A doctrinal approach to modal/temporal Heyting logic and non-determinism in processes

The study of algebraic modelling of labelled non-deterministic concurrent processes leads us to consider a category LB , obtained from a complete meet-semilattice B and from B-valued equivalence relations. We prove that, if B has enough properties, then LB presents a two-fold internal logical structure, induced by two doctrines definable on it: one related to its families of subobjects and one to its families of regular subobjects. The first doctrine is Heyting and makes LB a Heyting category, the second one is Boolean. We will see that the difference between these two logical structures, namely the different behaviour of the negation operator, can be interpreted in terms of a distinction between non-deterministic and deterministic behaviours of agents able to perform computations in the context of the same process. Moreover, the sorted first-order logic naturally associated with LB can be extended to a modal/temporal logic, again using the doctrinal setting. Relations are also drawn to other computational model

Coalgebras, braidings, and distributive laws

We show, for a monad T, that coalgebra structures on a T-algebra can be described in terms of ‘braidings’, provided that the monad is equipped with an invertible distributive law satisfying the Yang-Baxter equation

Tree morphisms and bisimulations

A category of (action labelled) trees is defined that can be used to model unfolding of labelled transition systems and to study behavioural relations over them. In this paper we study five different equivalences based on bisimulation for our model. One, that we called resource bisimulation, amounts essentially to three isomorphism. Another, its weak counterpart, permits abstracting from silent actions while preserving the tree structure. The other three are the well known strong, branching and weak bisimulation equivalence. For all bisimulations, but weak, canonical representatives are constructed and it is shown that they can be obtained via enriched functors over our categories of trees, with and without silent actions. Weak equivalence is more problematic; a canonical minimal representative for it cannot be denned by quotienting our trees. The common framework helps in understanding the relationships between the various equivalences and the results provide support to the claim that branching bisimulation is the natural generalization of strong bisimulation to systems with silent moves and that resource and weak resource have an interest of their own

Innocent strategies as presheaves and interactive equivalences for CCS

Seeking a general framework for reasoning about and comparing programming languages, we derive a new view of Milner's CCS. We construct a category E of plays, and a subcategory V of views. We argue that presheaves on V adequately represent innocent strategies, in the sense of game semantics. We then equip innocent strategies with a simple notion of interaction. This results in an interpretation of CCS. Based on this, we propose a notion of interactive equivalence for innocent strategies, which is close in spirit to Beffara's interpretation of testing equivalences in concurrency theory. In this framework we prove that the analogues of fair and must testing equivalences coincide, while they differ in the standard setting.Comment: In Proceedings ICE 2011, arXiv:1108.014

A Relational Model of incomplete Data without nulls

The theoretical study of the relational model of data is ongoing and highly developed. Yet the vast majority of real databases include incomplete data, and the incomplete data is widely modelled using special values called {\em nulls}. As noted many times by Date and others, the inclusion of special values is not compatible with the relational model and invalidates many of the theoretical results. In category theoretic applications to computer science, partial functions are frequently modelled by using a special value approach (the {\em partial map classifier}), or by explicit reference to the {\em domain of definition subobject}. In a former edition of the CATS conference the first author and his colleague Rosebrugh proved a Morita equivalence theorem showing that for database modelling the two approaches are equivalent, {\em provided} the domain of definition subobject is complemented. In this paper we study the uncomplemented domain of definition approach (which is {\em not} equivalent to using special values). Our main results show that using uncomplemented domains of definition to model incomplete data is entirely compatible with the relational model and so leaves the well-developed theory applicable to real databases that use this approach. Furthermore, using uncomplemented domains of definition supports in-place updating, in stark contrast to special values, and, in a wide variety of circumstances, ensures the existence of cartesian and op-cartesian models which, as shown in a recent TCS article, are important for solving view update problems

Observational trees as models for concurrency

Given an automaton, its behaviour can be modelled as the sets of strings over an alphabet A that can be accepted from any of its states. When considering concurrent systems, we can see a concurrent agent as an automaton, where non-determinism derives from the fact that its states can offer a different behaviour at different moments in time. Non-deterministic computations between a pair of states can then no longer be described as a ‘set’ of strings in a free monoid. Consequently, between two states we will have a labelled structured set of computations, where the structure describes the possibility of two computations parting from each other while maintaining the same observable steps. In this paper, we shall consider different kinds of observation domains and related structured sets of computations. Structured sets of computations will be organised as a category of generalised trees built over a meet-semilattice monoid formalizing the observation domain. Theorems allowing us to introduce the usual concurrency operators in the models and relating different models will then be obtained by first considering ordinary functors (on and between the observation domains), and then lifting them to the categories of structured sets of computations

The topos of continuous trees as a model for distributed calculi

Defining the topos of continuous time processes and its logi