J. A. Goguen and J. Meseguer. Eqlog: Equality, types and generic modules for logic programming. In DeGroot and Lindstrom, editors, Logic Programming, pages 295--262. Prentice Hall, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....integration of existing languages, e.g. Loglisp [Robinson and Sibert, 1982] Qute [M. Sato, 1983] or LeFun [Ait Kaci et al. 1987] over higher order logic (e.g. Miller and Nadathur, 1987] to equational logic languages, where logic programs are augmented with equational theories, e.g. Eqlog [Goguen and Meseguer, 1986]. In Loglisp, for example, a Horn clause calculus called Logic is implemented by a set of Lisp functions such that Logic and Lisp are mutually embedded. Loglisp is not a logically complete system but smoothly integrates the advantages of logic and functional programming by giving the user the ....

....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 1 ; s n )g Gamma (D [ ....

J. A. Goguen and J. Meseguer. Eqlog: Equality, types and generic modules for logic programming. In DeGroot and Lindstrom, editors, Logic Programming, pages 295--262. Prentice Hall, 1986.

....di erence is that a predicate must be de ned in a single module whereas in our language predicate de nitions can be spread over several modules. In this respect the system of [22] is less exible than ours but safer in that predicates are better protected from unintended use. Goguen and Meseguer [23] present the EqLog language which combines functional and logic programming. The language provides a parameterised module system which o ers functionalities similar to those described in [22] The module system is however described in terms of EqLog rather than Prolog so that it is not directly ....

....be parameterised w.r.t. predicate and function symbols, and predicate de nitions can be split over di erent modules. The semantics of modules is de ned via a program transformation that demodularises a modular program into a module free program. It is worth noting that our approach di ers from [4, 22, 23] also because we extend the language of modules to include in formulae that support the dynamic construction of parametric module expressions and context switching. Finally, several works have been devoted to investigate the application of logic based languages to database interoperability. In ....

J.A. Goguen and J. Meseguer. Equality, types and generic modules for logic programming. In D. DeGroot and G. Lindstrom, editors, Logic Programming Functions, Relations and Equations. Prentice Hall, 1986.

....functions. Narrowing is used as operational semantics. This approach has been taken in [Re85, Re87] and the language BABEL [MR88, MR92] The alternative approach considers the combination predicates defined by a set of Horn clauses with functions defined by a set of equations. Examples are EQLOG [GM84], FUNLOG [SY86] and K LEAF [BGLMP87, BCGMP89] The operational semantics is SLD resolution guided by semantic unification. Under some conditions, this strategy is equivalent to narrowing and it could be simulated by transforming clauses and equations into a flat form and then using pure ....

J.A. Goguen, J. Meseguer: EQLOG: Equality, Types and Generic Modules for Logic Programming, Journal of Logic Programming 1(2), 1984, pp. 179-210; also in [dGL86].

....Equational logic programming in a proper sense, i.e. languages that are not based on a constructor discipline and allow equations with user defined functions as formal parameters, is also a very interesting model, but very difficult to implement efficiently. Examples of such languages are EQLOG [GM84], LPG [BE86] etc. An alternative way to integrate functions, predicates and constraints can be done in the framework of concurrent constraint languages. The language OZ also includes object orientation. Laziness can be simulated by using perpetual processes and the synchronization facilities of ....

J.A. Goguen, J. Meseguer: EQLOG: Equality, Types and Generic Modules for Logic Programming, Journal of Logic Programming 1(2), 1984, pp. 179-210; new version in [dGL86].

....paper and gives some directions of future work. 2 Syntax Many frameworks have been used to define the integration of functional and logic programming paradigms. Without loss of generality, we consider here Horn clause logic with equality (HCLE) as the basis of a functional logic language, e.g. [12, 3]. Our approach of concurrent functional logic programs can be generalised easily to other frameworks such as Curry [16] or Escher [20] Hereafter, we assume that the reader is familiar with the classical notions about HCLE. Let P = #, Cl) be a HCLE theory presentation where # = S,# , #) is a ....

