J.H. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using E-unification. Journal of Logic Programming, 6:3--43, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the applicability of concrete proof strategies such as Prolog has not yet been assessed. A major di#culty is caused by the use of an underlying equational theory, which requires a non standard unification procedure in conjunction with an extended resolution principle called SLDE resolution [8, 13]. In this paper, we follow an alternative direction and investigate a particular program where a unification algorithm for our special equational theory is integrated by means of additional program clauses while otherwise standard unification is used. On the basis of this logic program, we ....

J. H. Gallier and S. Raatz. Extending SLD-Resolution to Equational Horn Clauses Using E-Unification. Journal of Logic Programming, 6:3--44, 1989.

....an appropriate E unification procedure. We follow the idea of [31] and adopt the definition used in [50] The selection rule is constrained such that negative literals are only selected if they are ground. If the selected literal is positive then the derivation step is done as for SLDE resolution [31, 18, 26]. If the selected literal is negative and if the SLDENF evaluation of the corresponding positive literal succeeds then the derivation fails, otherwise the derivation continues with the selected literals removed from the actual set of goal literals. The following two definitions are similar to the ....

J. H. Gallier and S. Raatz. Extending SLD-Resolution to Equational Horn Clauses Using E-Unification. Journal of Logic Programming, 6:3--44, 1989.

....equational axioms into the unification procedure. In the case considered herein this means to build a universal unification procedure into the SLD resolution rule. Soundness and completeness results for this technique often called SLDE resolution have been achieved by Gallier Raatz [13] for unconditional equational theories and by Holldobler [20] for conditional theories. However, this technique has also a major disadvantage. Unification problems under equational theories may have infinitely many independent solutions. Thus, the use of SLDE resolution may lead to a search space, ....

J. H. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using Eunification. Journal of Logic Programming, 6:3--44, 1989.

....of equality. For practical reasons, however, this technique is inappropriate. Plotkin [Plotkin, 1972] proposed to build troublesome axioms into the unification procedure and, then, to use resolution equipped with such a generalized unification procedure. Gallier Raatz [Gallier and Raatz, 1986; Gallier and Raatz, 1989] and Holldobler [Holldobler, 1989] formally demonstrated that Plotkin s idea can be adapted to SLD resolution if a logic program is augmented with an unconditional (resp. conditional) equational theory. Before we review these results we need some technicalities. Terms, atoms, and clauses are ....

Gallier, J. H. and Raatz, S. (1989). Extending SLD-resolution to equational Horn clauses using E-unification. Journal of Logic Programming, 6:3--44.

....idea [1967] to remove troublesome (equational) axioms from the data base and to build them into the deductive machinery, Plotkin [1972] has shown that these axioms can be handled by a sound and complete E unification procedure. These results have been adapted for equational logic programming by Gallier and Raatz [1986; 1989] (resp. Holldobler [1988b] in case a logic program is augmented by a unconditional (resp. conditional) equational theory. Though the main 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 ....

....for it. Only recently Hsiang Jouannaud [1988] have announced such a proof for unconditional theories. In this paper we consider only first order equational theories. Gallier Snyder [1988b] have defined a complete higher order 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 ....

J. H. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using E-unification. Journal of Logic Programming, 6:3--44, 1989.

....model of data deals with simple values, namely tuples consisting of atomic components. Various generalizations and formalisms have been proposed to handle more complex values like nested tuples, tuples of sets, etc. 1] Most of these formalisms can be expressed in terms of LP with equality [62, 63, 74, 72, 39] and constraint logic programming considered in Section 9. 8.1. Equational theories Let L be a language containing the equality predicate = By an equation over L we mean an atom s = t where s and t are terms in L. An equational theory E over L is a set of equations closed under the logical ....

J. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using E-unification. J. Logic Programming, 6(3):3--44, 1989.

.... but it has been shown that equational theories can be successfully built into the uni cation procedure [20] To allow a general treatment, SLD resolution has been extended to SLDE resolution which uses a universal uni cation procedure based on the proper ties common to all equational theories [9, 7]. In contrast to other techniques, SLDE resolution allows to cut down the often tremendous search space by merging equation solving and standard resolution steps. Narrowing [8] is an ecient approach to solve certain equational theories, and can be integrated as part of the uni cation into ....

