Jayaraman, B., and Plaisted, D. A. Programming with Equations, Subsets and Relations. In Proceedings of NACLP89 (1989), E. Lusk and R. Overbeek, Eds., The MIT Press, Cambridge, Mass., pp. 1051-1068. Cleveland.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....hereditarily finite sets. Hybrid means that they can involve free Herbrand functors, i.e. distinct from the set constructor and the emptyset symbol ; In the literature two main approaches have been adopted for the (extensional) representation of sets: ACI terms [2, 5, 21] and list like terms [4, 15, 7, 13]. Each has advantages and disadvantages over the other. Regarding operations on sets, they include primarily the ability to compare sets. This, when considering the most general case of non ground and partially specified sets, amounts to solving a (non trivial) set unification problem [2, 7, 1, ....

....equational theory T. There are three main ways of representing sets as terms: ffl using the binary union symbol [ as the set constructor, as done, for instance, in [2, 5, 21] ffl using the binary element insertion operator f Delta j Deltag as the set constructor, as done, for instance, in [4, 15, 7, 13]; A few other proposals for logic programming with sets can be hardly classified as following one of these approaches. In CLPS [19] all three approaches appear viable, as no choice is explicitly made. In [12] instead, sets are intended as subsets of a finite domain D of objects. At the ....

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B. Jayaraman and D. A. Plaisted. Programming with Equations, Subsets and Relations. In Proc. of NACLP'89, 1051--1068. The MIT Press, 1989.

....that is, unification under associativity, commutativity, and idempotence. This approach, however, limits the kind of sets that can be represented to be flat sets. The listlike representation used in this paper (and in most papers dealing with sets in logic programming languages, such as [Kup90, BNST91, DOPR96, HL91, JP89]) on the contrary, naturally accommodates for nested sets (such as #, a, # ) 3 On the other hand, the # based representation allows us to write set terms that cannot be directly expressed using a listlike representation, for instance, the term X # a # Y . The listlike ....

B. Jayaraman and D. A. Plaisted. Programming with equations, subsets and relations. In E. Lusk and R. Overbeek, editors, Proceedings of NACLP89, pages 1051--1068. The MIT Press, Cambridge, 1989.

....ACI uni cation, i.e. uni cation under Associativity, Commutativity, and Idempotence. This approach, however, limits the kind of sets that can be represented to be at sets. The listlike representation used in this paper (and in most papers dealing with sets in logic programming languages, such as [25, 11, 18, 22, 24]) on the contrary, naturally accommodates for nested sets (such as f; fa; f;ggg) 3 On the other hand, the [ based representation allows to write set terms that cannot be directly expressed using a list like representation, as for instance the term X [ a [ Y ; the list like representation, in ....

B. Jayaraman and D. A. Plaisted. Programming with Equations, Subsets and Relations. In E. Lusk and R. Overbeek, editors, Proceedings of NACLP89, pages 1051-1068. The MIT Press, Cambridge, Mass., 1989. Cleveland.

....techniques known from or parallel Prolog implementations cannot be applied without modifications, because Escher and Prolog differ considerably in their functionality, especially concerning the handling of Boolean expressions. Different approaches towards set processing have been explored, see [14, 13, 15, 24, 12, 11, 9, 4, 2, 3] for a representative selection. Ultimately, they all use a constructor representation of sets (in various forms) on the language level. Even though it is claimed, e.g. in [14] that no distinction is made between functions and constructors, set matching is still performed within the constraints ....

B. Jayaraman and D. A. Plaisted. Programming with Equations, Subsets, and Relations. In Proceedings of the North American Conference on Logic Programming, pages 1051--1068, Cleveland, Ohio, USA, 1989.