J. A. Goguen and J. Meseguer. EQLOG: Equality, types and generic modules for logic programming. In DeGroot and Lindstrom, editors, Functional and Logic Programming. Prentice Hall, 1986.

....Sect. 2 and 3 can be found in the full paper [6] 2 Syntax Many frameworks have been used to de ne the integration of functional and logic programming paradigms. Without loss of generality, we consider here Horn clause logic with equality (HCLE) as the basis of a functional logic language, e.g. [8, 2]. Our approach of concurrent functional logic programs can be generalized easily to other frameworks such as Curry [11] or Escher [15] Hereafter, we assume that the reader is familiar with the classical notions about HCLE. Let P = Cl ) be a HCLE theory presentation where = S; is a ....

J. A. Goguen and J. Meseguer. EQLOG: Equality, types and generic modules for logic programming. In DeGroot and Lindstrom, editors, Functional and Logic Programming. Prentice Hall, 1986.

....in a computation sequence, how to achieve incrementality in the computation process and how to profit from finitely failed derivations as a heuristic for optimizing the algorithms. In the last few years several approaches to the integration of logic and equational programming have been developed [6, 15, 19, 24, 26, 31, 35, 11, 50, 51]. One relevant approach [31, 39] defines equational logic programs as logic programs which are augmented by Horn equational theories. These programs admit least model and fixpoint semantics. Interpreted function symbols can appear as arguments of relations and existentially quantified variables ....

....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 ....

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J. Goguen and J. Meseguer. Eqlog: Equality, Types and Generic Modules for Logic Programming. In D. de Groot and G. Lindstrom, editors, , pages 295--363. Prentice-Hall, 1986.

....is an obstacle to their effective use. 5 There have been several attempts to introduce new logic languages offering constructs for declaring abstract data types, generic modules, and overloading; the interested reader will find a discussion of theoretical problems introduced by these concepts in [73]. 2.1.2 Prolog as support for rapid prototyping An executable specification can be considered a prototype; indeed, historically the first applications of logic programming in software engineering were for rapid prototyping of applications [43, 115, 68, 160] Given a Prolog specification, the ....

....of different specification formalisms according to Mekly and Yuhas coherent language which is then able to handle each subproblem with the most adequate formalism. An example is the integration of Horn clause logic and equational theories in a number of logic functional languages: e.g. Eqlog [73], KLEAF [70] Babel [138] etc. An approach to integrating different specification notations in a single language framework is Constraint Logic Programming (CLP) a general language integration mechanism built around Horn clause logic and consistent constraints [86, 87] The CLP mechanisms ....

J. Goguen and J. Meseguer. Eqlog: Equality, Types and Generic Modules for Logic Programming. In D. de Groot and G. Lindstrom, editors, Logic Programming, Functions, Relations and Equations, pages 295--363. Prentice-Hall, 1986.

....difference is that a predicate must be defined in a single module, whereas, in our system, predicates that are parameterised can be defined in more than one module. The system is less flexible than the one described here but safer in that the predicates are better protected from unintended use. [Goguen and Meseguer, 1984] present EqLog which combines functional and logic programming. The language provides a parameterised module system which appears to be similar in function to the module system described in [Sannella and Wallen, 1992] However, it is described in the framework of EqLog rather than Prolog so that ....

J.A. Goguen and J. Meseguer. EQLOG: Equality, types and generic modules for logic programming. Journal of Logic Programming, 1:68--131, 1984.

....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 ....

J.A. Goguen and J. Meseguer. EQLOG: Equality, types and generic modules for logic programming. In D. DeGroot and G. Lindstrom, editors, Logic Programming, Functions, Relations and Equations, pages 295--363. Prentice Hall, 1986.

....the logic and functional programming paradigms in a single language. The computational mechanism underlying many of these amalgamated languages is conditional narrowing. Examples include alf (Hanus [19] babel (Moreno Navarro and Rodr iguez Artalejo [35] eqlog (Goguen and Meseguer [18]) k leaf (Giovannetti et al. 16] and slog (Fribourg [15] Narrowing was first studied in the context of semantic or E unification. Fay [14] and Hullot [23] showed that narrowing is a complete method for solving equations in the theory defined by a confluent and terminating term rewriting ....

