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Kripke Semantics for Martin-Löf’s Extensional Type Theory

  • Steve Awodey
  • Florian Rabe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5608)

Abstract

It is well-known that simple type theory is complete with respect to non-standard models. Completeness for standard models only holds when increasing the class of models, e.g., to cartesian closed categories. Similarly, dependent type theory is complete for locally cartesian closed categories. However, it is usually difficult to establish the coherence of interpretations of dependent type theory, i.e., to show that the interpretations of equal expressions are indeed equal. Several classes of models have been used to remedy this problem.

We contribute to this investigation by giving a semantics that is both coherent and sufficiently general for completeness while remaining relatively easy to compute with. Our models interpret types of Martin-Löf’s extensional dependent type theory as sets indexed over posets or, equivalently, as fibrations over posets. This semantics can be seen as a generalization to dependent type theory of the interpretation of intuitionistic first-order logic in Kripke models. This yields a simple coherent model theory with respect to which simple and dependent type theory are sound and complete.

Keywords

Type Theory Natural Transformation Kripke Model Coherence Property Categorical Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Steve Awodey
    • 1
  • Florian Rabe
    • 2
  1. 1.Department of PhilosophyCarnegie Mellon UniversityPittsburghUSA
  2. 2.School of Engineering and ScienceJacobs University BremenGermany

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