....from the availability of sets constructs. And, recently, a number of papers have addressed the problem also in a wider setting. General purpose set constructs and basic 1 Read setlog . operations on sets are inserted into some general logic based framework: an equational logic language in [15, 16], a pure logic programming language in [7, 9] and a CLP language in [17] The development of such kind of extensions raises several interesting theoretical as well as practical problems (some of them are discussed also in [23] ffl which kind of objects one should aim at dealing with: ....

....(i.e. X withY )withZ = X withZ)withY ) and a Right absorption property (i.e. X with Y ) with Y = X with Y ) Representation ii is quite usual when dealing with sets in logic programming. It is used for instance in [14] in [4] where with is called scons) and also in the Godel language [12] [15] uses the [ operator but actually its behaviour is that of the with operator of approach ii. Representation ii is also adopted, for instance, in [20] Representation i, on the contrary, is often used when dealing with the problem of set unification on its own, e.g. 6] and [18] where set ....

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B. Jayaraman and D.A. Plaisted. Programming with Equations, Subsets and Relations. Proc. of NACLP89, Cleveland, 1989.

....[1, 6, 7, 33, 42] Recently, however, a number of papers have addressed the problem also in a wider setting. General purpose set constructs and basic operations on sets have been added to general logic based frameworks: pure logic programming languages [16, 17] equational logic languages [28, 29], and Constraint Logic Programming languages [22, 35, 36] The importance of sets as a high level data structure in a programming language has been also recognized in Godel [26] which supplies the user with a few basic facilities to define (both extensionally and intensionally) finite sets, and ....

.... A perhaps preferable formulation of action 9(b) less uniform with the rest of the algorithm but more efficient as for the number of generated solutions, can be found in [20, 44] To end, let us remark that our unification algorithm is akin to the one sketched by Jayaraman and Plaisted in [28], but it solves a larger number of cases. In particular, the algorithm in [28] intentionally does not take into account the idempotency property of sets (i.e. our absorption property) While this restriction enables a simplification of the unification algorithm, it leads, on the other hand, to a ....

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Jayaraman, B. and Plaisted, D. A., Programming with Equations, Subsets and Relations, in: Proceedings of the North American Conference on Logic Programming '89, The MIT Press, Cambridge, MA, 1989.

....etc. 43, 71, 77] require recursion through inductive aggregate operations such as sum. These cannot be expressed in the current formulation of preference logic programs. We are investigating the issue of extending the paradigm with the ability to manipulate sets as in subset logic languages [41, 43, 44] to enable us to reason about generalized aggregation. Many traditional database applications such as configuration of systems and travel planning benefit from a constraint solving ability. Constraint Query Languages (CQL) were introduced by Kannelakis et al. [47] and straightforward modifications ....

....sub optimal solutions. Traditional top down evaluaton strategies for logic programming languages such as Prolog sacrifice completeness and they incur non termination for extremely simple programs. Memoization is a technique that is used to prevent infinite loops in top down evaluation mechanisms [99, 44]. In our framework, memoization will be necessary not only for avoiding non termination but also for efficient pruning of sub optimal solutions. The first implementation will most likely be a meta interpreter written in a traditional logic programming language such as Prolog. To test out the ....

B. Jayaraman and D. Plaisted. Programming with Equations, Subsets and Relations. In Proc. North American Conference on Logic Programming, pages 1051--1068, 1989.

....omitting the second equation, we obtain a datatype MSet(ff) for polymorphic multisets. Data structures based on non free constructors, specially sets and multisets, play an important role in several recent proposals for extended logic programming and multiparadigm declarative programming; see e.g. [13, 4, 8, 15]. As a This research has been partially supported by the the Spanish National Project TIC95 0433 C03 01 CPD and the Esprit BRA Working Group EP 22457 CCLII . 2 Note that user defined datatypes are also called algebraic in Haskell. In spite of this terminology, Haskell s data ....

