J. Jaffar, J.-L. Lassez, M.J. Maher, A Theory of Complete Logic Programs with Equality, J. Logic Programming 3 (1984) 211-223  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in Section 5.1.2 by discussing the problem of finding equationally complete solutions to equations and queries. The terminology surrounding equational theories follows that of [Huet 80] while the terminology concerning the theory of definite clause programs follows that of [Emden 76] and [Jaffar 84] 5.1.1 Definite clause programs with equality Let V be a countably infinite set of variables, F a finite set of function symbols, P a finite set of predicate symbols, and S a finite set of sorts. Associated with every variable v 2 V is a sort from S. Let V S be the set of all variables of ....

J. Jaffar, J.-L. Lassez, and M. J. Maher. A Theory of Complete Logic Programs with Equality. Journal of Logic Programming, 1(3):211--223, November 1984.

....by the equational logic programming approach [28, 23] augmented by specificity. We also illustrate the approach by examples motivated by the Broken Item and the Yale Shooting domain. Equational logic programs for computing change and specificity as well as their completion in the sense of [13] and [31, 50] are presented in Section 4. Section 5 focuses on models which interpret terms representing situations as multisets. In addition, we show how these models are related to the intended meaning of causality and specificity. Section 6 introduces SLDENF resolution as SLDNF resolution extended by a ....

....consistent situation descriptions in the extended Yale Shooting scenario together with all action descriptions of the three actions load , shoot , and wait . section. We will present a completed logic program in the sense of [13] along with a unification complete equational theory in the sense of [31] which can be used to represent objects or situations and to define actions, change, and specificity. Recall that a situation is a multiset of facts. Facts itself are represented by terms. In order to represent multisets, a binary function symbol ffi is introduced such that ffi is associative, ....

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J. Jaffar, J.-L. Lassez, and M. J. Maher. A theory of complete logic programs with equality. Journal of Logic Programming, 1(3):211--223, 1984.

....or methods are preferred. We formally define situations, objects, and specificity and illustrate the approach by examples motivated by the Broken Object and the Yale Shooting domain. Equational logic programs for computing change and specificity as well as their completion in the sense of [3] and [10] are presented, and we demonstrate that the models for these programs exhibit the intended meaning of causality and specificity. Finally, we show that SLDENF resolution is a sound and complete inference rule for the defined class of equational logic programs. In the sequel we will briefly define ....

J. Jaffar, J-L. Lassez, and M. J. Maher. A theory of complete logic programs with equality. In Proceedings of the International Conference on Fifth Generation Computer Systems, pages 175--184. ICOT, 1984.

.... machinery so that the elegance and simplicity of the semantics of predicate logic gets lost (see e.g. 42] On the other hand the semantics of a logic programming language like Prolog [31] can smoothly be extended to logic programs which are augmented by a (conditional) equational theory [25, 26, 20]. Equational axioms play a vital role in many applications of logic. However, they are often the reason for a tremendous inefficiency. One solution to this problem was to identify troublesome axioms and to build them into the deductive machiney [36] notably into the unification procedure [35] In ....

....with related work in this area. The reader is assumed to be familiar with the foundations of logic programming as for example presented in [31] Though the extension of these foundations to equational logic programming are discussed in great length some familiarity with these results (e.g. [25, 26]) would be helpful. As far as the operational semantics is concerned the results used in this paper can be found in [10] and [20] Efforts have been made to omit technical details whenever possible and the interested reader is refered to the given literature. Throughout the paper x; y : ....

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J. Jaffar, J-L. Lassez, and M. J. Maher. A theory of complete logic programs with equality. In Proceedings of the International Conference on Fifth Generation Computer Systems, pages 175--184. ICOT, 1984.

....atoms as well as equations. EP and LP are called equational and logic program, respectively. We distinguish between atoms of the form P (t 1 ; t n ) and equations of the form s =t, where P is an n ary predicate symbol different from = and s and t (possibly indexed) are terms. Jaffar et al. [Jaffar et al. 1984; Jaffar et al. 1986] defined a theoretical framework for equational logic programming and showed that the principal semantic properties of logic programming such as the existence of a canonical domain of computation, the existence of a least and greatest model semantics, or the the existence of ....

