Abstract Variants of Kleene algebra have been used to provide foundations of reasoning about programs, for instance by representing Hoare Logic (HL) in algebra. That work has generally emphasised program correctness, i.e., proving the absence of bugs. Recently, Incorrectness Logic (IL) has been advanced as a formalism for the dual problem: proving the presence of bugs. IL is intended to underpin the use of logic in program testing and static bug finding. Here, we use a Kleene algebra with diamond operators and countable joins of tests, which embeds IL, and which also is complete for reasoning about the image of the embedding. Next to embedding IL, the algebra is able to embed HL, and allows making connections between IL and HL specifications. In this sense, it unifies correctness and incorrectness reasoning in one formalism.

The aim of algebraic logic is to compact series of small steps of general logical inference into larger (in)equational steps. Algebraic structures that have proved very useful in this context are modal semirings and modal Kleene algebras. We show that they can also model knowledge and belief logics as well as games without additional effort; many of the standard logical properties are theorems rather than axioms in this setting. As examples of the first area, we treat the classical puzzles of the Wise Men and the Muddy Children. Moreover, we show possibilities of handling knowledge update and revision algebraically. For the area of games, we generalize the well-known connection between game logic and dynamic logic to the setting of modal semirings and link it to predicate transformer semantics, in particular to demonic refinement algebra. We think that our study provides evidence that modal semirings are capable of handling a wide variety of (multi-)modal logics in a uniform algebraic fashion.

Abstract Around 1960, Dijkstra, Floyd and Warshall published papers on algorithms for solving single-source and all-sources shortest path problems, respectively. These algorithms, nowadays named after their inventors, are well known and well established. This paper sheds an algebraic light on these algorithms. We combine the shortest path problems with Kleene algebra, also known as Conway’s regular algebra. This view yields a purely algebraic version of Dijkstra’s shortest path algorithm and the one by Floyd/Warshall. Moreover, the algebraic abstraction yields applications of these algorithms to structures different from graphs and pinpoints the mathematical requirements on the underlying cost algebra that ensure their correctness.

Embedded systems such as automotive systems are very complex to specify. Since it is difficult to capture all their requirements or their design in one single model, approaches working with several system views are adopted. The main problem there is to keep these views coherent; the issue is known as view reconciliation. This paper proposes an algebraic solution. It uses sets of integration constraints that link (families of) system features in one view to other (families of) features in the same or a different view. Both, families and constraints, are formalised using a feature algebra. Besides presenting a constraint relation and its mathematical properties, the paper shows in several examples the suitability of this approach for a wide class of integration constraint formulations.

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.

Formal program construction by transformations is a method of software development in which a program is derived from a formal problem specification by manageable, controlled transformation steps which guarantee that the final product meets the initial specification. This methodology has been investigated in the Munich project CIP (computer-aided intuition-guided programming). The research includes the design of a wide-spectrum language specifically tailored to the needs of transformational programming, the construction of a transformation system to support the methodology, and the study of transformation rules and other methodological issues. Particular emphasis has been laid on developing a sound theoretical basis for the overall approach.>