Journal ArticleDOI
Categorical logic of names and abstraction in action calculi
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TLDR
The well-known categorical semantics of the λ-calculus is generalised to the action calculus, and a suitable functional completeness theorem for symmetric monoidal categories is proved.Abstract:
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 Zlxr; the latter lead to algebraic extensions like Zl√2r or Zlir.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.read more
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References
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Book
Categories for the Working Mathematician
TL;DR: In this article, the authors present a table of abstractions for categories, including Axioms for Categories, Functors, Natural Transformations, and Adjoints for Preorders.
Journal ArticleDOI
A calculus of mobile processes, II
TL;DR: The a-calculus is presented, a calculus of communicating systems in which one can naturally express processes which have changing structure, including the algebraic theory of strong bisimilarity and strong equivalence, including a new notion of equivalence indexed by distinctions.
Journal ArticleDOI
Notions of computation and monads
TL;DR: Calculi are introduced, based on a categorical semantics for computations, that provide a correct basis for proving equivalence of programs for a wide range of notions of computation.
Book
Introduction to higher order categorical logic
Joachim Lambek,Philip J. Scott +1 more
TL;DR: In this article, Cartesian closed categories and Calculus are used to represent Numerical functions in various categories and to describe the relation between categories. But they do not specify the topology of the categories.