.... n ) If we augment a logic program with a conditional equational theory EP , then SLD resolution becomes SLDE resolution: if oe is an EP unifier of P (s 1 ; s n ) and P (t 1 ; t n ) then (7) is an SLDE resolvent of (5) and (6) As an immediate consequence of the results of Jaffar et al. [Jaffar et al. 1984; Jaffar et al. 1986] we find that the success set with respect to SLDE resolution is equal to the least Herbrand E model of an equational logic program. As Fages Huet [Fages and Huet, 1986] showed, the concept of a most general unifier can only be generalized to a complete set of EP unifiers ....

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Jaffar, J., Lassez, J.-L., and Maher, M. J. (1984). A theory of complete logic programs with equality. In Proceedings of the International Conference on Fifth Generation Computer Systems, pages 175--184.

.... such as the existence of a canonical domain of computation, the existence of a least and greatest model semantics, or the soundness and strong completeness for successful and finitely failed derivations of the underlying implementation model hold also for equational logic programming (see [Jaffar et al. 1984; Jaffar et al. 1986] the main problem remains of how the E unifiers of two expressions can be computed. This can be done by flattening and SLD resolution (e.g. Barbuti et al. 1986] by paramodulation or special forms of it (e.g. Robinson and Wos, 1969; Fribourg, 1985; Reddy, 1985; Furbach ....

....unification procedure based on sets of transformations. Moreover, Gallier et al. 1989] have extended this result to hold also for higher order E unification. The transformations rules presented herein can be used as a computational method for equational logic programs as proposed by Jaffar et al. 1984; 1986] Goguen Meseguer [1986] by adding a lazy resolution rule as suggested in [Holldobler, 1987a] This rule applied to a selected atom of the form P (s 1 ; s n ) and a program clause of the form P (t 1 ; t n ) D forces the comparison of corresponding arguments, i.e. D [ fP (s ....

J. Jaffar, J-L. Lassez, and M. J. Maher. A theory of complete logic programs with equality. In Proceedings of the International Conference on Fifth Generation Computer Systems, pages 175--184, 1984.

....we parametrize the semantics of the traditional Horn clause language w.r.t. an equality theory E. To this aim the problem is that, whenever a semantics is defined over the Herbrand universe U , equality is interpreted by default as syntactic identity. To overcome this restriction, Jaffar et al. [43] proposed the use of quotient universes. Here we adapt this technique to our context. Definition 17 Let R be a congruence relation. The quotient universe of U with respect to R, indicated as U=R, is the set of the equivalence classes of U under R, i.e. the partition given by R in U . Given an ....

....23 A ground atom A is a logical E consequence of a logic program (P; E) if, for every E interpretation I , I is an E model of (P; E) implies that dAe 2 I . The least E model of a logic program (P; E) can be characterized as the least fixed point of a mapping T (P;E) over E interpretations [43], written as lfp(T (P;E) Let ground(P ) be the set of all ground instances of clauses in P . Definition 24 Let I be an E interpretation of a logic program (P; E) Then T (P;E) is defined as follows: T (P;E) I) f dAe : A e 1 ; e q ; A 1 ; Am ) 2 ground(P ) E j= e i for ....

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J. Jaffar, J.-L. Lassez, and M. J. Maher. A theory of complete logic programs with equality. J. Logic Programming, 3:211--223, 1984.

.... that compute complete sets of uni ers are, for example, applied in theorem proving with built in theories (see, e.g. 55, 68] in generalizations of the Knuth Bendix completion procedure to rewriting modulo theories (see, e.g. 34, 13] and in logic programming with equality (see, e.g. [32]) With the development of constraint approaches to theorem proving (see, e.g. 18, 51] term rewriting (see, e.g. 41] and logic programming (see, e.g. 31, 22] decision procedures for E uni cation have been gaining in importance. The combination problem for E uni cation algorithms is ....

