.v. for further bibliography) and others. This approach is logically impeccable, but does not address the issue of how the compiler or the interpreter of the general logic program should treat atoms (goals) whose positive and negative literals are neither logical consequences of the ....

J. C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 19--88. Morgan Kaufmann, Los Altos, CA, 1988.

....completion does not always result in a satisfactory semantics. For many programs, it leads to a semantics which appears too weak. This problem applies both to standard Clark s semantics as well as to its 3 valued extensions and it has been extensively discussed in the literature (see e.g. [She88, She84, Prz89c, VGRS90]) We illustrate it on the following three examples. Example 6.1 Suppose that to the program P 1 defined before we add a seemingly meaningless clause: natural number(X) natural number(X) It appears that the newly obtained program P 0 1 should have the same semantics. However, Clark s ....

J.C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 19--88. Morgan Kaufmann, Los Altos, CA., 1988.

....symbol. Definition 3. 19 The extended completion of P , Comp L (P ) is defined as IFF (P ) CETL , where IFF (P ) is the collection of completed definitions of predicates in P (see Apt, 1990) and CET L is the set of the equality and freeness axioms (EA[FA) for L, as defined, for instance, in Shepherdson (1988). Comp L (P ) is an extension of Comp(P ) as shown by the following proposition. Proposition 3.20 Let F be a (first order) formula. Then if Comp(P ) j= F then Comp L (P ) j= F: Proof L contains all symbols which occur in P , hence the axioms of Comp(P ) are a subset of the axioms of Comp L (P ....

....of the NAI rule we use some results from the literature on the threevalued logic and the Kunen Fitting immediate consequence operator Phi P . In the following, we indicate by j= 3 the truth in a three valued model. For the basic definitions of these notions see, fot instance, Fitting (1985) or Shepherdson (1988). These results are formulated within a framework which assumes an underlying language containing countably infinite sets of n adic function symbols and of n adic predicate symbols, for each n 0. Anyway, for our purposes, it is not necessary to consider such an extension, since the only ....

[Article contains additional citation context not shown here]

Shepherdson, J. C. (1988), Negation in logic programming, in "Foundations of Deductive Databases and Logic Programming" (J. Minker, Ed.), pp. 19--88, Morgan Kaufmann, Los Altos, CA.

....variables that are easily grounded, i.e. converted to propositional logic programs without variables. The reason for allowing the use of variables is to simplify the writing of logic programs. The reason for limiting this use is to avoid the floundering problems of ungrounded negative literals [She88]. If the use of variables is suitably limited it is also possible to avoid the generation of grounded rules that do not influence the set of stable models of the program. In Section 8.1 we describe logic programs with variables and how they are grounded. In Section 8.2 we describe the class of ....

J.C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 19--88. Morgan Kaufmann Publishers, Los Altos, 1988. -- 66 --

....by total computable functions. Indeed, we only ever construct such frameworks. To construct such a framework Fn for axiomatising a type T , we proceed incrementally. We start with a small framework F 0 with an obvious isoinitial model I 0 . 6 Thus an isoinitial model is a generic model (see [16]) 7 This holds only for reachable models. For non reachable models initiality and isoinitiality are independent properties (see [1] 7 For instance, if the freeness axioms (see [16] hold in T , then F 0 consists of just the constructor symbols in Fn together with their freeness axioms. The ....

....small framework F 0 with an obvious isoinitial model I 0 . 6 Thus an isoinitial model is a generic model (see [16] 7 This holds only for reachable models. For non reachable models initiality and isoinitiality are independent properties (see [1] 7 For instance, if the freeness axioms (see [16]) hold in T , then F 0 consists of just the constructor symbols in Fn together with their freeness axioms. The term model generated by the constructors is of course just the set of ground terms of T . It is an isoinitial model of their freeness axioms, i.e. it is an isoinitial model of F 0 . Then ....

J.C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming , pages 19--88. Morgan Kaufmann, 1988.

....there appears to be no problem with allowing atomic negative queries. It is not clear how to deal with more complex negative queries, however, since the GCWA, which we model in our queryanswering, is not complete with respect to its minimal model semantics for more complex negative queries [ Shepherdson, 1988 ] Non Uniform Clauses: We have shown how to handle shared predicate and shared argument uniform clauses in HKBs. There is an obvious question of whether these techniques can be combined to represent clauses of the form Pa Qb. While we see no reason why this should not be possible, we have not ....

