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Alloy: A Lightweight Object Modelling Notation
, 2001
"... Alloy is a little language for describing structural properties. It offers a declaration syntax compatible with graphical object models, and a setbased formula syntax powerful enough to express complex constraints and yet amenable to a fully automatic semantic analysis. Its meaning is given by tr ..."
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Cited by 339 (13 self)
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Alloy is a little language for describing structural properties. It offers a declaration syntax compatible with graphical object models, and a setbased formula syntax powerful enough to express complex constraints and yet amenable to a fully automatic semantic analysis. Its meaning is given by translation to an even smaller (formally defined) kernel. This paper presents the language in its entirety, and explains its motivation, contributions and deficiencies.
On the Relative Expressiveness of Description Logics and Predicate Logics
 ARTIFICIAL INTELLIGENCE JOURNAL
, 1996
"... It is natural to view concept and role definitions in Description Logics as expressing monadic and dyadic predicates in Predicate Calculus. We show that the descriptions built using the constructors usually considered in the DL literature are characterized exactly as the predicates definable by form ..."
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Cited by 145 (3 self)
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It is natural to view concept and role definitions in Description Logics as expressing monadic and dyadic predicates in Predicate Calculus. We show that the descriptions built using the constructors usually considered in the DL literature are characterized exactly as the predicates definable by formulas in ¨L³, the subset of First Order Predicate Calculus with monadic and dyadic predicates which allows only three variable symbols. In order to handle “number bounds”, we allow numeric quantifiers, and for transitive closure of roles we use infinitary disjunction. Using previous results in the literature concerning languages with limited numbers of variables, we get as corollaries the existence of formulae of FOPC which cannot be expressed as descriptions. We also show that by omitting role composition, descriptions express exactly the formulae in ¨L², which is known to be decidable.
Automating FirstOrder Relational Logic
, 2000
"... An analysis is described that can automatically find models of firstorder formulas with relational operators and scalar quantifiers. The formula is translated to a quantifierfree boolean formula that has a model exactly when the original formula has a model within a given scope (that is, involving ..."
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Cited by 116 (19 self)
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An analysis is described that can automatically find models of firstorder formulas with relational operators and scalar quantifiers. The formula is translated to a quantifierfree boolean formula that has a model exactly when the original formula has a model within a given scope (that is, involving no more than some finite number of atoms). The paper presents a simple logic and gives a compositional translation scheme. It reports on the use of Alcoa, a tool based on the scheme, to analyze a variety of specifications expressed in Alloy, an object modelling notation based on the logic.
On Binary Constraint Problems
 Journal of the ACM
, 1994
"... The concepts of binary constraint satisfaction problems can be naturally generalized to the relation algebras of Tarski. The concept of pathconsistency plays a central role. Algorithms for pathconsistency can be implemented on matrices of relations and on matrices of elements from a relation algeb ..."
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Cited by 87 (2 self)
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The concepts of binary constraint satisfaction problems can be naturally generalized to the relation algebras of Tarski. The concept of pathconsistency plays a central role. Algorithms for pathconsistency can be implemented on matrices of relations and on matrices of elements from a relation algebra. We give an example of a 4by4 matrix of infinite relations on which no iterative local pathconsistency algorithm terminates. We give a class of examples over a fixed finite algebra on which all iterative local algorithms, whether parallel or sequential, must take quadratic time. Specific relation algebras arising from interval constraint problems are also studied: the Interval Algebra, the Point Algebra, and the Containment Algebra. 1 Introduction The logical study of binary relations is classical [8], [9], [51], [52], [56], [53], [54]. Following this tradition, Tarski formulated the theory of binary relations as an algebraic theory called relation algebra [59] 1 . Constraint satis...
PairDense Relation Algebras
 Transactions of the American Mathematical Society
, 1991
"... The central result of this paper is that every pairdense relation algebra is completely representable. A relation algebra is said to be pairdense if every nonzero element below the identity contains a "pair". A pair is the relation algebraic analogue of a relation of the form fha; ai ; hb; big ..."
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Cited by 62 (8 self)
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The central result of this paper is that every pairdense relation algebra is completely representable. A relation algebra is said to be pairdense if every nonzero element below the identity contains a "pair". A pair is the relation algebraic analogue of a relation of the form fha; ai ; hb; big (with a = b allowed). In a simple pairdense relation algebra, every pair is either a "point" (an algebraic analogue of fha; aig) or a "twin" (a pair which contains no point). In fact, every simple pairdense relation algebra A is completely representable over a set U iff jU j = + 2, where is the number of points of A and is the number of twins of A.