....This is shown by the existence of free models for programs (Theorem 10) the adequateness of the rewriting calculi w.r.t. models (Theorem 8) and type preservation results (Theorems 2, 4 and 10) Related works dealing with non free data constructors in declarative programming languages include [13, 4, 8, 15]. The main novelty here has been to include polymorphic data types and lazy (possibly non deterministic) defined functions. The combination of algebraic constructors and lazy defined functions precludes a direct use of equational reasoning to deal with the equational theories for constructors. ....

Jayaraman B., Plaisted D.A.: Programming with Equations, Subsets, and Relations. In Proc. ICLP'89, Vol. 2, the MIT Press, pp. 1051--l068, 1989.

....clp(sc) subsumes ordinary logic programming over an Herbrand domain, since ground terms can be identified with singleton sets, and singleton sets are definable in clp(sc) There have been several previous approaches to augmenting logic programming languages with sets. Jayaraman and Plaisted [18] present a language in the equational programming style which combines relational, subset, and equational assertions. Operational and fixpoint semantics are given. A collect all property is posed as part of the semantics, which plays the same role as minimal models or least fixpoints in logic ....

B. Jayaraman and D. A. Plaisted. Programming with equations, subsets, and relations. In E. L. Lusk and R. A. Overbeek, editors, Proc. North Amer. Conf. Logic Programming 1989, volume 2, pages 1051--1068. MIT Press, 1989.

....Herbrand structure, logical semantics. y This manuscript is available as Technical Report TR 94 030, Department of Computer Science, SUNYBuffalo, August 1994. 1 Introduction The use of sets has been advocated by several authors in the literature on logic programming ( DOPR91] Jay92] [JP89], Kup90] and deductive databases ( AG91] BNST91] NT89] In studying the inclusion of sets in logic programs, it is natural to study finite sets at first. A representation often chosen for finite sets is that of scons, parallel to the list constructor cons. The use of this constructor for ....

....(from fxg y = Both constructors find natural uses in specifying sets in logic programs. The scons is used for specifying sets in terms of parts that may well overlap with each other. The dscons is used for specifying sets in terms of an element and remainder. The scons constructor in [JP89] has been used in both senses on the left hand sides of rules it was used to mean dscons, while on the right hand sides of rules it was used to mean scons. Two factors complicate the design of SetAx: First, scons is a primitive symbol in our theory, rather than a defined one. This is natural in ....

[Article contains additional citation context not shown here]

Jayaraman, B. and Plaisted, D. A.: Programming with Equations, Subsets, and Relations, Proc. North American Conf. Logic Programming, Cleveland, Oct. 1989, pp. 1051--1068.

....This paper describes the implementation of a logic programming paradigm based upon three kinds of program clauses: equational, subset, and general relational clauses. This paradigm is called subset logic programming, and it has been the subject of our investigations over the past several years [9, 10, 11, 12, 13, 14, 15, 18, 23, 25]. The particular language that was implemented is called SuRE, which stands for Subsets, Relations, and Equations 1 . While equational and relational clauses are well known in functional and logic programming respectively, subset clauses are relatively a recent development. Hence we first ....

....our proposed solutions. 1.1. Subset Logic Programming Our interest in subset logic programming stems from the fact that it provides a declarative and efficient means of working with sets, a feature that has received considerable recent interest in logic programming and deductive databases [2, 4, 5, 15, 16]. While sets are ubiquitous in applications of logical reasoning, practical functional and logic languages (such as ML or Prolog) do not support bona fide sets, apparently due to the difficulty of implementing them efficiently. For example, Prolog s setof actually constructs an ordered list, not a ....

[Article contains additional citation context not shown here]

B. Jayaraman and D.A. Plaisted, Programming with Equations, Subsets, and Relations, Proc. N. Amer. Conference on Logic Programming, pp. 1051--1068, MIT Press, 1989.

