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21
Embedding Extensional Finite Sets in CLP
, 1993
"... In this paper we review the definition of {log} 1, a logic language with sets, from the viewpoint of CLP. We show that starting with a CLPscheme allows a more uniform treatment of the builtin set operations (namely, =, ∈ and their negative counterparts), and allows all the theoretical results of ..."
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Cited by 25 (17 self)
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In this paper we review the definition of {log} 1, a logic language with sets, from the viewpoint of CLP. We show that starting with a CLPscheme allows a more uniform treatment of the builtin set operations (namely, =, ∈ and their negative counterparts), and allows all the theoretical results of CLP to be immediately exploitable. We prove this by precisely defining the privileged interpretation domain and the axioms of the selected set theory. Then we define a nondeterministic procedure for checking constraint satisfiability based on the reduction of a given constraint to a collection of constraint in a suitable canonical form, which is provable to be sound and complete w.r.t. the given theory. Algorithms for trasforming each one of the set constraints the language provides (=, �=, ∈ and �∈) into their corresponding canonical forms are described in details. It is also shown that the resulting language is powerful enough to allow all the usual operations on sets (such as ⊆, ∪, etc.) to be effectively programmed in the language itself. 1
Algebraic Approaches to Nondeterminism  an Overview
 ACM Computing Surveys
, 1997
"... this paper was published as Walicki, M.A. and Meldal, S., 1995, Nondeterministic Operators in Algebraic Frameworks, Tehnical Report No. CSLTR95664, Stanford University ..."
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Cited by 23 (3 self)
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this paper was published as Walicki, M.A. and Meldal, S., 1995, Nondeterministic Operators in Algebraic Frameworks, Tehnical Report No. CSLTR95664, Stanford University
Birewriting, a Term Rewriting Technique for Monotonic Order Relations
 Rewriting Techniques and Applications, LNCS 690
, 1993
"... We propose an extension of rewriting techniques to derive inclusion relations $a \subseteq b$ between terms built from monotonic operators. Instead of using only a rewriting relation $\REa$ and rewriting $a$ to $b$, we use another rewriting relation $\REb$ as well and seek a common expression $c$ su ..."
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Cited by 22 (6 self)
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We propose an extension of rewriting techniques to derive inclusion relations $a \subseteq b$ between terms built from monotonic operators. Instead of using only a rewriting relation $\REa$ and rewriting $a$ to $b$, we use another rewriting relation $\REb$ as well and seek a common expression $c$ such that $a \REa^* c$ and $b \REb^* c$. Each component of the birewriting system $\pair{\REa}{\REb}$ is allowed to be a subset of the corresponding inclusion $\subseteq$ or $\superseteq$. In order to assure the decidability and completeness of the proof procedure we study the commutativity of $\REa$ and $\REb$. We also extend the existing techniques of rewriting modulo equalities to birewriting modulo a set of inclusions. We present the canonical birewriting system corresponding to the theory of nondistributive lattices.
Deductive Database Languages: Problems and Solutions
, 1999
"... Deductive databases result from the integration of relational database and logic programming techniques. However, significant problems remain inherent in this simple synthesis from the language point of view. In this paper, we discuss these problems from four different aspects: complex values, objec ..."
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Cited by 18 (4 self)
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Deductive databases result from the integration of relational database and logic programming techniques. However, significant problems remain inherent in this simple synthesis from the language point of view. In this paper, we discuss these problems from four different aspects: complex values, object orientation, higherorderness, and updates. In each case, we examine four typical languages that address the corresponding issues.
Oracle Semantics for Prolog
 ALGEBRAIC AND LOGIC PROGRAMMING, PROCEEDINGS OF THE THIRD INTERNATIONAL CONFERENCE, VOLUME 632 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1992
"... This paper proposes to specify semantic definitions for logic programming languages such as Prolog in terms of an oracle which specifies the control strategy and identifies which clauses are to be applied to resolve a given goal. The approach is quite general. It is applicable to Prolog to specify b ..."
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Cited by 15 (5 self)
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This paper proposes to specify semantic definitions for logic programming languages such as Prolog in terms of an oracle which specifies the control strategy and identifies which clauses are to be applied to resolve a given goal. The approach is quite general. It is applicable to Prolog to specify both operational and declarative semantics as well as other logic programming languages. Previous semantic definitions for Prolog typically encode the sequential depthfirst search of the language into various mathematical frameworks. Such semantics mimic a Prolog interpreter in the sense that following the "leftmost" infinite path in the computation tree excludes computation to the right of this path from being considered by the semantics. The basic idea in this paper is to abstract away from the sequential control of Prolog and to provide a declarative characterization of the clauses to apply to a given goal. The decision whether or not to apply a clause is viewed as a query to an oracle wh...
A Minimality Study for Set Unification
, 1997
"... A unification algorithm is said to be minimal for a unification problem if it generates exactly a (minimal) complete set of mostgeneral unifiers, without instances, and without repetitions. The aim of this paper is to present a combinatorial minimality study for a significant collection of sample p ..."