J. Jaar, J.-L. Lassez, and M. Maher. A theory of complete logic programs with equality. J. Logic Programming, 1, 1984.

....cUAC1 (t 1 ; t 2 ) then 8 x 2 4 t 1 = t 2 oe 2cUAC1 (t 1 ;t 2 ) 9 y: 3 5 where y denotes the variables which occur in = but not in x . The axioms of item 3, in conjunction with the standard uniqueness of names assumption in item 2, ensure that EUNA is unification complete [ Jaffar et al. 1984; Shepherdson, 1992 ] wrt. state terms and the equational theory AC1. The latter axiomatizes the arbitrary re arranging of the fluent terms that occur in a state term; hence, the following observation, which will be needed below, is a consequence of EUNA being AC1 unification complete: ....

Joxan Jaffar, Jean-Louis Lassez, and Michael J. Maher. A theory of complete logic programs with equality. Journal of Logic Programming, 1(3):211--223, 1984. 25

....terms are required. In the goal member(E,append( 1] 2] the second argument is equal to the list [1,2] and therefore the two answers to this goal are E=1 and E=2. This kind of amalgamated language is known as logic programming with equality and has a clearly defined declarative semantics [50, 71, 106]. It is similar to the well known Horn clause logic [83] but with the difference that the equality predicate = is always interpreted as the identity on the carrier sets in all interpretations. Therefore we omit the details of the declarative semantics in this survey. The definition of the ....

J. Jaffar, J.-L. Lassez, and M.J. Maher. A theory of complete logic programs with equality. Journal of Logic Programming, Vol. 1, pp. 211--223, 1984.

.... languages by generalizing unification to E unification , i.e. unification with respect to an equational theory E (see, for example, 3, 17, 20, 39, 40] a survey is given in [5] Theoretical aspects of extended unification in the context of logic programming have been studied by Jaffar et al. [19] and Gallier and Raatz [14] Here we consider how logic programming languages extended to deal with certain kinds of equality theories can be viewed as instances of our framework. Let D denote the set of terms of the language under consideration, augmented with a distinguished element . An ....

J. Jaffar, J.-L. Lassez, and M. Maher, "A Theory of Complete Logic Programs with Equality", J. Logic Programming vol. 1 no. 3, Oct. 1984, pp. 211-224. 23

.... fg = 21 In order that the inequality of two state terms follows whenever they consist in different collections of fluent literals, an extension of the standard unique name assumption is needed, namely, the concept of unification completeness known from logic programming (see, e.g. Jaffar et al. 1984; Shepherdson, 1992; Thielscher, 1996 ] Let E be an equational theory, that is, a set of universally quantified equations. Two terms s and t are said to be E equal , written s =E t , iff s = t is entailed by E plus the standard axioms of equality (see (25) below) A substitution oe is called ....

Joxan Jaffar, Jean-Louis Lassez, and Michael J. Maher. A theory of complete logic programs with equality. Journal of Logic Programming, 1(3):211--223, 1984.

....unifiers cUAC1 (t 1 ; t 2 ) then 8 x 2 4 t 1 = t 2 oe 2cUAC1 (t 1 ;t 2 ) 9 y: 3 5 where y denotes the variables which occur in = but not in x . The axioms of item 3, in conjunction with the standard uniqueness of namesassumption in item 2, ensure that EUNA is unification complete [13, 19] wrt. state terms and the equational theory AC1. These axioms entail inequality of two state terms (or effect terms, resp. whenever these are composed of different fluent terms. The assertion that some fluent f holds (resp. does not hold) in some situation s is formalized as 9z: State(s) f ffi ....

Joxan Jaffar, Jean-Louis Lassez, and Michael J. Maher. A theory of complete logic programs with equality. Journal of Logic Programming, 1(3):211--223, 1984.

