Categorical logic of Names and Abstraction in Action Calculi

Received Milner's action calculus implements abstraction in monoidal categories, so that familiar-calculi can be subsumed together with the-calculus and the Petri nets. Variables are generalised to names: only a restricted form of substitution is allowed. In the present paper, the well-known categorical semantics of the-calculus is generalised to the action calculus. A suitable functional completeness theorem for symmetric monoidal categories is proved: we determine the conditions under which the abstraction is deenable. Algebraically, the distinction between the variables and the names boils down to the distinction between the transcendental and the algebraic elements. The former lead to polynomial extensions, like e.g. the ring Zx], the latter to algebraic extensions like Z p 2] or Zi]. Building upon the work of P. Gardner, we introduce action categories, and show that they are related to the static action calculus exactly as cartesian closed categories are related to the-calculus. Natural examples of this structure arise from allegories and cartesian bicategories. On the other hand, the free algebras for any commutative Moggi monad form an action category. The general correspondence of action calculi and Moggi monads will be worked out in a sequel to this work.