Georg Struth

Kleene algebra with domain

We propose Kleene algebra with domain (KAD), an extension of Kleene algebra by simple equational axioms for a domain and a codomain operation. KAD considerably augments the expressiveness of Kleene algebra, in particular for the specification and analysis of programs and state transition systems. We develop the basic calculus, present the most interesting models and discuss some related theories. We demonstrate applicability by two examples: algebraic reconstructions of Noethericity and propositional Hoare logic based on equational reasoning.

Concurrent Kleene Algebra and its Foundations

A Concurrent Kleene Algebra oers two composition operators, related by a weak version of an exchange law: when applied in a trace model of program semantics, one of them stands for sequential execution and the other for concurrent execution of program components [22]. After introducing this motivating concrete application, we investigate its abstract background in terms of a primitive independence relation between the traces. On this basis, we develop a series of richer algebras; the richest validates a proof calculus for programs similar to that of a Jones style rely/guarantee calculus. On the basis of this abstract algebra, we nally reconstruct the original trace model, using the notion of atoms from lattice theory.

Algebras of modal operators and partial correctness

Modal Kleene algebras are Kleene algebras enriched by forward and backward box and diamond operators. We formalise the symmetries of these operators as Galois connections, complementarities and dualities. We study their properties in the associated operator algebras and show that the axioms of relation algebra are theorems at the operator level. Modal Kleene algebras provide a unifying semantics for various program calculi and enhance efficient cross-theory reasoning in this class, often in a very concise pointfree style. This claim is supported by novel algebraic soundness and completeness proofs for Hoare logic and by connecting this formalism with an algebraic decision procedure.

Concurrent Kleene Algebra

A concurrent Kleene algebra offers, next to choice and iteration, operators for sequential and concurrent composition, related by an inequational form of the exchange law. We show applicability of the algebra to a partially-ordered trace model of program execution semantics and demonstrate its usefulness by validating familiar proof rules for sequential programs (Hoare triples) and for concurrent ones (Joness rely/guarantee calculus). This involves an algebraic notion of invariants; for these the exchange inequation strengthens to an equational distributivity law. Most of our reasoning has been checked by computer.

Relational and Kleene-Algebraic Methods in Computer Science

Invited Papers.- Choice Procedures in Pairwise Comparison Multiple-Attribute Decision Making Methods.- Kleene Algebra with Relations.- Contributed Papers.- Integrating Model Checking and Theorem Proving for Relational Reasoning.- Fixed-Point Characterisation of Winning Strategies in Impartial Games.- Checking the Shape Safety of Pointer Manipulations.- Applying Relational Algebra in 3D Graphical Software Design.- Investigating Discrete Controllability with Kleene Algebra.- Tracing Relations Probabilistically.- Pointer Kleene Algebra.- Kleene Modules.- The Categories of Kleene Algebras, Action Algebras and Action Lattices Are Related by Adjunctions.- Towards a Formalisation of Relational Database Theory in Constructive Type Theory.- SCAN Is Complete for All Sahlqvist Formulae.- Relations and GUHA-Style Data Mining II.- A Note on Complex Algebras of Semigroups.- Calculational Relation-Algebraic Proofs in Isabelle/Isar.- A Calculus of Typed Relations.- Greedy-Like Algorithms in Modal Kleene Algebra.- Rasiowa-Sikorski Style Relational Elementary Set Theory.- Relational Data Analysis.- Two Proof Systems for Peirce Algebras.- An Institution Isomorphism for Planar Graph Colouring.- Decomposing Relations into Orderings.

wp Is wlp

Using only a simple transition relation one cannot model commands that may or may not terminate in a given state. In a more general approach commands are relations enriched with termination vectors. We reconstruct this model in modal Kleene algebra. This links the recursive definition of the do od loop with a combination of the Kleene star and a convergence operator. Moreover, the standard wp operator coincides with the wlp operator in the modal Kleene algebra of commands. Therefore our earlier general soundness and relative completeness proof for Hoare logic in modal Kleene algebra can be re-used for wp. Although the definition of the loop semantics is motivated via the standard Egli-Milner ordering, the actual construction does not depend on Egli-Milner-isotony of the constructs involved.3. Substance is volatile: YES NO 4. Procedure will be performed at Standard Temperature and Pressure (STP) YES NO 5. Protocol: Provide a brief description of the procedure (for routine procedures, it is appropriate to state “triphosphates for DNA sequencing & transcription” or “orthophosphates for cell labeling” or “RIA kits.” Also include information on chemical and physical form generated, any special equipment used to handle, shield or contain the radioactivity, and unusual hazards associated with the procedure.

Internal axioms for domain semirings

New axioms for domain operations on semirings and Kleene algebras are proposed. They generalise the relational notion of domain-the set of all states that are related to some other state-to a wide range of models. They are internal since the algebras of state spaces are induced by the domain axioms. They are simpler and conceptually more appealing than previous two-sorted external approaches in which the domain algebra is determined through typing. They lead to a simple and natural algebraic approach to modal logics based on equational reasoning. The axiomatisations have been developed in a new style of computer-enhanced mathematics by automated theorem proving, and the approach itself is suitable for automated systems analysis and verification. This is demonstrated by a fully automated proof of a modal correspondence result for Lobs formula that has applications in termination analysis.

Termination in Modal Kleene Algebra

Modal Kleene algebras (MKAs) are Kleene algebras with forward and backward modal operators defined via domain and codomain operations. The paper formalizes and compares different notions of termination, including Lob’s formula, in MKA. It studies exhaustive iteration and gives calculational proofs of two fundamental termination-dependent statements from rewriting theory: the well-founded union theorem by Bachmair and Dershowitz and Newman’s lemma. These results are also of general interest for the termination analysis of programs and state transition systems.

Automating algebraic methods in isabelle

We implement a large Isabelle/HOL repository of algebras for application in modelling computing systems. They subsume computational logics such as dynamic and Hoare logics and form a basis for various software development methods. Isabelle has recently been extended by automated theorem provers and SMT solvers. We use these integrated tools for automatically proving several rather intricate refinement and termination theorems. We also automate a modal correspondence result and soundness and relative completeness proofs of propositional Hoare logic. These results show, for the first time, that Isabelles tool integration makes automated algebraic reasoning particularly simple. This is a step towards increasing the automation of formal methods.

On locality and the exchange law for concurrent processes

This paper studies algebraic models for concurrency, in light of recent work on Concurrent Kleene Algebra and Separation Logic. It establishes a strong connection between the Concurrency and Frame Rules of Separation Logic and a variant of the exchange law of Category Theory. We investigate two standard models: one uses sets of traces, and the other is state-based, using assertions and weakest preconditions. We relate the latter to standard models of the heap as a partial function. We exploit the power of algebra to unify models and classify their variations.