The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations
, 1991
"... ..."
On Binary Constraint Networks
, 1988
"... It is wellknown that general constraint satisfaction problems (CSPs) may be reduced to the binary case (BCSPs) [Pei92]. CSPs may be represented by binary constraint networks (BCNs), which can be represented by a graph with nodes for variables for which values are to be found in the domain of intere ..."
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Cited by 39 (5 self)
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It is wellknown that general constraint satisfaction problems (CSPs) may be reduced to the binary case (BCSPs) [Pei92]. CSPs may be represented by binary constraint networks (BCNs), which can be represented by a graph with nodes for variables for which values are to be found in the domain of interest, and edges labelled with binary relations between the values, which constrain the choice of solutions to those which satisfy the relations (e.g. [Mac77]). We formulate networks and algorithms in a general algebraic setting, that of Tarski's relation algebra [JonTar52], and obtain a parallel O(n log n) upper bound for pathconsistency, and give a class of examples on which reductiontype algorithms (which include the standard serial algorithms [Mac77, MacFre85, MohHen86] and all possible parallelisations of them) are O(n ). We then consider BCNs over various classes of relations that arise from an underlying linearly ordered set, the most wellknown being the interval algebra [All83, LadMad88.1]. There are three main consequences of the algebraic approach. Firstly, it puts the theory of BCNs on a firm (and classical) theoretical footing, enabling, for example, the complexity results. Secondly, we can apply techniques from relation algebra to show that consistency checking for a large class of relations on intervals ([All83]) is serial cubic, or parallel log time, significantly extending previous results (the problem is NPhard in general [VilKau86]). Thirdly, results are obtained via a new construction of relation algebras from other algebras which is of independent mathematical interest.
Dynamic Algebras as a wellbehaved fragment of Relation Algebras
 In Algebraic Logic and Universal Algebra in Computer Science, LNCS 425
, 1990
"... The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect ..."
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Cited by 35 (5 self)
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The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect to representable relation algebras, when expressed in their DA form are complete with respect to representable dynamic algebras. Moreover, whereas the theory of RA is undecidable, that of DA is decidable in exponential time. These results follow from representability of the free intensional dynamic algebras. Dept. of Computer Science, Stanford, CA 94305. This paper is based on a talk given at the conference Algebra and Computer Science, Ames, Iowa, June 24, 1988. It will appear in the proceedings of that conference, to be published by SpringerVerlag in the Lecture Notes in Computer Science series. This work was supported by the National Science Foundation under grant number CCR8814921 ...
Relation algebras in qualitative spatial reasoning
 Fundamenta Informaticae
, 1999
"... The formalization of the “part – of ” relationship goes back to the mereology of S. Le´sniewski, subsequently taken up by Leonard & Goodman (1940), and Clarke (1981). In this paper we investigate relation algebras obtained from different notions of “part–of”, respectively, “connectedness” in various ..."
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Cited by 34 (13 self)
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The formalization of the “part – of ” relationship goes back to the mereology of S. Le´sniewski, subsequently taken up by Leonard & Goodman (1940), and Clarke (1981). In this paper we investigate relation algebras obtained from different notions of “part–of”, respectively, “connectedness” in various domains. We obtain minimal models for the relational part of mereology in a general setting, and when the underlying set is an atomless Boolean algebra. 1
Back and Forth Between Modal Logic and Classical Logic
, 1994
"... Model Theory. That is, we have a nonempty family I of partial isomorphisms between two models M and N, which is closed under taking restrictions to smaller domains, and where the standard BackandForth properties are now restricted to apply only to partial isomorphisms of size at most k. Proof. ..."
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Cited by 30 (3 self)
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Model Theory. That is, we have a nonempty family I of partial isomorphisms between two models M and N, which is closed under taking restrictions to smaller domains, and where the standard BackandForth properties are now restricted to apply only to partial isomorphisms of size at most k. Proof. (A complete argument is in [16].) An outline is reproduced here, for convenience. First, kvariable formulas are preserved under partial isomorphism, by a simple induction. More precisely, one proves, for any assignment A and any partial isomorphism I 2 I which is defined on the Avalues for all variables x 1 ; : : : ; x k , that M;A j= OE iff N; I ffi A j= OE: The crucial step in the induction is the quantifier case. Quantified variables are irrelevant to the assignment, so that the relevant partial isomorphism can be restricted to size at most k \Gamma 1, whence a matching choice for the witness can be made on the opposite side. This proves "only if". Next, "if" has a proof analogous to...