Trakhtenbrot's Theorem in Coq: Finite Model Theory through the Constructive Lens release_3mzlh7crvvhndapmxqypnhej5a

by Dominik Kirst, Dominique Larchey-Wendling

Released as a article .

2021  

Abstract

We study finite first-order satisfiability (FSAT) in the constructive setting of dependent type theory. Employing synthetic accounts of enumerability and decidability, we give a full classification of FSAT depending on the first-order signature of non-logical symbols. On the one hand, our development focuses on Trakhtenbrot's theorem, stating that FSAT is undecidable as soon as the signature contains an at least binary relation symbol. Our proof proceeds by a many-one reduction chain starting from the Post correspondence problem. On the other hand, we establish the decidability of FSAT for monadic first-order logic, i.e. where the signature only contains at most unary function and relation symbols, as well as the enumerability of FSAT for arbitrary enumerable signatures. To showcase an application of Trakhtenbrot's theorem, we continue our reduction chain with a many-one reduction from FSAT to separation logic. All our results are mechanised in the framework of a growing Coq library of synthetic undecidability proofs.
In text/plain format

Archived Files and Locations

application/pdf  711.0 kB
file_sovq2vsbxvgmfgnnvl5ovwmqqq
arxiv.org (repository)
web.archive.org (webarchive)
Read Archived PDF
Preserved and Accessible
Type  article
Stage   submitted
Date   2021-12-13
Version   v3
Language   en ?
arXiv  2104.14445v3
Work Entity
access all versions, variants, and formats of this works (eg, pre-prints)
Catalog Record
Revision: 7dd0ecdc-61dc-4be2-ad18-a4d154fd23dd
API URL: JSON