....can be integrated as part of the uni cation into SLDE. 1 This is in contrast to [10] where containment of goal properties is encoded in the program. 3 SLDE resolution Formally, simple Fluent Calculus domains are (de nite) E programs (P; E) i.e. logic programs P with an equational theory E, [9, 7]. An equational theory E is de ned as a set of universally closed formulas of the form 8(s=E t) for some predicate =E complemented by the standard axioms of equality. 2 Consequently, if E = we obtain the standard equational theory, i.e. only syntactically identical terms are considered to be ....

J. H. Gallier and S. Raatz. Extending SLD resolution to equational horn clauses using E-unication. J. Logic Program., 6(1-2):3-43, 1989.

....most important operation in logic programming systems. Unification in the presence of an equational theory, also known as E unification, is necessary if the computational domain in a theorem prover enjoys certain equational properties [26] or if functions should be integrated into a logic language [10]. Therefore the development of Eunification algorithms is an active research topic during recent years (see, for instance, 29] Since E unification is a complex problem even for simple equational axioms, we are interested in efficient E unification methods in order to incorporate such methods ....

J.H. Gallier and S. Raatz. Extending SLD-Resolution to Equational Horn Clauses Using E-Unification. Journal of Logic Programming (6), pp. 3--43, 1989.

....But how can we compute such solutions In general, we have to 2 compute unifiers w.r.t. the given equational axioms which is known as E unification [44] Replacing standard unification by E unification in a resolution step yields a computational mechanism to deal with functions in logic programs [43, 49]. Unfortunately, E unification can be a very hard problem even for simple equations (see [116] for a survey) For instance, if we state the associativity of the append function by the equation append(append(L,M) N) append(L,append(M,N) then it is known that the corresponding E unification ....

J.H. Gallier and S. Raatz. Extending SLD-Resolution to Equational Horn Clauses Using E-Unification. Journal of Logic Programming (6), pp. 3--43, 1989.

....cope with the equational theory, the conventional SLD resolution mechanism based on a (syntactical) unification algorithm of logic programs has to be modified. The operational semantics of an equational logic language is based on some special form of equational resolution (such as SLDE resolution [9, 22, 26, 31]) where SLD resolution is usually kept as the only inference rule but the syntactical unification algorithm is replaced by equational (semantic) unification ( 16, 56, 57] means the general 1 E E E E E E = X X H H E E H H E unifiers unifiers unification = unification w.r.t. an ....

J.H. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using Eunification. , 6:3--43, 1989.

....1 ; t n g will be denoted by the term ft 1 j : ft n j;g : g. The only changes to the semantics of logic programming are the changes in the treatment of equality, since new predicate symbols are not free constructors. Such an approach is considered in a number of papers, for example (Gallier Raatz 1989, Kuper 1990, Beeri, Naqvi, Schmueli Tsur 1991, Schmueli, Tsur Zaniolo 1992, Dovier, Omodeo, Pontelli Rossi 1996, Dantsin Voronkov 1997b, Dantsin Voronkov 1997a) 2 Of course, a combination of the two approaches is also possible. This paper studies complexity of nonrecursive query ....

Gallier, J. & Raatz, S. (1989), `Extending SLD-resolution to equational Horn clauses using E-unification', Journal of Logic Programming 6(3), 3--44.

....set ft1 ; tng will be denoted by the term ft1 j : ftnj;g : g. The only changes to the semantics of logic programming are the changes in the treatment of equality, since new predicate symbols are not free constructors. Such an approach is considered in a number of papers, for example [25, 37, 9, 50, 22, 21, 19]. Of course, a combination of the two approaches is also possible. This paper studies complexity of nonrecursive query answering in logic databases with complex values. By complexity we mean expression complexity formalized as the following decision problem: given both the database and a query, ....

J.H. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using E-unification. Journal of Logic Programming, 6(3):3--44, 1989.

....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 is an equational theory (a set of equations) and P is a logic program where equality can only ....