....is incomplete in a technical sense. However, the operational and declarative semantics coincide exactly for terminating partial order programs. Finally, we note that partial order clauses are a generalization and an extension of the concept of subset clauses described in our previous papers [12, 13, 14]. The significance of generalizing subset clauses to partial order clauses is that it provides a simple and efficient way of programming aggregate operations. In a recent paper [23] we showed how to translate partial order clauses into normal program clauses [17] whose meaning is formalized using ....

Jayaraman, B. and D. A. Plaisted, Programming with Equations, Subsets, and Relations, Proc. N. American Conf. on Logic Prog., pp. 1051-1068, MIT Press, 1989.

....the realization that a subset of first order predicate logic, namely definite clauses, can be given a procedural interpretation. It is well known that the paradigm of pure definite clauses has serious limitations as a programming language, and this has prompted variations of this basic framework [21, 30]. An important development in this vein is the paradigm of constraint logic programming (CLP) 17, 18] Essentially, a CLP scheme is parameterized by a domain, such as reals, and certain function symbols ( Gamma, etc. and predicates ( etc. have pre defined meanings. Computationally, the ....

B. Jayaraman and D. Plaisted. Programming with Equations, Subsets and Relations. In Proc. North American Conference on Logic Programming, pages 1051--1068, 1989.

....finite sets, Zermelo Fraenkel set theory, freeness axioms, set unification, Herbrand structure, logical semantics. y This is an expanded version of the paper [JJ92] 1 Introduction The use of sets has been advocated by several authors in the literature on logic programming ( DOPR96] Jay92] [JP89], Kup90] and deductive databases ( AG91] BNST91] NT89] In studying the inclusion of sets in logic programs, it is natural to study finite sets at first. A representation often chosen for finite sets is that of scons, parallel to the list constructor cons. The use of this constructor for ....

....(from fxg y = Both constructors find natural uses in specifying sets in logic programs. The scons is used for specifying sets in terms of parts that may well overlap with each other. The dscons is used for specifying sets in terms of an element and remainder. The scons constructor in [JP89] has been used in both senses on the left hand sides of rules it was used to mean dscons, while on the right hand sides of rules it was used to mean scons. Two factors complicate the design of SetAx: First, scons is a primitive symbol in our theory, rather than a defined one. This is natural ....

[Article contains additional citation context not shown here]

Jayaraman, B. and Plaisted, D. A.: Programming with Equations, Subsets, and Relations, Proc. North American Conf. Logic Programming, Cleveland, Oct. 1989, pp. 1051--1068, MIT Press.

....This paper describes the implementation of a logic programming paradigm based upon three kinds of program clauses: equational, subset, and general relational clauses. This paradigm is called subset logic programming, and it has been the subject of our investigations over the past several years [JP87, JN88, JP89, Jay91, Jay92, OJ93, Jan94, JJ94, JOM95]. The particular language that was implemented is called SuRE, which stands for Subsets, Relations, and Equations 1 . While equational and relational clauses are well known in functional and logic programming respectively, subset clauses are relatively a recent development. We therefore first ....

....outline our proposed solutions. 1. 1 Subset Logic Programming Our interest in subset logic programming stems from the fact that it provides a declarative and efficient means of working with sets, a feature that has received considerable recent interest in logic programming and deductive databases [AG91, BNST91, DOPR91, JP89, Kup90]. While sets are ubiquitous in applications of logical reasoning, practical functional and logic languages (such as ML or Prolog) do not support bona fide sets, apparently due to the difficulty of implementing them efficiently. For example, Prolog s setof actually constructs an ordered list, not a ....

[Article contains additional citation context not shown here]

B. Jayaraman and D.A. Plaisted, Programming with Equations, Subsets, and Relations, Proc. NACLP, pp. 1051-1068, MIT Press, 1989.

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Jayaraman, B., and Plaisted, D. A. Programming with Equations, Subsets and Relations. In Proceedings of NACLP89 (1989), E. Lusk and R. Overbeek, Eds., The MIT Press, Cambridge, Mass., pp. 1051-1068. Cleveland.

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