Shepherdson, J. C., "Negation in Logic Programming", in J. Minker (ed), Foundations of Deductive Databases and Logic Programming, Morgan Kaufmann, Los Altos, CA, 19--88.

....to be fixed and to contain all the functions symbols occurring in all the modules we consider. A known problem that semantics based on program completion face is that when LB is finite (that is, when it contains only a finite number of functions symbols) CETLB is not a complete theory (see [She88] Typically, this problem is solved by adopting one of the following solutions: a) adding to CETLB some domain closure axioms which are intended to restrict the interpretation of the quantification to LB terms (as in [She88] or (b) assuming that the language contains always an infinite set ....

.... number of functions symbols) CETLB is not a complete theory (see [She88] Typically, this problem is solved by adopting one of the following solutions: a) adding to CETLB some domain closure axioms which are intended to restrict the interpretation of the quantification to LB terms (as in [She88] or (b) assuming that the language contains always an infinite set of function symbols (as in [Kun87] or (c) by considering only interpretations and models over a specific fixed domain D (as in [Fit85] This latter solution requires the adoption of axioms which are usually not first order ....

[Article contains additional citation context not shown here]

J. C. Shepherdson. Negation in logic programming. In editor J. Minker, editor, Foundation of Deductive Databases and Logic Programming, pages 19-- 88. Morgan Kaufmann, 1988.

....positive clause. Then, 1) C is in MS P iff CIRC(P; C 2) A 2 GCWA(P ) iff CIRC(P; A 14 Proof: Part 1 is true since for disjunctive logic programs, MS P and CIRC(P; capture the minimal model semantics and hence are equivalent. For proving Part 2 we need Theorem 32 from Shepherdson [41] which shows that every minimal model of P is a minimal model of P [ GCWA(P ) and vice versa. Hence, A can be inferred from CIRC(P; if and only if A is not in any minimal model of P if and only if :A is in GCWA(P ) 2 For funtion free theories, the above result reduces to Proposition 2.13. ....

J.C. Shepherdson. Negation in Logic Programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 19--88. Morgan Kaufman Pub., 1988.

....goes through systems equivalent in CL, i.e. such that hAx (P 1 ) D;CLi i hAx(P 2 ) D;CLi where D is a set of definition axioms, linking the predicate symbols of the two programs. 3 Note that if S is finite, then there exists a logic program P whose completion is equivalent to S (see, e.g. [9, 11]) and this difference disappears. For example, from the program P 1 (to compute Fibonacci numbers) f(0; s(0) f(s(0) s(0) f(s(s(I) Z) f(I ; Z 1 ) f(s(I) Z 2 ) Z 1 ; Z 2 ; Z) using the definition axiom D : 8x; y; z: ff(x; y; z) f(x; y) f(s(x) z) we can get the program P ....

J.C. Shepherdson. Negation in Logic Programming. in J. Minker, editor, Foundations of Deductive Databases and Logic Programming , pages 19-88. Morgan Kaufmann, 1988.

....operator often referred to as negation as failure. As a result, the problem of finding a suitable declarative or intended semantics for logic programs is one of the most important and difficult problems in logic programming and is frequently discussed in the literature (see e.g. S] S2] [S3], JLM] C] R] R2] LM] K] ABW] VG] L] N] BH] P] GPP] GMN] E] Mi] F] G] Due to the non monotonic character of the negation operator, this problem, however, can be viewed as the problem of finding a suitable formalization of the type of non monotonic reasoning used ....

....the realm of logic programming and therefore does not play a major role in formalizing general non monotonic reasoning in AI. In addition, Clark s semantics is considered by many researchers to be too weak and has some other important drawbacks often discussed in the literature (see e.g. S] S2] [S3]) In this paper we describe a semantics based on the class PERF(P) of all perfect not necessarily Herbrand models of a program P. This approach leads to a natural intended semantics of logic programs, which is shown to combine many desirable features of previously defined semantics, at the ....

[Article contains additional citation context not shown here]

Shepherdson, J., "Negation in Logic Programming", in: Foundations of Deductive Databases and Logic Programming, (ed. J.Minker), Morgan Kaufmann 1987, 19-88.

....and therefore does not play a major role in formalizing general non monotonic reasoning in AI. More importantly, Clark s semantics is considered by many researchers to be too weak and to have various unintuitive features and drawbacks, frequently discussed in recent literature (see e.g. [She87, She84, Prz88b, VGRS88]) The situation has changed recently, following the introduction by Apt, Blair and Walker [ABW87] and by Van Gelder [VG87] see also [CH85, Naq86] of the class of stratified logic programs, later extended by Przymusinski [Prz87, Prz88b] to the class of (locally) stratified disjunctive databases, ....