J.A. Goguen and J. Meseguer, EQLOG: Equality, Types and Generic Modules for Logic Programming, in: "Logic Programming: Functions, Relations and Equations", (eds. D. DeGroot and G. Lindstrom), Prentice Hall, pp. 295--363, 1986.

....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 ....

J.A. Goguen and J. Meseguer. EQLOG: Equality, types and generic modules for logic programming. In D. DeGroot and G. Lindstrom, editors, Logic Programming, Functions, Relations and Equations, pages 295--363. Prentice Hall, 1986.

....rewrite rules (see e.g. 21] In other words, narrowing not only extends the basic computational mechanism of FP but also that of LP. It is then not surprising that narrowing based techniques have so far prevailed as operational mechanisms for functional logic interpreters [1] 12] 15] [17] [23] 32] Such a choice can often be nicely justified by rigorous completeness results. For instance, narrowing is a complete inference procedure in the case of a canonical system E [20] In this paper, we shall explain how to efficiently implement various strategies of narrowing using Prolog ....

J. Goguen and J. Meseguer. EQLOG: equality, types and generic modules for logic programming. In D. DeGroot and G. Lindstrom, editors, Functional and Logic Programming, pages 295--363. Prentice-Hall, 1986.

....rules to compute the value of an expression. These languages are related to logic programming languages like Prolog [CM 81] which has also been touted as an executable specification language. One view of the relationship between rewrite rule based programming and logic programming is given in [GM 86] 2.4.1 HOPE HOPE [BMS 80] is a purely applicative programming language having a rich but secure type system which allows new data types to be defined by the user. For example, binary trees with integer labels on nodes and leaves may be defined as follows (we assume that the type int has ....

Goguen, J.A. and Meseguer, J. Eqlog: equality, types and generic modules for logic programming. Functional and Logic Programming (D. DeGroot and G. Lindstrom, eds.), Prentice-Hall (1986).

....language. Not that these features are exclusive to this approach: overloading existed long before object oriented languages (FORTRAN already used it) while subtyping, even if it was first suggest by object oriented paradigms, has been included in other different paradigms (e.g. EQLOG [GM85] LIFE [AKP91] or Quest [CL91] But their combination is peculiar to object oriented 6 Note that, in this system, fg is not a supertype of any non empty overloaded type; this is not a problem, since the empty overloaded type is only used to type j, which is only used only to start overloaded ....

Joseph Goguen and Jos'e Meseguer. EQLOG: equality, types and generic modules for logic programming. In deGroot and Lindstrom, editors, Functional and Logic Programming. Prentice-Hall, 1985.

....# 3020 (INTEGRATION) y Department of Computer and Information Science, Linkoping University, S 581 83 Linkoping, Sweden z Institute of Informatics, Warsaw University, Poland ffl Completely new languages that allow definitions of both functions and relations. Examples: SLOG [Fri85] EQLOG [GM86] LEAF [BBLM86] KLEAF [BGL 87] BABEL [MNRA92] FUNLOG [SY86] Le Fun [AKLN87] Aflog [SNHM87] and ALF [Han90] The major drawback of the first approach is that unrestricted access to features of the underlying language often makes it hard to give a declarative reading to the programs. The ....

....view any amalgamated program P as an equational logic program, namely: P [ E The problem of assigning a declarative semantics to amalgamated programs has thus been reduced to assigning such a semantics to equational logic programs. This problem has been studied by several authors (e.g. JLM84, GM86, Hol89] For our purposes only a few declarative notions are needed. One is the equivalence relation = E induced by E . It is defined on the set of terms by: t = E u ( E j= t : u That is, t = E u iff every interpretation (of our alphabet) which interprets equality as identity and satisfies ....

J. Goguen and J. Meseguer. EQLOG: Equality, types and generic modules for logic programming. In D. DeGroot and G. Lindstrom, editors, Logic programming, functions, relations and equations, pages 295--364. Prentice-Hall, 1986.

