Truly concurrent constraint programming

Abstract

Concurrent Constraint Programming (CCP) is a powerful computation model for concurrency obtained by internalizing the notion of computation via deduction over (first-order) systems of partial information (constraints). In [SRP91] a semantics for indeterminate CCP was given via sets of bounded trace operators; this was shown to be fully abstract with respect to observing all possible quiescent stores (=final states) of the computation. Bounded trace operators constitute a certain class of (finitary) “invertible” closure operators over a downward closed sublattice. They can be thought of as generated via the grammar: t::=c ¦ c → t¦ c∧t where c ranges over primitive constraints, ∧ is conjunction and → intuitionistic implication.

We motivate why it is interesting to consider as observable a “causality” relation on the store: what is observed is not just the conjunction of constraints deposited in the store, but also the causal dependencies between these constraints — what constraints were required to be present in the computation before others could be generated. We show that the same construction used to give the “interleaving” semantics in [SRP91] can be used to give a true-concurrency semantics provided that denotations are taken to be sets of bounded closure operators, which can be generated via the grammar k::=c ¦ c → k ¦ k∧k

Thus we obtain a denotational semantics for CCP fully-abstract with respect to observing this “causality” relation on constraints. This semantics differs from the earlier semantics in preserving more fine-grained structure of the computation; in particular the Interleaving Law (a → P) ∥ (b → Q)=(a → (P ∥ (b → Q))) p[ (b → (Q ∥ (a → P))) (1) is not verified (□ is indeterminate choice). Relationships between such a denotational approach to true concurrency and different powerdomain constructions are explored.