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Action Logic and Pure Induction
 Logics in AI: European Workshop JELIA '90, LNCS 478
, 1991
"... In FloydHoare logic, programs are dynamic while assertions are static (hold at states). In action logic the two notions become one, with programs viewed as onthefly assertions whose truth is evaluated along intervals instead of at states. Action logic is an equational theory ACT conservatively ex ..."
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Cited by 67 (6 self)
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In FloydHoare logic, programs are dynamic while assertions are static (hold at states). In action logic the two notions become one, with programs viewed as onthefly assertions whose truth is evaluated along intervals instead of at states. Action logic is an equational theory ACT conservatively extending the equational theory REG of regular expressions with operations preimplication a!b (had a then b) and postimplication b/a (b ifever a). Unlike REG, ACT is finitely based, makes a reflexive transitive closure, and has an equivalent Hilbert system. The crucial axiom is that of pure induction, (a!a) = a!a. This work was supported by the National Science Foundation under grant number CCR8814921. 1 Introduction Many logics of action have been proposed, most of them in the past two decades. Here we define action logic, ACT, a new yet simple juxtaposition of old ideas, and show off some of its attractive aspects. The language of action logic is that of equational regular expressio...
A Survey of Residuated Lattices
"... Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of latticeordered ..."
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Cited by 64 (6 self)
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Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of latticeordered groups, ideal lattices of rings, linear logic and multivalued logic. Our exposition aims to cover basic results and current developments, concentrating on the algebraic structure, the lattice of varieties, and decidability. We end with a list of open problems that we hope will stimulate further research.
Diagram Groups
, 1996
"... this paper, we study 2dimensional analogies of this idea: semigroup diagrams, monoid pictures, annular diagrams, cylindric pictures and braided pictures. While the groups of linear diagrams are all free, we get a large class of groups which are representable by 2dimensional semigroup diagrams. Sem ..."
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Cited by 52 (8 self)
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this paper, we study 2dimensional analogies of this idea: semigroup diagrams, monoid pictures, annular diagrams, cylindric pictures and braided pictures. While the groups of linear diagrams are all free, we get a large class of groups which are representable by 2dimensional semigroup diagrams. Semigroup diagrams, are wellknown geometrical objects used in the study of Thue systems (=semigroup presentations). They were first formally introduced by Kashintsev [16], see also Remmers [29], Stallings [34] or Higgins [13]. The role of semigroup diagrams in the study of semigroups is similar to the role of van Kampen diagrams in the study of groups (see [22] or [26])
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 44 (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 ...
Topology on the spaces of orderings of groups
, 2003
"... A natural topology on the space of left orderings of an arbitrary semigroup is introduced. It is proved that this space is compact and that for free abelian groups it is homeomorphic to the Cantor set. An application of this result is a new proof of the existence of universal Gröbner bases. ..."
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Cited by 43 (0 self)
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A natural topology on the space of left orderings of an arbitrary semigroup is introduced. It is proved that this space is compact and that for free abelian groups it is homeomorphic to the Cantor set. An application of this result is a new proof of the existence of universal Gröbner bases.
Lambek Grammars Based on Pregroups
 Logical Aspects of Computational Linguistics, LNAI 2099
, 2001
"... Lambek [13] introduces pregroups as a new framework for syntactic structure. In this paper we prove some new theorems on pregroups and study grammars based on the calculus of free pregroups. We prove that these grammars are equivalent to contextfree grammars. We also discuss the relation of pregrou ..."
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Cited by 34 (5 self)
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Lambek [13] introduces pregroups as a new framework for syntactic structure. In this paper we prove some new theorems on pregroups and study grammars based on the calculus of free pregroups. We prove that these grammars are equivalent to contextfree grammars. We also discuss the relation of pregroups to the Lambek calculus. 1 Introduction and
Measurement Of Membership Functions: Theoretical And Empirical Work
, 1995
"... This chapter presents a review of various interpretations of the fuzzy membership function together with ways of obtaining a membership function. We emphasize that different interpretations of the membership function call for different elicitation methods. We try to make this distinction clear u ..."
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Cited by 32 (1 self)
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This chapter presents a review of various interpretations of the fuzzy membership function together with ways of obtaining a membership function. We emphasize that different interpretations of the membership function call for different elicitation methods. We try to make this distinction clear using techniques from measurement theory.
The equivalence problem of multitape finite automata
 Theoretical Computer Science
, 1991
"... Using a result of B.H. Neumann we extend Eilenberg’s Equality Theorem to a general result which implies that the multiplicity equivalence problem of two (nondeterministic) multitape finite automata is decidable. As a corollary we solve a long standing open problem in automata theory, namely, the equ ..."
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Cited by 32 (2 self)
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Using a result of B.H. Neumann we extend Eilenberg’s Equality Theorem to a general result which implies that the multiplicity equivalence problem of two (nondeterministic) multitape finite automata is decidable. As a corollary we solve a long standing open problem in automata theory, namely, the equivalence problem for multitape deterministic finite automata. The main theorem states that there is a finite test set for the multiplicity equivalence of finite automata over conservative monoids embeddable in a fully ordered group. 1
A theory of risk
 Journal of Mathematical Psychology
, 1970
"... A psychological theory of perceived risk is developed. The theory is formulated in terms of an ordering of options, conceived of as probability distributions with respect to risk. It is shown that, under the assumptions of the theory, the risk of an option is expressible as a linear combination of i ..."
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Cited by 31 (0 self)
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A psychological theory of perceived risk is developed. The theory is formulated in terms of an ordering of options, conceived of as probability distributions with respect to risk. It is shown that, under the assumptions of the theory, the risk of an option is expressible as a linear combination of its mean and variance. The relationships to other theories of risk and preference are explored. I.
Noncommutative logic II: sequent calculus and phase semantics
, 1998
"... INTRODUCTION Noncommutative logic is a unication of :  commutative linear logic (Girard 1987) and  cyclic linear logic (Girard 1989; Yetter 1990), a classical conservative extension of the Lambek calculus (Lambek 1958). In a previous paper with Abrusci (Abrusci and Ruet 1999) we presented the mu ..."
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Cited by 27 (6 self)
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INTRODUCTION Noncommutative logic is a unication of :  commutative linear logic (Girard 1987) and  cyclic linear logic (Girard 1989; Yetter 1990), a classical conservative extension of the Lambek calculus (Lambek 1958). In a previous paper with Abrusci (Abrusci and Ruet 1999) we presented the multiplicative fragment of noncommutative logic, with proof nets and a sequent calculus based on the structure of order varieties, and a sequentialization theorem. Here we consider full propositional noncommutative logic. Noncommutative logic. Let us rst review the basic ideas. Consider the purely noncommutative fragment of linear logic, obtained by removing the exchange rule entirely : ` ; ; ; , ` ; ; ; y This work has been partly carried out at LIENSCNRS, Ecole Normale Superieure (Paris), at McGill University