Relevance logic and concurrent composition

The operation of relativizing properties with respect to parallel environments often used in obtaining compositionality in theories for concurrency corresponds to a notion of (contraction-free) relevant deduction. The author considers program logics in which this notion of deduction is internalized by the corresponding implication. The idea is carried through for safety properties of a simple system of SCCS-type synchronous processes with an internal choice operator. They present two completeness results: first for a modal extension of positive propositional linear logic with respect to the equational class of algebras containing the safety testing quotient of the author's process system as its free member, and second for the free algebra itself.<>