Symmetry between the construction of programs and generating their behaviour can be properly formulated in terms of category theory by using significant property of duality. On one hand, this consists of the algebraic construction of static data structures and relations between them (programs) and on the other hand it consists of the coalgebraic generating of dynamic behavioural structures of p...

In this paper we present methods of transition from one perspective on logic to others, and apply this in particular to obtain a coalgebraic presentation of logic. The central ingredient in this process is to view consequence relations as morphisms in a category.

We present the concept of a disjunctive basis as a generic framework for normal forms in modal logic based on coalgebra. Disjunctive bases were defined in previous work on completeness for modal fixpoint logics, where they played a central role in the proof of a generic completeness theorem for coalgebraic mu-calculi. Believing the concept has a much wider significance, here we investigate it m...

Building upon Vaughan Pratt's work on applications of Chu space theory to Stone duality, we develop a general categorical dualities the basis and closure conditions, which encompasses variety for topological spaces, convex measurable spaces (some are new duality results their own). It works as method generate analogues between frames (locales) beyond topology, e.g., spaces. After establishing t...

The purpose of this dissertation is to further previous work on coalgebras as infinite statebased transition systems and their logical characterisation with particular focus on infinite regular behaviour and branching. Finite trace semantics is well understood [DR95] for nondeterministic labelled transition systems, and has recently [Jac04, HJS06] been generalised to a coalgebraic level where m...

In the setting of enriched category theory, we describe dual adjunctions of the form L R : Spa −→ Alg between the dual of the category Spa of “spaces” and the category Alg of “algebras” that arise from a schizophrenic object , which is both an “algebra” and a “space”. We call such adjunctions logical connections. We prove that the exact nature of is that of a module that allows to lift optimall...

Timed automata, and machines alike, currently lack a general mathematical characterisation. In this paper we provide a uniform coalgebraic understanding of these devices. This framework encompasses known behavioural equivalences for timed automata and paves the way for the extension of these notions to new timed behaviours and for the instantiation of established results from the coalgebraic th...

The adjunction of a unit to an algebraic structure with a given binary associative operation is discussed by interpreting such structures as semigroups and monoids respectively in a monoidal category. This approach then allows for results on the adjunction of counits to coalgebraic structures with a binary co-associative co-operation as well. Special attention is paid to situations where a give...

The operational semantics of interactive systems is usually described by labeled transition systems. Abstract semantics (that is defined in terms of bisimilarity) is characterized by the final morphism in some category of coalgebras. Since the behaviour of interactive systems is for many reasons infinite, symbolic semantics were introduced as a mean to define smaller, possibly finite, transitio...

We propose a coalgebraic definition of weak bisimulation for classes of coalgebras obtained from bifunctors in the category Set. Weak bisimilarity for a system is obtained as strong bisimilarity of a transformed system. The particular transformation consists of two steps: First, the behavior on actions is lifted to behavior on finite words. Second, the behavior on finite words is taken modulo t...