We show that Kleene algebra with tests (KAT) subsumes propositional Hoare logic (PHL). Thus the specialized syntax and deductive apparatus of Hoare logic are inessential and can be replaced by simple equational reasoning. In addition, we show that all relationally valid inference rules are derivable in KAT and that deciding the relational validity of such rules is PSPACE-complete.

. 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 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...

We prove Church-Rosser theorems for non-symmetric transitive relations, quasiorderings and equations in Kleene algebra. Proofs are simple, rigorous and general, using solely algebraic properties of the regular operations. They are fixed point-based, induction-free and often amenable to automata. They are mere calculations as opposed to deduction and in particular suited to automation. In the Church-Rosser proofs for the -calculus, the term and algebra part are cleanly separated. In all our considerations, Kleene algebra is an excellent means of abstraction.

Abstract. The mechanisation of proofs for probabilistic systems is particularly challenging due to the verification of real-valued properties that probability entails: experience indicates [12, 4, 11] that there are many difficulties in automating real-number arithmetic in the context of other program features. In this paper we propose a framework for verification of probabilistic distributed systems based on the generalisation of Kleene algebra with tests that has been used as a basis for development of concurrency control in standard programming [7]. We show that verification of real-valued properties in these systems can be considerably simplified, and moreover that there is an interpretation which is susceptible to counterexample search via state exploration, despite the underlying real-number domain. 1

We show that propositional Hoare logic is subsumed by the type calculus of typed Kleene algebra augmented with subtypes and typecasting. Assertions are interpreted as typecast operators. Thus Hoare-style reasoning with partial correctness assertions reduces to typechecking in this system. 1

Abstract. It is well known that finite square matrices over a Kleene algebra again form a Kleene algebra. This is also true for infinite matrices under suitable restrictions. One can use this fact to solve certain infinite systems of inequalities over a Kleene algebra. Automatic systems are a special class of infinite systems that can be viewed as infinite-state automata. Automatic systems can be collapsed using Myhill–Nerode relations in much the same way that finite automata can. The Brzozowski derivative on an algebra of polynomials over a Kleene algebra gives rise to a triangular automatic system that can be solved using these methods. This provides an alternative method for proving the completeness of Kleene algebra. 1

Abstract. Monads are a well-established tool for modelling various computational effects. They form the semantic basis of Moggi’s computational metalanguage, the metalanguage of effects for short, which made its way into modern functional programming in the shape of Haskell’s do-notation. Standard computational idioms call for specific classes of monads that support additional control operations. Here, we introduce Kleene monads, which additionally feature nondeterministic choice and Kleene star, i.e. nondeterministic iteration, and we provide a metalanguage and a sound calculus for Kleene monads, the metalanguage of control and effects, which is the natural joint extension of Kleene algebra and the metalanguage of effects. This provides a framework for studying abstract program equality focussing on iteration and effects. These aspects are known to have decidable equational theories when studied in isolation. However, it is well known that decidability breaks easily; e.g. the Horn theory of continuous Kleene algebras fails to be recursively enumerable. Here, we prove several negative results for the metalanguage of control and effects; in particular, already the equational theory of the unrestricted metalanguage of control and effects over continuous Kleene monads fails to be recursively enumerable. We proceed to identify a fragment of this language which still contains both Kleene algebra and the metalanguage of effects and for which the natural axiomatisation is complete, and indeed the equational theory is decidable. 1