2006 was a special year for both mathematical logic and computer science, as it celebrated Gödel's centenary. Although Gödel's work was mainly concerned with mathematics and metamathematics, the crucial role it had in the foundation of modern theoretical computer science is undeniable: for instance, one only has to remember Gödel's contributions to the birth of recursion theory as well as his part in the debate in the nineteen thirties on the subject of the Church Thesis.

This paper studies the equational theory of bisimulation equivalence over the process algebra BCCSP extended with the priority operator of Baeten, Bergstra and Klop. We prove that, in the presence of an infinite set of actions, bisimulation equivalence has no finite, sound, ground-complete equational axiomatisation over that language. This negative result applies even if the syntax is extended with an arbitrary collection of auxiliary operators, and motivates the study of axiomatisations using equations with action predicates as conditions. In the presence of an infinite set of actions, it is shown that, in general, bisimulation equivalence has no finite, sound, ground-complete axiomatisation consisting of equations with action predicates as conditions over the language studied in this paper. Finally, sufficient conditions on the priority structure over actions are identified that lead to a finite, ground-complete axiomatisation of bisimulation equivalence using equations with action predicates as conditions.

Behavioural equivalences are a means of establishing whether computing systems possess the same properties. The specific set of properties that are preserved by a specific behavioural equivalence clearly depends on how the system behaviour is observed and can usually be characterised by means of a modal logic. In this paper we consider three different approaches to the definition of behavioural equivalences – bisimulation, testing and trace – applied to three different classes of systems – non-deterministic, probabilistic and Markovian – and we survey the nine resulting modal logic characterisations, each of which stems from the Hennessy–Milner logic. We then compare the nine characterisations with respect to the logical operators, in order to emphasise the differences between the three approaches in the definition of behavioural equivalences and the regularities within each of them. In the probabilistic and Markovian cases we also address the issue of whether the probabilistic and temporal aspects should be treated in a local or global way and consequently whether the modal logic interpretation should be qualitative or quantitative.

In recent decades, qualitative approaches to probabilistic uncertainty have received more and more attention. We propose a characterisation of partial preference orders through a uniform axiomatic treatment of a variety of qualitative uncertainty notions. To this end, we prove a representation result that connects qualitative notions of partial uncertainty to their numerical counterparts. We describe an executable specification, in the declarative framework of Answer Set Programming, that constitutes the core engine for qualitative management of uncertainty. Some basic reasoning tasks are also identified.

We investigate cut elimination in propositional substructural logics. The problem is to decide whether a given calculus admits (reductive) cut elimination. We show that for commutative single-conclusion sequent calculi containing generalised knotted structural rules and arbitrary logical rules the problem can be decided by resolution-based methods. A general cut-elimination proof for these calculi is also provided.

Action-labelled transition systems (LTSs) have proved to be a fundamental model for describing and proving properties of concurrent systems. In this paper we introduce Multiple-Labelled Transition Systems (MLTSs) as generalisations of LTSs that enable us to deal with system features that are becoming increasingly important when considering languages and models for network-aware programming. MLTSs enable us to describe not only the actions that systems can perform but also their usage of resources and their handling (creation, revelation . . .) of names; these are essential for modelling changing evaluation environments. We also introduce MoMo, which is a logic inspired by Hennessy–Milner Logic and the μ-calculus, that enables us to consider state properties in a distributed environment and the impact of actions and movements over the different sites. MoMo operators are interpreted over MLTSs and both MLTSs and MoMo are used to provide a semantic framework to describe two basic calculi for mobile computing, namely μKlaim and the asynchronous π-calculus.

This paper proposes a new relevant logic B+⊓⊔, which is obtained by adding two binary connectives, intensional conjunction ⊓ and intensional disjunction ⊔, to Meyer–Routley minimal positive relevant logic B+, where ⊓ and ⊔ are weaker than fusion ˚ and fission +, respectively. We give Kripke-style semantics for B+⊓⊔, with →, ⊓ and ⊔ modelled by ternary relations. We prove the soundness and completeness of the proposed semantics. A number of axiomatic extensions of B+⊓⊔, including negation-extensions, are also considered, together with the corresponding semantic conditions required for soundness and completeness to be maintained.

The Symmetric Circulant Travelling Salesman Problem asks for the minimum cost tour in a symmetric circulant matrix. The computational complexity of this problem is not known – only upper and lower bounds have been determined. This paper provides a characterisation of the two-stripe case. Instances where the minimum cost of a tour is equal to either the upper or lower bound are recognised. A new construction providing a tour is proposed for the remaining instances, and this leads to a new upper bound that is closer than the previous one.

This paper is the journal version of Lenzi (2006). We consider the positive set theory with equality described these and propose a candidate model.

This paper presents an analysis of definitions and decidability for exponential functions on various matrix algebras. The main idea is to show that, generically, the entries of the exponential (or logarithm) of a matrix are Pfaffian functions of the entries of the matrix.

The density matrices of graphs are combinatorial laplacians normalised to have trace one (Braunstein et al. 2006b). If the vertices of a graph are arranged as an array, its density matrix carries a block structure with respect to which properties such as separability can be considered. We prove that the so-called degree-criterion, which was conjectured to be necessary and sufficient for the separability of density matrices of graphs, is equivalent to the PPT-criterion. As such, it is not sufficient for testing the separability of density matrices of graphs (we provide an explicit example). Nonetheless, we prove the sufficiency when one of the array dimensions has length two (see Wu (2006) for an alternative proof). Finally, we derive a rational upper bound on the concurrence of density matrices of graphs and show that this bound is exact for graphs on four vertices.