Catálogo de publicaciones - libros

The Kleene algebra axioms are too strong for some program models of interest (e.g. models that mix demonic choice with angelic or probabilistic choice). This has led to proposals that weaken the right distributivity axiom to monotonicity, and possibly weaken or eliminate the right induction and left annihilation axioms (e.g. lazy Kleene algebra, probabilistic Kleene algebra, monodic tree Kleene algebra, etc.). We’ll address some of the basic metatheoretic properties of these theories using rational trees modulo simulation equivalence.

The class of finite symmetric integral relation algebras with no 3-cycles is a particularly interesting and easily analyzable class of finite relation algebras. For example, it contains algebras that are not representable, algebras that are representable only on finite sets, algebras that are representable only on infinite sets, algebras that are representable on both finite and infinite sets, and there is an algorithm for determining which case holds.

We explore the view of a computation as a relational section of a (trivial) fibre bundle: initial states lie in the base of the bundle and final states lie in the fibres located at their initial states. This leads us to represent a computation in ‘fibre-form’ as the angelic choice of its behaviours from each initial state. That view is shown to have the advantage also of permitting final states to be of different types, as might be used for example in a semantics of probabilistic computations, and of providing a natural setting for refinement of computations.

However we apply that view in a different direction. On computations more general than code the two standard models, the relational and the predicate-transformer models, obey different laws. One way to understand that difference is to study the laws of more refined models, like the semantics of probabilistic computations. Another is to characterise each model by its laws. In spite of their differences, the relational model is embedded in the transformer model by a Galois connection which can be used to transfer much of the structure on transformers to the relational model. We investigate the extent to which the conjugate on predicate transformers translates to relations and use the result to motivate a characterisation of relational computations, achieved by using fibre-forms.

The formal analysis of programs with arrays is a notoriously difficult problem due largely to aliasing considerations. In this paper we augment the rules of Kleene algebra with tests (KAT) with rules for the equational manipulation of arrays in the style of schematic KAT. These rules capture and make explicit the essence of , where two array accesses can be to the same element. We prove the soundness of our rules, as well as illustrate their usefulness with several examples, including a complete proof of the correctness of heapsort.

Most previous work on the semantics of programs with local state involves complex storage modeling with pointers and memory cells, complicated categorical constructions, or reasoning in the presence of context. In this paper, we explore the extent to which relational semantics and axiomatic reasoning in the style of Kleene algebra can be used to avoid these complications. We provide (i) a fully compositional relational semantics for a first-order programming language with a construct for local variable scoping; and (ii) an equational proof system based on Kleene algebra with tests for proving equivalence of programs in this language. We show that the proof system is sound and complete relative to the underlying equational theory without local scoping. We illustrate the use of the system with several examples.

We model groups as relational systems and develop relation-algebraic specifications for direct products of groups, quotient groups, and the enumeration of all subgroups and normal subgroups. The latter two specifications immediately lead to specifications of the lattices of subgroups and normal subgroups, respectively. All specifications are algorithmic and can directly be translated into the language of the computer system . Hence, the system can be used for constructing groups and for computing and visualizing their lattices of subgroups and normal subgroups. This is demonstrated by some examples.

Pratt [22] defines action algebras as Kleene algebras with residuals. In [9] it is shown that the equational theory of *-continuous action algebras (lattices) is Π–complete. Here we show that the equational theory of relational action algebras (lattices) is Π –hard, and some its fragments are Π–complete. We also show that the equational theory of action algebras (lattices) of regular languages is Π–complete.

We first recall the concept of Kleene algebra with domain (KAD). Then we explain how to use the operators of KAD to define a demonic refinement ordering and demonic operators (many of these definitions come from the literature). Then, taking the properties of the KAD-based demonic operators as a guideline, we axiomatise an algebra that we call (DAD). The laws of DAD not concerning the domain operator agree with those given in the 1987 CACM paper by Hoare et al. Finally, we investigate the relationship between demonic algebras with domain and KAD-based demonic algebras. The question is whether every DAD is isomorphic to a KAD-based demonic algebra. We show that it is not the case in general. However, if a DAD is isomorphic to a demonic algebra based on a KAD , then it is possible to construct a KAD isomorphic to using the operators of . We also describe a few open problems.

The theory of Boolean contact algebras has been used to represent a region based theory of space. Some of the primitives of Boolean algebras are not well motivated in that context. One possible generalization is to drop the notion of complement, thereby weakening the algebraic structure from Boolean algebra to distributive lattice. The main goal of this paper is to investigate the representation theory of that weaker notion, i.e., whether it is still possible to represent each abstract algebra by a substructure of the regular closed sets of a suitable topological space with the standard Whiteheadean contact relation.

We give a brief overview of the axiomatic development of betweenness relations, and investigate the connections between these and comparability graphs. Furthermore, we characterize betweenness relations induced by reflexive and antisymmetric binary relations, thus generalizing earlier results on partial orders. We conclude with a sketch of the algorithmic aspects of recognizing induced betweenness relations.