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62
A completeness theorem for Kleene algebras and the algebra of regular events
 Information and Computation
, 1994
"... We givea nitary axiomatization of the algebra of regular events involving only equations and equational implications. Unlike Salomaa's axiomatizations, the axiomatization given here is sound for all interpretations over Kleene algebras. 1 ..."
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Cited by 185 (22 self)
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We givea nitary axiomatization of the algebra of regular events involving only equations and equational implications. Unlike Salomaa's axiomatizations, the axiomatization given here is sound for all interpretations over Kleene algebras. 1
Regular Path Queries with Constraints
 SYMPOSIUM ON PRINCIPLES OF DATABASE SYSTEMS
, 1997
"... The evaluation of path expression queries on semistructured data in a distributed asynchronous environment is considered. The focus is on the use of local information expressed in the form of path constraints in the optimization of path expression queries. In particular, decidability and complexity ..."
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Cited by 147 (6 self)
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The evaluation of path expression queries on semistructured data in a distributed asynchronous environment is considered. The focus is on the use of local information expressed in the form of path constraints in the optimization of path expression queries. In particular, decidability and complexity results on the implication problem for path constraints are established.
Verification on Infinite Structures
, 2000
"... In this chapter, we present a hierarchy of infinitestate systems based on the primitive operations of sequential and parallel composition; the hierarchy includes a variety of commonlystudied classes of systems such as contextfree and pushdown automata, and Petri net processes. We then examine the ..."
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Cited by 69 (2 self)
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In this chapter, we present a hierarchy of infinitestate systems based on the primitive operations of sequential and parallel composition; the hierarchy includes a variety of commonlystudied classes of systems such as contextfree and pushdown automata, and Petri net processes. We then examine the equivalence and regularity checking problems for these classes, with special emphasis on bisimulation equivalence, stressing the structural techniques which have been devised for solving these problems. Finally, we explore the model checking problem over these classes with respect to various linear and branchingtime temporal logics.
Automata and coinduction (an exercise in coalgebra
 LNCS
, 1998
"... The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which ..."
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Cited by 62 (16 self)
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The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which coinduction proof methods for language equality and language inclusion. At the same time, the present treatment of automata theory may serve as an introduction to coalgebra.
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 50 (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...
Behavioural Differential Equations: A Coinductive Calculus of Streams, Automata, and Power Series
, 2000
"... Streams, (automata and) languages, and formal power series are viewed coalgebraically. In summary, this amounts to supplying these sets with a deterministic automaton structure, which has the universal property of being final. Finality then forms the basis for both definitions and proofs by coinduct ..."
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Cited by 50 (17 self)
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Streams, (automata and) languages, and formal power series are viewed coalgebraically. In summary, this amounts to supplying these sets with a deterministic automaton structure, which has the universal property of being final. Finality then forms the basis for both definitions and proofs by coinduction, the coalgebraic counterpart of induction. Coinductive definitions take the shape of what we have called behavioural differential equations, after Brzozowski's notion of input derivative. A calculus is developed for coinductive reasoning about all of the afore mentioned structures, closely resembling (and at times generalising) calculus from classical analysis. 2000 Mathematics Subject Classification: 68Q10, 68Q55, 68Q85 1998 ACM Computing Classification System: F.1, F.3 Keywords & Phrases: Coalgebra, automaton, finality, coinduction, stream, formal language, formal power series, differential equation, input derivative, behaviour, semiring, maxplus algebra 1 Contents 1 Introductio...
The Equational Theory of Pomsets
, 1988
"... Pomsets have been introduced as a mode2 of concurrency. Since a pomset is a string in which the total order has been relaxed to be a partial order, in this paper we view them as a generalization cf Strings, and investigate their algebraic properties. In particular, we investigate the axiomatic prope ..."
