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...

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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: 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.

We introduce and investigate new methods to define parallel composition of words and languages. The operation of parallel composition leads to new shuffle-like operations defined by syntactic constraints on the usual shuffle operation. The approach is applicable to concurrency, providing a method to define parallel composition of processes. It is also applicable to parallel computation. The operations are introduced using a uniform method based on the notion of trajectory. As a consequence, we obtain a very intuitive geometrical interpretation of the parallel composition operation. These operations lead in a natural way to a large class of semirings. The approach is amazingly flexible, diverse concepts from the theory of concurrency can be introduced and studied in this framework. For instance, we provide examples of applications to fairness property and to parallelization of non-context-free languages in terms of context-free and even regular languages. This paper concetrates on syntactic constraints. Semantic constraints will be dealt with in a forthcoming contribution. TUCS Research Group

Parikh's Theorem says that the commutative image of every context free language is the commutative image of some regular set. Pilling has shown that this theorem is essentially a statement about least solutions of polynomial inequalities. We prove the following general theorem of commutative Kleene algebra, of which Parikh's and Pilling's theorems are special cases: Every system of polynomial inequalities f i (x 1 ; : : : ; xn ) x i , 1 i n, over a commutative Kleene algebra K has a unique least solution in K n ; moreover, the components of the solution are given by polynomials in the coefficients of the f i . We also give a closed-form solution in terms of the Jacobian matrix. 1 Introduction Parikh's theorem [8] says that every context-free language is "letter-equivalent" to a regular set; formally, the commutative image of any context-free language is also the commutative image of some regular set. The commutative image of a string x over the alphabet fa 1 ; : : : ; a k g is ...

Action algebras have been proposed by Pratt [22] as an alternative to Kleene algebras [8, 9]. Their chief advantage over Kleene algebras is that they form a finitely-based equational variety, so the essential properties of (iteration) are captured purely equationally. However, unlike Kleene algebras, they are not closed under the formation of matrices, which renders them inapplicable in certain constructions in automata theory and the design and analysis of algorithms. In this paper we consider a class of action algebras called action lattices. An action lattice is simply an action algebra that forms a lattice under its natural order. Action lattices combine the best features of Kleene algebras and action algebras: like action algebras, they form a finitely-based equational variety; like Kleene algebras, they are closed under the formation of matrices. Moreover, they form the largest subvariety of action algebras for which this is true. All common examples of Kleene algebras appeari...

We study the complexity of reasoning in Kleene algebra and *-continuous Kleene algebra in the presence of extra equational assumptions E; that is, the complexity of deciding the validity of universal Horn formulas E ! s = t, where E is a finite set of equations. We obtain various levels of complexity based on the form of the assumptions E. Our main results are: for *- continuous Kleene algebra, ffl if E contains only commutativity assumptions pq = qp, the problem is \Pi 0 1 -complete; ffl if E contains only monoid equations, the problem is \Pi 0 2 -complete; ffl for arbitrary equations E, the problem is \Pi 1 1 - complete. The last problem is the universal Horn theory of the *-continuous Kleene algebras. This resolves an open question of Kozen (1994). 1 Introduction Kleene algebra (KA) is fundamental and ubiquitous in computer science. Since its invention by Kleene in 1956, it has arisen in various forms in program logic and semantics [17, 28], relational algebra [27, 32], aut...

In previous work we have found it necessary to argue that certain theorems of Kleene algebra hold even when the symbols are interpreted as nonsquare matrices. In this note we define and investigate typed Kleene algebra, a typed version of Kleene algebra in which objects have types s ! t. Although nonsquare matrices are the principal motivation, there are many other useful interpretations: traces, binary relations, Kleene algebra with tests. We give a set of typing rules and show that every expression has a unique most general typing (mgt). Then we prove the following metatheorem that incorporates the abovementioned results for nonsquare matrices as special cases. Call an expression 1-free if it contains only the Kleene algebra operators (binary) +, (unary) + , 0, and \Delta, but no occurrence of 1 or . Then every universal 1-free formula that is a theorem of Kleene algebra is also a theorem of typed Kleene algebra under its most general typing. The metatheorem is false without the...

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