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Cited by 10 (7 self)
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A unification algorithm is said to be minimal for a unification problem if it generates exactly a (minimal) complete set of mostgeneral unifiers, without instances, and without repetitions. The aim of this paper is to present a combinatorial minimality study for a significant collection of sample problems that can be used as benchmarks for testing any setunification algorithm. Based on this combinatorial study, a new SetUnification Algorithm (named SUA) is also described and proved to be minimal for all the analyzed problems. Furthermore, an existing nave setunification algorithm has also been tested to show its bad behavior for most of the sample problems.
Set Unification
, 2001
"... The goal of this paper is to provide a uniform overview of the unification problem in algebras capable of describing sets. The problem has been tackled, directly and indirectly, by many researchers and it can find important applications in various research areas  e.g., deductive databases, theorem ..."
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Cited by 8 (4 self)
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The goal of this paper is to provide a uniform overview of the unification problem in algebras capable of describing sets. The problem has been tackled, directly and indirectly, by many researchers and it can find important applications in various research areas  e.g., deductive databases, theorem proving, static analysis, rapid software prototyping. The problem has been explored in depth, but the various solutions proposed are spread across a large literature, and some of the approaches have been ignored and/or rediscovered by different researchers. In this
Theory of PartialOrder Programming
, 1995
"... This paper shows the use of partialorder program clauses and lattice domains for declarative programming. This paradigm is particularly useful for expressing concise solutions to problems from graph theory, program analysis, and database querying. These applications are characterized by a need to s ..."
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Cited by 7 (4 self)
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This paper shows the use of partialorder program clauses and lattice domains for declarative programming. This paradigm is particularly useful for expressing concise solutions to problems from graph theory, program analysis, and database querying. These applications are characterized by a need to solve circular constraints and perform aggregate operations, a capability that is very clearly and efficiently provided by partialorder clauses. We present a novel approach to their declarative and operational semantics, as well as the correctness of the operational semantics. The declarative semantics is modeltheoretic in nature, but the least model for any function is not the classical intersection of all models, but the greatest lower bound/least upper bound of the respective terms defined for this function in the different models. The operational semantics combines topdown goal reduction with memotables. In the partialorder programming framework, however, memoization is primarily nee...
Set Constructors, Finite Sets, and Logical Semantics
 Journal of Logic Programming
, 1994
"... The use of sets in declarative programming has been advocated by several authors in the literature. A representation often chosen for finite sets is that of scons, parallel to the list constructor cons. The logical theory for such constructors is usually tacitly assumed to be some formal system of c ..."
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Cited by 7 (1 self)
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The use of sets in declarative programming has been advocated by several authors in the literature. A representation often chosen for finite sets is that of scons, parallel to the list constructor cons. The logical theory for such constructors is usually tacitly assumed to be some formal system of classical set theory. However, classical set theory is formulated for a general setting, dealing with both finite and infinite sets, and not making any assumptions about particular set constructors. In giving logicalconsequence semantics for programs with finite sets, it is important to know exactly what connection exists between sets and set constructors. The main contribution of this paper lies in establishing these connections rigorously. We give a formal system, called SetAx, designed around the scons constructor. We distinguish between two kinds of set constructors, scons(x; y) and dscons(x; y), where both represent fxg [ y, but x 2 y is possible in the former, while x 62 y holds in the latter. Both constructors find natural uses in specifying sets in logic programs. The design of SetAx is guided by our choice of scons as a primitive symbol of our theory rather than as a defined one, and by the need to deduce nonmembership relations between terms, to enable the use of dscons. After giving the axioms SetAx, we justify it as a suitable theory for finite sets in logic programming with the aid of the classical theory and unification, and formulate Herbrand structure within it. Together, these provide a rigorous foundation for the set constructors in the context of logical semantics. KEYWORDS: set constructors, finite sets, ZermeloFraenkel set theory, freeness axioms, set unification, Herbrand structure, logical semantics.
A General Theory of Confluent Rewriting Systems for Logic Programming and its Applications
, 2001
"... Recently, Brass and Dix showed (Journal of Automated Reasoning 20(1), 1998) that the wellfounded semantics WFS can be defined as a conuent calculus of transformation rules. This lead not only to a simple extension to disjunctive programs (Journal of Logic Programming 38(3), 1999), but also to a new ..."
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Cited by 7 (4 self)
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Recently, Brass and Dix showed (Journal of Automated Reasoning 20(1), 1998) that the wellfounded semantics WFS can be defined as a conuent calculus of transformation rules. This lead not only to a simple extension to disjunctive programs (Journal of Logic Programming 38(3), 1999), but also to a new computation of the wellfounded semantics which is linear for a broad class of programs. We take this approach as a starting point and generalize it considerably by developing a general theory of Confluent LPSystems CS. Such a system CS is a rewriting system on the set of all logic programs over a fixed signature L and it induces in a natural way a canonical semantics. Moreover, we show four important applications of this theory: (1) most of the wellknown semantics are induced by confluent LPsystems, (2) there are many more transformation rules that lead to confluent LPsystems, (3) semantics induced by such systems can be used to model aggregation, (4) the new systems can be ...