....of positive programs is essentially different from the least model semantics. ffl The perfect model semantics is more powerful than Clark s semantics, at the same time eliminating various unintuitive features of the latter, which have been extensively discussed in recent literature (see e.g. [She87, She84, Prz88b, VGRS88]) In particular, the perfect model semantics is sufficiently expressive to naturally represent transitive closures, while Clark s semantics lacks this capability [Kun88] ffl Perfect model semantics admits a natural sound and complete procedural mechanism, called SLS resolution (SL resolution ....

J.C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 19--88, Morgan Kaufmann, Los Altos, CA., 1987.

.... a few but simple initial definitions rather than keeping up with front line research on negation in logic programming: in particular we shall only marginally deal with new developments in the field about constructive negation, completion, answers sets and completeness of NF (see for example [44, 6] and the recent survey by Apt Bol [3] Similarly, our review of related work is meant to help the reader to situate and compare our approach to the aforementioned issues, rather than contend in details about its edge over those others proposals. This paper, therefore, has in part a pedagogical ....

J.C. Shepherdson. Negation in Logic Programming. In Foundations of Deductive Databases and Logic Programming, J. Minker, ed., pp. 19--88, Morgan Kaufmann, 1988.

....contains only a finite number of functions symbols) CETL is not a complete theory. Typically, this problem is solved by adopting one of the following solutions: a) adding to CETL some domain closure axioms which are intended to restrict the interpretation of the quantification to L terms (as in [28]) or (b) assuming that the language contains always an infinite set of predicate symbol (as in [17] or (c) by considering only interpretations and models over a specific fixed domain D (as in [10] This latter solution requires the adoption of axioms which are usually not first order (unless ....

....the functions symbols are 0 ary, i.e. constants) and consequently leads to a semantics which is (usually) noncomputable. For these reasons we adopt either solutions (a) or (b) Luckily, these two solutions yield basically the same semantics. For an extended discussion of the subject, we refer to [17, 28]. Let L be a finite language (i.e. a language with a finite set of predicate symbols) The Domain Closure Axiom for the language L, DCAL , is 9 y 1 (x = f 1 ( y 1 ) 9 y r (x = f r ( y r ) where f 1 ; f r are all the function symbols in L and y i are tuples of variables of ....

J. C. Shepherdson. Negation in logic programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 19--88. Morgan Kaufmann,

....that select only ground negative literals) Jaffar, Lassez and Lloyd [JLL83] prove the completeness result and show that if a goal G is a logical consequence of comp(P) then there is a finitely failed SLD tree for G. For a comprehensive description of negation in logic programs, see Shepherdson [She88]. The excellent research monograph by John Lloyd [Llo87] pulls together the theoretical developments in logic programming up to the time of the publication date. In early 1980 I devoted my attention to trying to extend Reiter s work on the CWA for disjunctive theories. I developed a procedural ....

J.C. Shepherdson. Negation in Logic Programming. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 19--88. Morgan Kaufman Pub., 1988.

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J. C. Shepherdson, Negation in logic programming, in Foundations of Deductive Databases and Logic Programming, J. Minker editor, Morgan Kaufmann, Los Altos, CA (1988), pp. 19-88.

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J. C. Shepherdson, Negation in Logic Programming, in: J. Minker (ed.), Foundations of Deductive Databases and Logic Programming, Morgan Kaufmann, 1988, pp.19-88.

No context found.

) Shepherdson, J. C. Negation in Logic Programming. In J. Minker ed., Foundations of Deductive Databases and Logic Programming, pp. 19-88, Morgan Kaufmann, 1988.

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Shepherdson, J., "Negation in Logic Programming," pp. 19-88 in Foundations of Deductive

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J. C. Shepherdson, Negation in Logic Programming, in J. Minker (ed), Foundations of Deductive Databases and Logic Programs, Morgan Kaufmann, Los Altos, CA, 1988, pp. 19-88.

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Shepherdson, J. C., Negation in Logic Programming, in: Minker, J. (ed), Foundations of Deductive Databases and Logic Programs, Morgan Kaufmann, Los Altos, CA, 1988.

No context found.

Shepherdson, J. C., Negation in Logic Programming, in: Minker, J. (ed), Foundations of Deductive Databases and Logic Programs, Morgan Kaufmann, Los Altos, CA, 1988.

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