On synthetic undecidability in Coq, with an application to the Entscheidungsproblem

Yannick Forster, Dominik Kirst, Gert Smolka

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

35 Scopus citations

Abstract

We formalise the computational undecidability of validity, satisfiability, and provability of first-order formulas following a synthetic approach based on the computation native to Coq's constructive type theory. Concretely, we consider Tarski and Kripke semantics as well as classical and intuitionistic natural deduction systems and provide compact many-one reductions from the Post correspondence problem (PCP). Moreover, developing a basic framework for synthetic computability theory in Coq, we formalise standard results concerning decidability, enumerability, and reducibility without reference to a concrete model of computation. For instance, we prove the equivalence of Post's theorem with Markov's principle and provide a convenient technique for establishing the enumerability of inductive predicates such as the considered proof systems and PCP.

Original languageEnglish
Title of host publicationCPP 2019 - Proceedings of the 8th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with POPL 2019
EditorsAssia Mahboubi
PublisherAssociation for Computing Machinery, Inc
Pages38-51
Number of pages14
ISBN (Electronic)9781450362221
DOIs
StatePublished - 14 Jan 2019
Externally publishedYes
Event8th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2019 - Cascais, Portugal
Duration: 14 Jan 201915 Jan 2019

Publication series

NameCPP 2019 - Proceedings of the 8th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with POPL 2019

Conference

Conference8th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2019
Country/TerritoryPortugal
CityCascais
Period14/01/1915/01/19

Keywords

  • Coq
  • Entscheidungsproblem
  • Markov's principle
  • Post's theorem
  • first-order logic
  • synthetic undecidability

ASJC Scopus subject areas

  • Computer Science Applications
  • Software

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