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

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

Algebraic Principles for Rely-Guarantee Style Concurrency Verification Tools

We provide simple equational principles for deriving rely-guarantee-style inference rules and refinement laws based on idempotent semirings. We link the algebraic layer with concrete models of programs based on languages and execution traces. We have implemented the approach in Isabelle/HOL as a lightweight concurrency verification tool that supports reasoning about the control and data flow of concurrent programs with shared variables at different levels of abstraction. This is illustrated on two simple verification examples

Straight-line instruction sequence completeness for total calculation on cancellation meadows

A combination of program algebra with the theory of meadows is designed leading to a theory of computation in algebraic structures which use in addition to a zero test and copying instructions the instruction set $\{x \Leftarrow 0, x \Leftarrow 1, x\Leftarrow -x, x\Leftarrow x^{-1}, x\Leftarrow x+y, x\Leftarrow x\cdot y\}$. It is proven that total functions on cancellation meadows can be computed by straight-line programs using at most 5 auxiliary variables. A similar result is obtained for signed meadows.Comment: 24 page