.... Equational theories which are of unification type unitary or finitary play an important role in automated theorem provers with built in theories (see e.g. Pl72, St85] in generalizations of the Knuth Bendix algorithm (see e.g. JK86, Bc87] and in logic programming with equality (see e.g. [JL84]) The reason is that these applications usually require algorithms which compute finite complete sets of unifiers, i.e. finite sets of unifiers from which all unifiers can be generated by instantiation. However, with the recent development of constraint approaches to theorem proving (see e.g. ....

J. Jaffar, J.L. Lassez, M. Maher, "A Theory of Complete Logic Programs with Equality," J. Logic Programming 1, 1984.

....and a complete set of E unifiers were introduced in the area of automated theorem proving with equality [18] They generalize the notions of a unifier and a most general unifier, respectively, for the case of built in equational theories. In logic programming, E unifiers were introduced in [13]. The complete sets of unifiers in logic programming have been considered in [9, 6, 7, 11, 19, 20] All these papers except [6, 7] considered restricted classes of logic programs with equality. Despite different formulations, these restricted programs can be characterized as pairs (P ; E) where E ....

J. Jaffar, J.-L. Lassez, and M.J. Maher. A theory of complete logic programs with equality. Journal of Logic Programming, 1(3):211--223, 1983.

....algorithm for first order logic solves for syntactic equality. However, when reasoning with objects such as sets, it is desirable for the (set) terms to unify with terms that are not necessarily syntactically identical, as there are equational theories that the unification must take into account [37, 40]. Furthermore, some applications may need unification to be generalized to solving inequality constraints over some domain. Motivated by such considerations, Jaffar and Lassez [36] proposed the paradigm of constraint logic programming (CLP) which integrates constraint solving and logic ....

J. Jaffar, J. L. Lassez, and M. J. Maher. A Theory of Complete Logic Programs with Equality. Journal of Logic Programming, 3:211--223, 1984.

....and efficient implementation. Thus, deterministic computations can be formulated in this framework without resort to extra logical operations such as the cut. However, this paradigm of equations is too restrictive to serve as a complete basis for logic programming. ffl Relations and Equations [GM84, JLM84, DP85] It seems natural to combine the efficiency of equations with the programming expressiveness of relations. However, existing combinations [GM84, JLM84, DP85] of these two logical forms extend the power of ordinary unification to unification modulo an equational theory, also known as E unification ....

....cut. However, this paradigm of equations is too restrictive to serve as a complete basis for logic programming. ffl Relations and Equations [GM84, JLM84, DP85] It seems natural to combine the efficiency of equations with the programming expressiveness of relations. However, existing combinations [GM84, JLM84, DP85] of these two logical forms extend the power of ordinary unification to unification modulo an equational theory, also known as E unification or narrowing. A thorough analysis of E unification is given by Gallier and Raatz [GR86] While the E unification approach is very elegant from a theoretical ....

J. Jaffar, J.-L. Lassez, M.J. Maher, "A theory of complete logic programs with equality," In J. of Logic Programming, pp. 211-223, 1984.

....where, due to our equational theory (AC1) standard unification is replaced by a theory unification procedure. Following [18] the semantics of our program is then given by its completion (cf. 3] P D ; AC1 ) where AC1 denotes a unification complete theory wrt. AC1 (see [13] or [18] The following theorem forms the basis of our soundness and completeness result regarding the completion of our constructed equational logic program. Theorem2. Let D be a domain description in AC determining a transition function Phi ; then,there exists a oe 0 such that the structure ....

J. Jaffar, J.-L. Lassez, and M. J. Maher. A theory of complete logic programs with equality. Journal of Logic Programming, 1(3):211--223, 1984.

....the rest of axioms. The separation means that the rest of axioms does not contain positive occurrences of equality. Equational logic programs are defined by this restriction as a pair hP; Ei, where P is a logic program without positive occurrences of equality, and E is a set of equality facts in [JLM 84, GM 86] or facts and rules [Hol 89, She 92] Jaffar et al. defined SLDE resolution as the procedural interpretation of such a program. An extension of SLD resolution called SLDE resolution has been proposed in [GR 86, GR 89] Completeness of SLDE resolution was only proved for ....

