Kleene algebra with domain

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 expressiveness 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.Comment: 40 page

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Monoids with tests and the algebra of possibly non-halting programs

We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, if-then-else and while-do defined in terms of a Boolean algebra of conditions. It has previously been shown that there is no finite axiomatisation of algebras of partial functions under these operations alone, and this holds even if one restricts attention to transformations (representing halting programs) rather than partial functions, and omits while-do from the signature. In the halting case, there is a natural “fix”, which is to allow composition of halting programs with conditions, and then the resulting algebras admit a finite axiomatisation. In the current setting such compositions are not possible, but by extending the notion of if-then-else, we are able to give finite axiomatisations of the resulting algebras of (partial) functions, with while-do in the signature if the state space is assumed finite. The axiomatisations are extended to consider the partial predicate of equality. All algebras considered turn out to be enrichments of the notion of a (one-sided) restriction semigrou

Modal Kleene algebra and applications - a survey

Modal Kleene algebras are Kleene algebras with forward and backward modal operators defined via domain and codomain operations. They provide a concise and convenient algebraic framework that subsumes various other calculi and allows treating quite a variety of areas. We survey the basic theory and some prominent applications. These include, on the system semantics side, Hoare logic and PDL (Propositional Dynamic Logic), wp calculus and predicate transformer semantics, temporal logics and termination analysis of rewrite and state transition systems. On the derivation side we apply the framework to game analysis and greedy-like algorithms

Well structured program equivalence is highly undecidable

We show that strict deterministic propositional dynamic logic with intersection is highly undecidable, solving a problem in the Stanford Encyclopedia of Philosophy. In fact we show something quite a bit stronger. We introduce the construction of program equivalence, which returns the value $\mathsf{T}$ precisely when two given programs are equivalent on halting computations. We show that virtually any variant of propositional dynamic logic has $\Pi_1^1$-hard validity problem if it can express even just the equivalence of well-structured programs with the empty program \texttt{skip}. We also show, in these cases, that the set of propositional statements valid over finite models is not recursively enumerable, so there is not even an axiomatisation for finitely valid propositions.Comment: 8 page

Certification of Compiler Optimizations using Kleene Algebra with Tests

We use Kleene algebra with tests to verify a wide assortment of common 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