Automated verification of refinement laws

Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs

Embedding Kozen-Tiuryn Logic into Residuated One-Sorted Kleene Algebra with Tests

Kozen and Tiuryn have introduced the substructural logic $\mathsf{S}$ for reasoning about correctness of while programs (ACM TOCL, 2003). The logic $\mathsf{S}$ distinguishes between tests and partial correctness assertions, representing the latter by special implicational formulas. Kozen and Tiuryn's logic extends Kleene altebra with tests, where partial correctness assertions are represented by equations, not terms. Kleene algebra with codomain, $\mathsf{KAC}$, is a one-sorted alternative to Kleene algebra with tests that expands Kleene algebra with an operator that allows to construct a Boolean subalgebra of tests. In this paper we show that Kozen and Tiuryn's logic embeds into the equational theory of the expansion of $\mathsf{KAC}$ with residuals of Kleene algebra multiplication and the upper adjoint of the codomain operator

Domain and Antidomain Semigroups

Abstract. We axiomatise and study operations for relational domain and antidomain on semigroups and monoids. We relate this approach with previous axiomatisations for semirings, partial transformation semi-groups and dynamic predicate logic.

Unified Foundations of Team Semantics via Semirings

Semiring semantics for first-order logic provides a way to trace how facts represented by a model are used to deduce satisfaction of a formula. Team semantics is a framework for studying logics of dependence and independence in diverse contexts such as databases, quantum mechanics, and statistics by extending first-order logic with atoms that describe dependencies between variables. Combining these two, we propose a unifying approach for analysing the concepts of dependence and independence via a novel semiring team semantics, which subsumes all the previously considered variants for first-order team semantics. In particular, we study the preservation of satisfaction of dependencies and formulae between different semirings. In addition we create links to reasoning tasks such as provenance, counting, and repairs

Kleene Algebra with Dynamic Tests: Completeness and Complexity

We study versions of Kleene algebra with dynamic tests, that is, extensions of Kleene algebra with domain and antidomain operators. We show that Kleene algebras with tests and Propositional dynamic logic correspond to special cases of the dynamic test framework. In particular, we establish completeness results with respect to relational models and guarded-language models, and we show that two prominent classes of Kleene algebras with dynamic tests have an EXPTIME-complete equational theory

Minimisation in Logical Form

Stone-type dualities provide a powerful mathematical framework for studying properties of logical systems. They have recently been fruitfully explored in understanding minimisation of various types of automata. In Bezhanishvili et al. (2012), a dual equivalence between a category of coalgebras and a category of algebras was used to explain minimisation. The algebraic semantics is dual to a coalgebraic semantics in which logical equivalence coincides with trace equivalence. It follows that maximal quotients of coalgebras correspond to minimal subobjects of algebras. Examples include partially observable deterministic finite automata, linear weighted automata viewed as coalgebras over finite-dimensional vector spaces, and belief automata, which are coalgebras on compact Hausdorff spaces. In Bonchi et al. (2014), Brzozowski's double-reversal minimisation algorithm for deterministic finite automata was described categorically and its correctness explained via the duality between reachability and observability. This work includes generalisations of Brzozowski's algorithm to Moore and weighted automata over commutative semirings. In this paper we propose a general categorical framework within which such minimisation algorithms can be understood. The goal is to provide a unifying perspective based on duality. Our framework consists of a stack of three interconnected adjunctions: a base dual adjunction that can be lifted to a dual adjunction between coalgebras and algebras and also to a dual adjunction between automata. The approach provides an abstract understanding of reachability and observability. We illustrate the general framework on range of concrete examples, including deterministic Kripke frames, weighted automata, topological automata (belief automata), and alternating automata