....as the procedural interpretation of such a program. An extension of SLD resolution called SLDE resolution has been proposed in [GR 86, GR 89] Completeness of SLDE resolution was only proved for well behaved programs, which are exactly equational logic programs in the sense of [JLM 84, GM 86, Hol 89] Completeness for arbitrary logic programs with equality has been left open. We refute completeness in Example 2.1) Dynamic separation of the equality theory was later used in tableaux methods [BH 92, Bec 94] and connection (or mating) methods [GNRS 92, Pet 93] The next major ....

J. Jaffar, J.-L. Lassez, and M.J. Maher. A theory of complete logic programs with equality. Journal of Logic Programming, 1(3):211--223, 1983.

....and a complete set of E unifiers were introduced in the area of automated theorem proving with equality [17] They generalize the notions of a unifier and a most general unifier, respectively, for the case of built in equational theories. In logic programming, E unifiers were introduced in [12]. The complete sets of unifiers in logic programming have been considered in [8, 5, 6, 10, 18, 19] All these papers except [5, 6] considered restricted classes of logic programs with equality. Despite different formulations, these restricted programs can be characterized as pairs (P; E) where E ....

J. Jaffar, J.-L. Lassez, and M.J. Maher. A theory of complete logic programs with equality. Journal of Logic Programming, 1(3):211--223, 1983.

....hfz = xg; fy = xg; fy = a 1 gi Gamma UN hfz = x; x = a; y = a 1 g; fg; fgi, the next state is the success state hfg; fz = x; x = a; y = a 1 g; fgi: 5 Declarative Semantics When using a Herbrand universe U , equality is syntactic identity. To generalize the equality, Jaffar et al. [16] proposed the use of quotient universes. Given an equality theory E and a congruence relation R consistent with E, the quotient universe of U with respect to R, indicated as U=R, is the set of the equivalence classes of U under R (i.e. the partition given by R in U ) In general, there is an ....

Jaffar, J., Lassez, J.-L. and Maher, M. J., A Theory of Complete Logic Programs with Equality, J. Logic Programming, 3:211--223 (1984).

....to the semantics of logic programs. One natural generalization of logic programs is to allow different unification mechanisms in the operational semantics. Such a generalization was welcomed since it promised the integration of the functional and logical programming paradigms. Jaffar et al. [10] generalized the theory of pure logic programs to a logic programming scheme which was parametric in the underlying equality theory, and proved that the main semantic results continued to hold. However, the theory of logic programs with equality was still not powerful enough to handle logic ....

J. Jaffar, J.-L. Lassez and M.J. Maher. A theory of complete logic programs with equality. The Journal of Logic Programming 3:211--223, 1984.

....list. In one variation from this standard we allow subscripts on program variables, to improve readability. Part I The Semantics of CLP Languages Many languages based on definite clauses have quite similar semantics. The crucial insight of the CLP Scheme [129, 128] and the earlier scheme of [130, 131] was that a logic based programming language, its operational semantics, its declarative semantics and the relationships between these semantics could all be parameterized by a choice of domain of computation and constraints. The resulting scheme defines the class of languages CLP (X ) obtained by ....

J. Jaffar, J-L. Lassez & M.J. Maher, A Theory of Complete Logic Programs with Equality, Journal of Logic Programming 1, 211--223, 1984.

....to the semantics of logic programs. One natural generalisation of logic programs is to allow different unification mechanisms in the operational semantics. Such a generalisation was welcomed since it promised the integration of the functional and logical programming paradigms. Jaffar et al. [10] generalised the theory of pure logic programs to a logic programming scheme which was parametric in the underlying equality theory, and proved that the main semantic results continued to hold. However, the theory of logic programs with equality was still not powerful enough to handle logic ....

J. Jaffar, J.-L. Lassez and M.J. Maher. A Theory of Complete Logic Programs with Equality. The Journal of Logic Programming 3:211--223, 1984.

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J. Jaffar, J.-L. Lassez, M.J. Maher, A Theory of Complete Logic Programs with Equality, J. Logic Programming 3 (1984) 211-223

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