Partiality and recursion in higher-order logic

  • Authors:

  • Łukasz Czajka

  • Affiliations:

  • Institute of Informatics, University of Warsaw, Warszawa, Poland

  • Venue:
  • FOSSACS'13 Proceedings of the 16th international conference on Foundations of Software Science and Computation Structures
  • Year:
  • 2013

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Abstract

We present an illative system $\ensuremath{{\cal I}}_s$ of classical higher-order logic with subtyping and basic inductive types. The system $\ensuremath{{\cal I}}_s$ allows for direct definitions of partial and general recursive functions, and provides means for handling functions whose termination has not been proven. We give examples of how properties of some recursive functions may be established in our system. In a technical appendix to the paper we prove consistency of $\ensuremath{{\cal I}}_s$. The proof is by model construction. We then use this construction to show conservativity of $\ensuremath{{\cal I}}_s$ over classical first-order logic. Conservativity over higher-order logic is conjectured, but not proven.