Presheaf models for concurrency (original) (raw)

1997, Lecture Notes in Computer Science

In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their work inspired this thesis by suggesting that presheaf categories could provide abstract models for concurrency with a built-in notion of bisimulation. We show how presheaf categories, in which traditional models of concurrency are embedded, can be used to deduce congruence properties of bisimulation for the traditional models. A key result is given here; it is shown that the homomorphisms between presheaf categories, i.e., colimit preserving functors, preserve open map bisimulation. We follow up by observing that presheaf categories and colimit preserving functors organise in what can be considered as a category of non-deterministic domains. Presheaf models can be obtained as solutions to recursive domain equations. We investigate properties of models given for a range of concurrent process calculi, including CCS, CCS with value-passing, π-calculus and a form of CCS with linear process passing. Open map bisimilarity is shown to be a congruence for each calculus. These are consequences of general mathematical results like the preservation of open map bisimulation by colimit preserving functors. In all but the case of the higher order calculus, open map bisimulation is proved to coincide with traditional notions of bisimulation for the process terms. In the case of higher order processes, we obtain a finer equivalence than the one one would normally expect, but this helps reveal interesting aspects of the relationship between the presheaf and the operational semantics. For a fragment of the language, corresponding to a form of λ-calculus, open map bisimulation coincides with applicative bisimulation. In developing a suitable general theory of domains, we extend results and notions, such as the limit-colimit coincidence theorem of Smyth and Plotkin, from the orderenriched case to a "fully" 2-categorical situation. Moreover we provide a domain theoretical analysis of (open map) bisimulation in presheaf categories. We present, in fact, induction and coinduction principles for recursive domains as in the works of Pitts and of Hermida and Jacobs and use them to derive a coinduction property based on bisimulation. vii Personal debts can never be adequately acknowledged. I am especially grateful to my supervisor Glynn Winskel. Not only has he taught me how to do research, but he also transmitted his enthusiasm for it. It has always been a pleasure and a source of learning to discuss ideas with him and this thesis owes much to his stimulating guidance. While leaving me the freedom of choosing the problems I wished to work on, he has always been very involved in what I was doing to the point that this thesis can, in fact, be regarded as the result of four years of joint work. I shall also heartily thank him for his friendship. Pino Rosolini gave unstinting support in more ways than one. He guided my first steps as a researcher when I was working on my 'tesi di laurea'. Later, when I decided to go on with postgraduate studies, he put me in contact and warmly suggested that I should study with Glynn. Ever since then he discreetly followed my progresses as a PhD student while always being available whenever I needed his help or advice. Thanks are due to Vladimiro Sassone, Ian Stark and Marcelo Fiore. They all showed me friendship and stimulated my research. Marcelo in particular has been very influential in the development of an important part of this thesis, Chapter 6.