Towards Proofs as Successful Executions of Processes

We further the understanding of the relationship between process calculi and non-commutative logic. This work focuses on, a first-order extension of the proof calculus BV featuring a de Morgan dual pair of nominal quantifiers, called BV1. An embedding of π-calculus process as predicates in BV1 is defined, and a procedure is provided for extracting successful executions from proofs of embedded processes. This procedure is used to establish the soundness of linear implication in BV1 with respect to trace inclusion in the π-calculus. We illustrate the expressive power of BV1, by demonstrating that these techniques extend also to the internal π-calculus, where privacy of inputs are guaranteed. We emphasise that linear implication is strictly finer than trace inclusion, providing a tight refinement semantics for processes respecting both causality and the scope of private names.