....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 is an equational theory (a set of equations) and P is a logic program where equality can only occur in the bodies of the clauses. ....

[Article contains additional citation context not shown here]

J.H. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using E-unification. Journal of Logic Programming, 6(3):3--44, 1989.

.... procedure, which decides the satisfiability of an equality e 1 =e 2 , simultaneously computes a valuation (substitution) for the logical variables contained in e 1 and e 2 [Llo87] Extending this procedure with respect to a rewriting relation leads to the notion of E unification [Hol89,GR89] The equality realized by Eunification incorporates the identities specified by the reduction relation that is induced by the defined functions. The operational challenge is to implement an efficient decision procedure for E unification. A common approach is the narrowing procedure, which is ....

J. H. Gallier and S. Raatz. Extending SLD-resolution to equational horn clauses using E-unification. Journal of Logic Programming, 6, 1989.

....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 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 ....

J.H. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using Eunification. Journal of Logic Programming, 6(3):3--44, 1989.

....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 is an equational theory (a set of equations) and P is a logic program where equality can only ....

....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 is an equational theory (a set of equations) and P is a logic program where equality can only occur in the bodies of the clauses. ....

[Article contains additional citation context not shown here]

J.H. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using E- unification. Journal of Logic Programming, 6(3):3--44, 1989.

....the applicability of concrete proof strategies such as Prolog has not yet been assessed. A major difficulty is caused by the use of an underlying equational theory, which requires a non standard unification procedure in conjunction with an extended resolution principle called SLDE resolution [8, 13]. In this paper, we follow an alternative direction and investigate a particular program where a unification algorithm for our special equational theory is integrated by means of additional program clauses while otherwise standard unification is used. On the basis of this logic program, we ....

J. H. Gallier and S. Raatz. Extending SLD-Resolution to Equational Horn Clauses Using E-Unification. Journal of Logic Programming, 6:3--44, 1989.

....that all solutions to a given query can be computed [25] The simplest solution would be to retain the principle of ordinary resolution and to add equality axioms to the program. Unfortunately, this leads to a combinatorial explosion of the search space. This has motivated several proposals [31] [14] to extend This work has been partially supported by ESPRIT project BRA 3020. CHEONG et al. IMPLEMENTATION OF NARROWING syntactic unification involved in resolution to unification modulo the equational theory defined by E (called semantic unification) However, semantic unification applies ....

J. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using E-unification. Journal of Logic Programming, 6:3--43, 1989.

....the applicability of concrete proof strategies such as Prolog has not yet been assessed. A major difficulty is caused by the use of an underlying equational theory, which requires a non standard unification procedure in conjunction with an extended resolution principle called SLDE resolution [15, 25]. In this paper, we follow an alternative direction and investigate a particular program where a unification algorithm which has been proposed in [20] for our special equational theory is integrated by means of additional program clauses while otherwise standard unification is used. On the ....

J. H. Gallier and S. Raatz. Extending SLD-Resolution to Equational Horn Clauses Using E-Unification. Journal of Logic Programming, 6:3--44, 1989.

....program without positive occurrences of equality, and E is a set of equality facts in [30, 22] or facts and rules [27, 47] 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 [18, 19]. Completeness of SLDE resolution was only proved for well behaved programs, which are exactly equational logic programs in the sense of [30, 22, 27] Completeness for arbitrary logic programs with equality has been left open. We refute completeness in Example 2.1) The next major idea is ....

J.H. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using E-unification. Journal of Logic Programming, 6(3):3--44, 1989.

No context found.

J.H. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using E-unification. Journal of Logic Programming, 6:3--43, 1989.

No context found.

J.H. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using E-unification. Journal of Logic Programming, 6:3--43, 1989.

No context found.

J.H. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using E-unification. Journal of Logic Programming, 6:3--43, 1989.

No context found.

J.H. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using Eunification. Journal of Logic Programming, 6(3):3--44, 1989.

No context found.

GR89 J.H. Gallier and S. Raatz. Extending SLD-resolution to equational Horn clauses using E-unification. Journal of Logic Programming, 6(3):3--44, 1989.

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