Introduction to higher order categorical logic
Introduction to higher order categorical logic
Information and Computation
Notions of computation and monads
Information and Computation
A calculus of mobile processes, I
Information and Computation
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Action Calculi, or Syntactic Action Structures
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A Reflective Higher-order Calculus
Electronic Notes in Theoretical Computer Science (ENTCS)
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Monoidal indeterminates and categories of possible worlds
Theoretical Computer Science
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Milner's action calculus implements abstraction in monoidal categories, so that familiar λ-calculi can be subsumed together with the π-calculus and the Petri nets. Variables are generalised to names, which allow only a restricted form of substitution.In the present paper, the well-known categorical semantics of the λ-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 definable. 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, for example, the ring ℤ[x]; the latter lead to algebraic extensions like ℤ[√2] or ℤ[i].Building upon the work of P. Gardner, we introduce action categories, and show that they are related to the static action calculus in exactly the same way as cartesian closed categories are related to the λ-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.