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Cited by 47 (0 self)
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Pomsets have been introduced as a mode2 of concurrency. Since a pomset is a string in which the total order has been relaxed to be a partial order, in this paper we view them as a generalization cf Strings, and investigate their algebraic properties. In particular, we investigate the axiomatic properties of pornsets, sets of pomsets and ideals of pornsets, under such operations as concatenation, parallel composition, union and their associated closure operations. We find that the equational theory of sets, pomsets under concatenation, parallel composition and union is finitely axiomatizable, whereas the theory of languages under the analogous operations is not. A similar result is obtained for ideals of pornsets, which incorporate the notion of subsumption which is also known as auaentation. Finally, we show that the addition of any closure operation (parallel or serial) Ieads to nonfinite axiomatizability of the resulting equational theory.
On Kleene Algebras and Closed Semirings
, 1990
"... Kleene algebras are an important class of algebraic structures that arise in diverse areas of computer science: program logic and semantics, relational algebra, automata theory, and the design and analysis of algorithms. The literature contains several inequivalent definitions of Kleene algebras and ..."
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Cited by 40 (6 self)
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Kleene algebras are an important class of algebraic structures that arise in diverse areas of computer science: program logic and semantics, relational algebra, automata theory, and the design and analysis of algorithms. The literature contains several inequivalent definitions of Kleene algebras and related algebraic structures [2, 14, 15, 5, 6, 1, 10, 7]. In this paper we establish some new relationships among these structures. Our main results are: There is a Kleene algebra in the sense of [6] that is not *continuous. The categories of *continuous Kleene algebras [5, 6], closed semirings [1, 10] and Salgebras [2] are strongly related by adjunctions. The axioms of Kleene algebra in the sense of [6] are not complete for the universal Horn theory of the regular events. This refutes a conjecture of Conway [2, p. 103]. Righthanded Kleene algebras are not necessarily lefthanded Kleene algebras. This verifies a weaker version of a conjecture of Pratt [15].
The AQUA Approach to Querying Lists and Trees in ObjectOriented Databases
 IN IEEE INTERNATIONAL CONFERENCE ON DATA ENGINEERING
, 1995
"... Relational database systems and most objectoriented database systems provide support for queries. Usually these queries represent retrievals over sets or multisets. Many new applications for databases, such as multimedia systems and digital libraries, need support for queries on complex bulk types s ..."
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Cited by 21 (0 self)
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Relational database systems and most objectoriented database systems provide support for queries. Usually these queries represent retrievals over sets or multisets. Many new applications for databases, such as multimedia systems and digital libraries, need support for queries on complex bulk types such as lists and trees. In this paper we describe an objectoriented query algebra for lists and trees. The operators in the algebra preserve the ordering between the elements of a list or tree, even when the result list or tree contains an arbitrary set of nodes from the original tree. We also present predicate languages for lists and trees which allow ordersensitive queries because they use pattern matching to examine groups of list or tree nodes rather than individual nodes. The ability to decompose predicate patterns enables optimizations that make use of indices.
On induction vs. *continuity
 Proc. Workshop on Logics of Programs 1981, SpringVerlag Lect. Notes in Comput
, 1981
"... Abstract. In this paper we study the relative expressibility of the infinitary *continuity condition (*cant) X ~ V n x and the equational but weaker induction axiom Ond) X ^ [a*](X =[alX) [a*]X in Propositional Dynamic Logic. We show: (1) under ind only, there is a firstorder sentence ..."
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Cited by 20 (10 self)
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Abstract. In this paper we study the relative expressibility of the infinitary *continuity condition (*cant) <a*>X ~ V n <an>x and the equational but weaker induction axiom Ond) X ^ [a*](X =[alX) [a*]X in Propositional Dynamic Logic. We show: (1) under ind only, there is a firstorder sentence distinguishing separable dynamic algebras from standard Kripke models; whereas (2) under the stronger axiom *cant, the class of separable dynamic algebras and the class of standard Kripke models are indistinguishable by any sentence of infinitary firstorder logic. I.