Automated Reasoning in Kleene Algebra

Automated Reasoning

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Abstract. Relation algebras provide abstract equational axioms for the calculus of binary relations. They name an established area of mathematics with various applications in computer science. We prove more than hundred theorems of relation algebras with off-the-shelf automated theorem provers. This yields a basic calculus from which more advance applications can be explored. Here, we present two examples from the formal methods literature. Our experiments not only further underline the feasibility of automated deduction in complex algebraic structures and provide theorem proving benchmarks, they also pave the way for lifting established formal methods such as B or Z to a new level of automation.

Section: Resultssupporting

confidence: 63%

On Automating the Calculus of Relations

Höfner

Automated Reasoning

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“…This impressively demonstrates the power of Isabelle's proof automation. Previous experience in theorem proving with algebra shows that the level of proof automation in algebra is often very high [15,13,12]. In this regard, our present proof experience is slightly underwhelming, as custom tactics and low-level proof techniques were needed for our step-by-step proof reconstruction.…”

Section: Verification Of Flowchart Equivalencementioning

confidence: 86%

“…Reasoning in Kleene algebras is based on first-order equational logic. It is therefore relatively simple, concise and well suited for automation [15,12,13,3]. The lightweight program semantics that Kleene algebras provide can further be specialised in various ways through their models, which include binary relations, program traces, paths in transition systems and (guarded string) languages [4].…”

Section: Introductionmentioning

confidence: 99%

Program Analysis and Verification Based on Kleene Algebra in Isabelle/HOL

Armstrong

Interactive Theorem Proving

Weber

2013

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Abstract. Schematic Kleene algebra with tests (SKAT) supports the equational verification of flowchart scheme equivalence and captures simple while-programs with assignment statements. We formalise SKAT in Isabelle/HOL, using the quotient type package to reason equationally in this algebra. We apply this formalisation to a complex flowchart transformation proof from the literature. We extend SKAT with assertion statements and derive the inference rules of Hoare logic. We apply this extension in simple program verification examples and the derivation of additional Hoare-style rules. This shows that algebra can provide an abstract semantic layer from which different program analysis and verification tasks can be implemented in a simple lightweight way.

“…Previous experiments with more than 500 theorems in Kleene algebras (e.g. [14,17]) suggest that an equational encoding is usually sufficient for finding ATP proofs of simple theorems. However, more complex theorems often succeed only with the inequational encoding.…”

Section: Formulas(sos)mentioning

confidence: 99%

“…Recent work already investigated the automation of computational logics [14], of relational reasoning [17], of rewriting and termination analysis [32,33], and of hybrid system analysis [13]. It has also been shown that ATP can be very helpful in developing consistent irredundant specifications and axiomatisations of algebraic theories [11,10].…”

mentioning

confidence: 99%

Automated verification of refinement laws

Höfner

Sutcliffe

2009

Ann Math Artif Intell

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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.