Ian Stark

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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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.

  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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