A proof-theoretic approach to semantics of concurrency
Abstract
A central problem in the area of concurrency is the very definition of concurrency. Despite several years of research, there is no general agreement whether one should take the interleaving approach, or the partial order approach is better. In this thesis we establish a precise relationship between different views of concurrency. This is done in the setting of Petri nets. In particular, we establish in a precise technical sense that a truly concurrent computation indeed represents a canonical object in an abstract class of all computations of the appropriate type on a net. To achieve this, we show that there is a tight correspondence between reachability in nets and provability in the associated tensor theory. Thus nets become theories and computations on a net become proofs. This also allows us to import decidability results from net theory to a fragment of linear logic. To relate various computations, we interpret cut as describing causal dependency and analyze the relevance of cut elimination in the context of a net. We give a normalizing system of rewrites which gives us a notion of canonicity. To understand the relevance of cut reduction to concurrency, we assign a suitable semantics to proofs. The semantics we assign is "dynamic" in the sense that it gives meaning to the very phenomena of cut reduction--the meaning being removal of spurious causal dependencies. Thus, we establish that a cut reduced proof is indeed a maximally concurrent computation on a net. The semantics we propose also gives us a notion of equivalence of proofs (or computations). We then analyze the dynamics of an induced notion of rewrite. We establish the Church-Rosser property for such rewrites which also gives another notion of equivalence. We then show that this equivalence coincides with another equivalence that has been proposed in the literature for which the motivations were entirely category theoretic. Using these rewrites, we induce a natural order on proofs to characterize transition sequences. We also give implementations of a theorem prover for a fragment of linear logic and of our cut reduction algorithm.
Subject Area
Computer science
Recommended Citation
Gehlot, Vijay, "A proof-theoretic approach to semantics of concurrency" (1992). Dissertations available from ProQuest. AAI9227666.
https://repository.upenn.edu/dissertations/AAI9227666