Presheaf Models for Concurrency

Gian Luca Cattani

April 1999


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, $\pi$-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 ner 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 $\lambda$-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 order-enriched 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

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