Relations and Kleene algebras in computer science

Rudolf Berghammer, Bernhard Möller, Georg Struth
2010 The Journal of Logic and Algebraic Programming  
Editorial Relations and Kleene algebras in computer science Relation algebras, Kleene algebras and related algebraic approaches are concerned with the abstraction and compaction of frequently occurring patterns of certain sets of formulas into a form amenable to simple algebraic manipulation, that is, (in)equational reasoning, and hence also to mechanised theorem proving. Among others, they cover the fundamental computer science concepts of choice, sequential composition and (in)finite
more » ... . These structures have received increasing interest over the last decade, notably through many joint conferences on Relational Methods in Computer Science (RelMiCS) and Applications of Kleene Algebra (AKA). Two special issues of JLAP devoted to these topics appeared in 2006 and 2008. This series is now continued with the present issue. Its seven research contributions underwent a thorough two round refereeing process. In the first round, a programme committee chose 28 technical contributions for the joint . In the second round, selected and substantially revised papers were reviewed again for this journal by new referees. The papers are cover both theory and application and reflect an interesting cross-section of current work in the field. In Boolean logics with relations, Philippe Balbiani and Tinko Tinchev introduce a Boolean language to which relation symbols have been added. This is useful for describing relational and algebraic structures. The paper introduces the concepts of Kripkean and Boolean semantics for that language and addresses the traditional issues of decidability and complexity; moreover it defines the new concepts of weak and strong canonicity. The paper Relation-algebraic specification and solution of special university timetabling problems by Rudolf Berghammer and Britta Kehden tackles a concrete practical problem. It was posed by a university administration and concerns the construction of a timetable with an even distribution of certain mandatory courses. Two relation-algebraic models and corresponding algorithmic solutions are developed; one of them is directly implementable in the Kiel RelView tool. Another paper on the foundational side is Abstract representation theorems for demonic refinement algebras by Jean-Lou De Carufel and Jules Desharnais. Its main result is that every demonic refinement algebra with enabledness and termination, as defined in earlier RelMiCS/AKA papers, is isomorphic to an algebra of ordered pairs of elements of a Kleene algebra with domain and with a divergence operator satisfying a mild condition. In addition, it is shown that every demonic refinement algebra with enabledness is also a demonic refinement algebra with termination. In Imperative abstractions for functional actions Walter Guttmann studies an application of algebraic concepts to the semantics of programming languages. It elaborates on an earlier relational model of non-strict, imperative computations by the same author. The resulting theory also supports infinite data structures and covers concepts such as procedures, parameters, partial application, algebraic data types, pattern matching and list comprehension. Moreover, a relational treatment of programming patterns -such as fold, unfold and divide-and-conquer -is given, including proofs of functional programming laws like fold-map fusion. The approach is validated by a number of examples. The paper Algebraic notions of nontermination: omega and divergence in idempotent semirings by Peter Höfner and Georg Struth is again on the foundational side. It compares two notions of nontermination in the setting of idempotent semirings. Their behaviour in various computational models is determined, and conditions for their existence and their coincidence are given. It confirms and deepens earlier results that the approach using divergence yields a simple and natural way of modelling infinite behaviours of programs and discrete systems, whereas the omega operator exhibits some anomalies. Next, Determinisation of relational substitutions in ordered categories with domain by Wolfram Kahl applies algebraic methods within the realm of term rewriting and unification. Rather than describing unification in terms of functional substitutions 1567-8326/$ -see front matter
doi:10.1016/j.jlap.2010.07.011 fatcat:lu2ocemoafcfngd5lrq7256hxy