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A Stratified Approach to Löb Induction

Authors Daniel Gratzer , Lars Birkedal



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Author Details

Daniel Gratzer
  • Aarhus University, Denmark
Lars Birkedal
  • Aarhus University, Denmark

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Daniel Gratzer and Lars Birkedal. A Stratified Approach to Löb Induction. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 23:1-23:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.FSCD.2022.23

Abstract

Guarded type theory extends type theory with a handful of modalities and constants to encode productive recursion. While these theories have seen widespread use, the metatheory of guarded type theories, particularly guarded dependent type theories remains underdeveloped. We show that integrating Löb induction is the key obstruction to unifying guarded recursion and dependence in a well-behaved type theory and prove a no-go theorem sharply bounding such type theories. Based on these results, we introduce GuTT: a stratified guarded type theory. GuTT is properly two type theories, sGuTT and dGuTT. The former contains only propositional rules governing Löb induction but enjoys decidable type-checking while the latter extends the former with definitional equalities. Accordingly, dGuTT does not have decidable type-checking. We prove, however, a novel guarded canonicity theorem for dGuTT, showing that programs in dGuTT can be run. These two type theories work in concert, with users writing programs in sGuTT and running them in dGuTT.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • Theory of computation → Denotational semantics
  • Theory of computation → Modal and temporal logics
Keywords
  • Dependent type theory
  • guarded recursion
  • modal type theory
  • canonicity
  • categorical gluing

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