....the operational semantics are the presentation of a sound inference system for equalities and disequalities, the use of uniform strategies and the completeness of such strategies for conditional term rewriting systems. Among the different proposals of functional logic programming languages, eqlog [13] is by no means the closest to lpg. Both are based on Horn clause logic with equality and provide generic modules. However, eqlog offers subsort facilities which lack in lpg whereas lpg allows the use of disequalities which is not possible in eqlog. As far as we know, eqlog has not been ....

....clause logic with equality and provide generic modules. However, eqlog offers subsort facilities which lack in lpg whereas lpg allows the use of disequalities which is not possible in eqlog. As far as we know, eqlog has not been implemented and its operational semantics has been only sketched in [13] without any seek for optimization or so. In lpg, predicates and functions are well distinguished. This fact allows the use of disequations within the bodies of Horn clauses without any problem. In other words, any completely instantiated lpg specification has an initial model. In languages based ....

J. A. Goguen and J. Meseguer. EQLOG: Equality, Types and Generic Modules for Logic Programming. In Functional and Logic Programming, eds. DeGroot and Lindstrom. Prentice-Hall, 1986.

....problem in view of functional logic programming than proving universally quantified equation. We will characterize narrowing as a proof method of this problem in the next section. For its semantical basis, we require Herbrand s theorem for equational logic proved by Goguen and Meseguer in [GM86] GM87] In this section, we review their result and show a semantical version of their result. They gave this theorem in the framework of order sorted equational logic. Here, we state a one sorted version of the theorem. Theorem 2.17 [Herbrand s theorem for equational logic [GM87] Let E be a ....

J. A. Goguen and J. Meseguer. Eqlog: equality, types and generic modules for logic programming. In DeGroot and Lindstrom, editors, Logic Programming, pages 295--364. Prentice Hall, 1986.

.... [Wan87] Red88] CHC90] GJ90] The database community has proposed some powerful type systems to manage complex objects with associated compiling and optimization techniques [Ku85] SZ87] Ja89] AK89] Investigations to merge logic and objects have also produced interesting type systems [Ait86][GM86][KL89] However, we have found necessary to develop yet another type system for our object oriented deductive language LAURE [Ca91a] which is strongly inspired by these various works, but still is different because of its interpretation of object oriented programming. We do not claim this type ....

J. Goguen, J. Meseguer: Eqlog: Equality, Types and Generic Modules for Logic Programming. In Logic Programming, Functions, Relations and Equations, Eds. D. DeGroot, G. Lindstrom, Prentice Hall, 1986.

....at the higher possible level. Moreover, the conflict resolution strategy can be enriched with new rules for method combination. An important corollary is that for each well defined feature, the set of method s domains is closed by intersection, which is close to the condition imposed in OBJ2 [GM86]. 5.2 Type Inference and Abstract Interpretation The LAURE language is an imperative language, built around the notion of message passing, which associates a function written [e] from S O S to each expression e of the language. The input parameter is the state of the system and the output is ....

J. Goguen, J. Meseguer: Eqlog: Equality, Types and Generic Modules for Logic Programming. In Logic Programming, Functions, Relations and Equations, Eds. D. DeGroot, G. Lindstrom, Prentice Hall, 1986.

....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 well behaved ....

....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 idea is ....

J.A. Goguen and J. Meseguer. EQLOG: Equality, types and generic modules for logic programming. In Logic Programming, Functions, Relations and Equations, pages 295--363. Prentice Hall, 1986.

....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 [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 ....

....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 the modified method by Brandt [7] He has shown how to get rid of the equality substitution axioms by pulling nonvariable terms out from argument lists. ....

J.A. Goguen and J. Meseguer. EQLOG: Equality, types and generic modules for logic programming. In D. DeGroot and G. Lindstrom, editors, Logic Programming, Functions, Relations and Equations, pages 295--363. Prentice Hall, 1986.

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J. Goguen and J. Meseguer. EQLOG: equality, types and generic modules for logic programming. In D. DeGroot and G. Lindstrom, editors, Logic Programming: Functions, Relations and Equations, pages 295--363. Prentice Hall, 1986.

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Joseph Goguen and Jose Meseguer. Eqlog: Equality, Types and Generic Modules for Logic Programming. Functional and Logic Programming, Prentice Hall, 1985.

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