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.

We propose Kleene algebra with domain (KAD), an extension of Kleene algebra with two equational axioms for a domain and a codomain operation, respectively. KAD considerably augments the expressibility of Kleene algebra, in particular for the specification and analysis of state transition systems. We develop the basic calculus, discuss some related theories and present the most important models of KAD. We demonstrate applicability by two examples: First, an algebraic reconstruction of Noethericity and well-foundedness. Second, an algebraic reconstruction of propositional Hoare logic.

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.

Guarded strings are like ordinary strings over a finite alphabet P, except that atoms of the free Boolean algebra on a set of atomic tests B alternate with the symbols of P. The regular sets of guarded strings play the same role in Kleene algebra with tests as the regular sets of ordinary strings do in Kleene algebra. In this paper we develop the elementary theory of finite automata on guarded strings, a generalization of the theory of finite automata on ordinary strings. We give several basic constructions, including determinization, state minimization, and an analog of Kleene’s theorem. We then use these results to verify a conjecture on the complexity of a complete Gentzen-style sequent calculus for partial correctness. We also show that a basic result of the theory of Boolean decision diagrams (BDDs), namely that minimal ordered BDDs are unique, is a special case of the Myhill-Nerode theorem for a class of automata on guarded strings. 1

We develop a coalgebraic theory of Kleene algebra with tests (KAT) along the lines of Rutten (1998) for Kleene algebra (KA) and Chen and Pucella (2003) for a limited version of KAT, resolving some technical issues raised by Chen and Pucella. Our treatment includes a simple definition of the Brzozowski derivative for KAT expressions and an automata-theoretic interpretation involving automata on guarded strings. We also give a complexity analysis, showing that an efficient implementation of coinductive equivalence proofs in this setting is tantamount to a standard automatatheoretic construction. It follows that coinductive equivalence proofs can be generated automatically in PSPACE. This matches the bound of Worthington (2008) for the automatic generation of equational proofs in KAT. 1

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...

Abstract. We describe an implementation of an interactive theorem prover for Kleene algebra with tests (KAT). The system is designed to reflect the natural style of reasoning with KAT that one finds in the literature. We illustrate its use with some examples. 1

We present a complete axiomatization of test algebra ([24, 18, 29]), the two-sorted algebraic variant of Propositional Dynamic Logic (PDL, [21, 7]). The axiomatization consists of adding a finite number of equations to any axiomatization of Kleene algebra ([15, 26, 17, 4]) and algebraic translations of the Segerberg ([27]) axioms for PDL. Kleene algebras are not finitely axiomatizable ([25, 6]), so our result does not give us a finite axiomatization of test algebra: in fact, no finite equational axiomatization exists. We also present a single-sorted version of test algebra, using the notion of dynamic negation ([9, 2, 11]), to which the previous results carry over. 1 Introduction Propositional Dynamic Logic (PDL, [21, 7]) is a widely studied modal logic, capable of reasoning about labeled transition systems (LTSs), and thus about any objects that use LTSs as their models, such as computational processes, the intended domain of application. The logic is set up by simultaneously definin...

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

Kleene Algebra with Tests is an extension of Kleene Algebra, the algebra of regular expressions, which can be used to reason about programs. We develop a coalgebraic theory of Kleene Algebra with Tests, along the lines of the coalgebraic theory of regular expressions based on deterministic automata. Since the known automata-theoretic presentation of Kleene Algebra with Tests does not lend itself to a coalgebraic theory, we define a new interpretation of Kleene Algebra with Tests expressions and a corresponding automatatheoretic presentation. One outcome of the theory is a coinductive proof principle, that can be used to establish equivalence of our Kleene Algebra with Tests expressions.