We give a finitary 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 Introduction Kleene algebras are algebraic structures with operators +, \Delta, , 0, and 1 satisfying certain axioms. They arise in various guises in a number of settings: relational algebra [22, 23], semantics and logics of programs [14, 24], automata and formal language theory [18, 19], and the design and analysis of algorithms [1, 21, 12]. An important example of a Kleene algebra is Reg \Sigma , the family of regular sets over a finite alphabet \Sigma. The equational theory of this structure has been called the algebra of regular events. This theory was first studied by Infor. and Comput. 110:2 (May 1994), 366--390. A preliminary version of this paper appeared as [16]. Kleene [13], who posed axiomatization as an open problem. Salomaa [2...

Abstract. We investigate conditions under which a given Kleene algebra with tests is isomorphic to an algebra of binary relations. Two simple separation properties are identified that, along with star-continuity, are sufficient for nonstandard relational representation. An algebraic condition is identified that is necessary and sufficient for the construction to produce a standard representation. 1

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 2-4, 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 CCR-8814921 ...

We combine the principles of the Floyd-Warshall-Kleene algorithm, enriched categories, and Birkhoff arithmetic, to yield a useful class of algebras of transitive vertex-labeled spaces. The motivating application is a uniform theory of abstract or parametrized time in which to any given notion of time there corresponds an algebra of concurrent behaviors and their operations, always the same operations but interpreted automatically and appropriately for that notion of time. An interesting side application is a language for succinctly naming a wide range of datatypes. 1 Introduction Posets, metric spaces, "closed" automata, and categories have in common the notion of a space of points with distances between points. These distances are respectively truth values, reals, languages, and sets. Distances have two facets, logical and metrical. The logical facet is expressed respectively via implications p ! q between truth values, comparisons x y between reals, inclusions L ` M between langua...

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: ffl There is a Kleene algebra in the sense of [6] that is not *-continuous. ffl The categories of *-continuous Kleene algebras [5, 6], closed semirings [1, 10] and S-algebras [2] are strongly related by adjunctions. ffl 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]. ffl Right-handed Kleene algebras are not necessarily left-handed Kleene algebras. This verifies a weaker version of a conjecture of Pratt [15]. In Rov...

We use Kleene algebra with tests to verify a wide assortment ofcommon compiler optimizations, including dead code elimination, common subexpression elimination, copy propagation, loop hoisting, induction variable elimination, instruction scheduling, algebraic simplification, loop unrolling, elimination of redundant instructions, array bounds check elimination, and introduction of sentinels. In each of these cases, we give a formal equational proof of the correctness of the optimizing transformation.

. Kleene algebras with tests provide a rigorous framework for equational specification and verification. They have been used successfully in basic safety analysis, source-to-source program transformation, and concurrency control. We prove the completeness of the equational theory of Kleene algebra with tests and *-continuous Kleene algebra with tests over language-theoretic and relational models. We also show decidability. Cohen's reduction of Kleene algebra with hypotheses of the form r = 0 to Kleene algebra without hypotheses is simplified and extended to handle Kleene algebras with tests. 1 Introduction A Kleene algebra with tests is an algebraic structure consisting of a Kleene algebra with an embedded Boolean subalgebra. This formalism provides a rigorous framework for equational specification and verification of programs. It has been applied successfully to problems in basic safety analysis, source-to-source program transformation, and concurrency control [3, 4, 5, 17]. Kleene ...

We concider an extened algebra of regular events (languages) with intersection besides the usual operations. This algebra has the structure of a distributive lattice with monotonic operations; the latter property is crucial for some applications. We give a new complete Horn equational axiomatiztion of the algebra and develop some termrewriting techniques for constructing logical inferences of valid equations. A shorter version of this paper is to appear in the proceedings of Developments in Language Theory, Univ. of Turku, July 1993, published by World Scientific. The present version has been submitted for publication elsewhere. 1 Introduction In this paper we consider an extended algebra of regular events (languages) on a given alphabet with intersection besides the usual operations (union, concatenation, Kleene star, empty, and the regular unit). This algebra has the structure of a distributive lattice (join is union, meet is intersection) with only monotonic operations. The latte...

Dynamic algebras combine the classes of Boolean (B 0 0) and regular (R [ ; ) algebras into a single finitely axiomatized variety (B R 3) resembling an R-module with "scalar" multiplication 3. The basic result is that is reflexive transitive closure, contrary to the intuition that this concept should require quantifiers for its definition. Using this result we give several examples of dynamic algebras arising naturally in connection with additive functions, binary relations, state trajectories, languages, and flowcharts. The main result is that free dynamic algebras are residually finite (i.e. factor as a subdirect product of finite dynamic algebras), important because finite separable dynamic algebras are isomorphic to Kripke structures. Applications include a new completeness proof for the Segerberg axiomatization of propositional dynamic logic, and yet another notion of regular algebra. Key words: Dynamic algebra, logic, program verification, regular algebra. This paper or...

The theory of flowchart schemes has a rich history going back to Ianov [6]; see Manna [22] for an elementary exposition. A central question in the theory of program schemes is scheme equivalence. Manna presents several examples of equivalence proofs that work by simplifying the schemes using various combinatorial transformation rules. In this paper we present a purely algebraic approach to this problem using Kleene algebra with tests (KAT). Instead of transforming schemes directly using combinatorial graph manipulation, we regard them as a certain kind of automaton on abstract traces. We prove a generalization of Kleene’s theorem and use it to construct equivalent expressions in the language of KAT. We can then give a purely equational proof of the equivalence of the resulting expressions. We prove soundness of the method and give a detailed example of its use. 1