A Dependent Type Theory with Names and Binding

Ulrich Schöpp and Ian Stark

In Computer Science Logic: Proceedings of the 18th International Workshop CSL 2004, Karpacz, Poland, September 20–24, 2004 , Lecture Notes in Computer Science 3210, pages 235–249. Springer-Verlag, 2004.

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Abstract

We consider the problem of providing formal support for working with abstract syntax involving variable binders. Gabbay and Pitts have shown in their work on Fraenkel-Mostowski (FM) set theory how to address this through first-class names: in this paper we present a dependent type theory for programming and reasoning with such names. Our development is based on a categorical axiomatisation of names, with freshness as its central notion. An associated adjunction captures constructions known from FM theory: the freshness quantifier И, name-binding, and unique choice of fresh names. The Schanuel topos — the category underlying FM set theory — is an instance of this axiomatisation. Working from the categorical structure, we define a dependent type theory which it models. This uses bunches to integrate the monoidal structure corresponding to freshness, from which we define novel multiplicative dependent products Π* and sums Σ* as well as a propositions-as-types generalisation H of the freshness quantifier.

@InProceedings{schoepp/stark:names+binding, author = {Ulrich {Sch\"opp} and Ian Stark}, title = {A Dependent Type Theory with Names and Binding}, booktitle = {Computer Science Logic: Proceedings of the 18th International Workshop CSL~2004}, pages = {235--249}, year = 2004, number = 3210, series = {Lecture Notes in Computer Science}, publisher = {Springer-Verlag}, url = {http://www.inf.ed.ac.uk/~stark/names+binding.html}, pdf = {http://www.inf.ed.ac.uk/~stark/names+binding.pdf} }

This research was supported by the EPSRC project GR/R04430/01 Reasoning with Names and Identity in Programming Languages.

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