From urelements to computation: A journey through applications of Fraenkel's permutation model in computer science

Ciancia V.

Semantics  Syntactics  Set theory 

Around 1922-1938, a new permutation model of set theory was defined. The permutation model served as a counterexample in the first proof of independence of the Axiom of Choice from the other axioms of Zermelo-Fraenkel set theory. Almost a century later, a model introduced as part of a proof in abstract mathematics fostered a plethora of research results, ranging from the area of syntax and semantics of programming languages to minimization algorithms and automated verification of systems. Among these results, we find Lawvere-style algebraic syntax with binders, final-coalgebra semantics with resource allocation, and minimization algorithms for mobile systems. These results are also obtained in various different ways, by describing, in terms of category theory, a number of models equivalent to the permutation model. We aim at providing both a brief history of some of these developments, and a mild introduction to the recent research line of "nominal computation theory", where the essential notion of name is declined in several different ways.

Source: History and Philosophy of Computing. HaPoC 2015. IFIP Advances in Information and Communication Technology, edited by Gadducci F.; Tavosanis M., pp. 141–155, 2016

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