Pawel Sobocinski

Van Kampen colimits as bicolimits in span

The exactness properties of coproducts in extensive categories and pushouts along monos in adhesive categories have found various applications in theoretical computer science, e.g. in program semantics, data type theory and rewriting. We show that these properties can be understood as a single universal property in the associated bicategory of spans. To this end, we first provide a general notion of Van Kampen cocone that specialises to the above colimits. The main result states that Van Kampen cocones can be characterised as exactly those diagrams in C that induce bicolimit diagrams in the bicategory of spans SpanC, provided that C has pullbacks and enough colimits.

Adhesive and quasiadhesive categories

We introduce adhesive categories, which are categories with structure ensuring that pushouts along monomorphisms are well-behaved, as well as quasiadhesive categories which restrict attention to regular monomorphisms. Many examples of graphical structures used in computer science are shown to be examples of adhesive and quasiadhesive categories. Double-pushout graph rewriting generalizes well to rewriting on arbitrary adhesive and quasiadhesive categories.

Reactive systems over cospans

The theory of reactive systems, introduced by Leifer and Milner and previously extended by the authors, allows the derivation of well-behaved labelled transition systems (LTS) for semantic models with an underlying reduction semantics. The derivation procedure requires the presence of certain colimits (or, more usually and generally, bicolimits) which need to be constructed separately within each model. In this paper, we offer a general construction of such bicolimits in a class of bicategones of cospans. The construction sheds light on as well as extends Ehrig and Konigs rewriting via borrowed contexts and opens the way to a unified treatment of several applications.

BEING VAN KAMPEN IS A UNIVERSAL PROPERTY

Colimits that satisfy the Van Kampen condition have interesting exactness properties. We show that the elementary presentation of the Van Kampen condition is actually a characterisation of a universal property in the associated bicategory of spans. The main theorem states that Van Kampen cocones are precisely those diagrams in a category that induce bicolimit diagrams in its associated bicategory of spans, provided that the category has pullbacks and enough colimits.

Representations of Petri net interactions

We introduce a novel compositional algebra of Petri nets, as well as a stateful extension of the calculus of connectors. These two formalisms are shown to have the same expressive power.

Deriving Structural Labelled Transitions for Mobile Ambients

We present a new labelled transition system ( lts ) for the ambient calculus on which ordinary bisimilarity coincides with contextual equivalence. The key feature of this lts is that it is the fruit of ongoing work on developing a systematic procedure for deriving lts s in the structural style from the underlying reduction semantics and observability. Notably, even though we have derived our lts for ambients systematically it compares very favourably with existing transition systems for the same calculus.

Semantic barbs and biorthogonality

We use the framework of biorthogonality to introduce a novel semantic definition of the concept of barb (basic observable) for process calculi. We develop a uniform basic theory of barbs and demonstrate its robustness by showing that it gives rise to the correct observables in specific process calculi which model synchronous, asynchronous and broadcast communication regimes.

A Congruence for Petri Nets

We introduce a way of viewing Petri nets as open systems. This is done by considering a bicategory of cospans over a category of p/t nets and embeddings. We derive a labelled transition system (LTS) semantics for such nets using GIPOs and characterise the resulting congruence. Technically, our results are similar to the recent work by Milner on applying the theory of bigraphs to Petri Nets. The two main differences are that we treat p/t nets instead of c/e nets and we deal directly with a category of nets instead of encoding them into bigraphs.

Full Abstraction for Signal Flow Graphs

Network theory uses the string diagrammatic language of monoidal categories to study graphical structures formally, eschewing specialised translations into intermediate formalisms. Recently, there has been a concerted research focus on developing a network theoretic approach to signal flow graphs, which are classical structures in control theory, signal processing and a cornerstone in the study of feedback. In this approach, signal flow graphs are given a relational denotational semantics in terms of formal power series. Thus far, the operational behaviour of such signal flow graphs has only been discussed at an intuitive level. In this paper we equip them with a structural operational semantics. As is typically the case, the purely operational picture is too concrete -- two graphs that are denotationally equal may exhibit different operational behaviour. We classify the ways in which this can occur and show that any graph can be realised -- rewritten, using the graphical theory, into an executable form where the operational behavior and the denotation coincides.

Deriving Bisimulation Congruences: A 2-categorical Approach

We introduce G-relative-pushouts (GRPO) which are a 2-categorical generalisation of relative-pushouts (RPO). They are suitable for deriving labelled transition systems (LTS) for process calculi where terms are viewed modulo structural congruence. We develop their basic properties and show that bisimulation on the LTS derived via GRPOs is a congruence, provided that sufficiently many GRPOs exist. The theory is applied to a simple subset of CCS and the resulting LTS is compared to one derived using a procedure proposed by Sewell.