TY - GEN
T1 - Undecidability of Dyadic First-Order Logic in Coq
AU - Hostert, Johannes
AU - Dudenhefner, Andrej
AU - Kirst, Dominik
N1 - Publisher Copyright:
© 2022 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.
PY - 2022/8/1
Y1 - 2022/8/1
N2 - We develop and mechanize compact proofs of the undecidability of various problems for dyadic first-order logic over a small logical fragment. In this fragment, formulas are restricted to only a single binary relation, and a minimal set of logical connectives. We show that validity, satisfiability, and provability, along with finite satisfiability and finite validity are undecidable, by directly reducing from a suitable binary variant of Diophantine constraints satisfiability. Our results improve upon existing work in two ways: First, the reductions are direct and significantly more compact than existing ones. Secondly, the undecidability of the small logic fragment of dyadic first-order logic was not mechanized before. We contribute our mechanization to the Coq Library of Undecidability Proofs, utilizing its synthetic approach to computability theory.
AB - We develop and mechanize compact proofs of the undecidability of various problems for dyadic first-order logic over a small logical fragment. In this fragment, formulas are restricted to only a single binary relation, and a minimal set of logical connectives. We show that validity, satisfiability, and provability, along with finite satisfiability and finite validity are undecidable, by directly reducing from a suitable binary variant of Diophantine constraints satisfiability. Our results improve upon existing work in two ways: First, the reductions are direct and significantly more compact than existing ones. Secondly, the undecidability of the small logic fragment of dyadic first-order logic was not mechanized before. We contribute our mechanization to the Coq Library of Undecidability Proofs, utilizing its synthetic approach to computability theory.
KW - Coq
KW - first-order logic
KW - synthetic computability
KW - undecidability
UR - https://www.scopus.com/pages/publications/85136312742
U2 - 10.4230/LIPIcs.ITP.2022.19
DO - 10.4230/LIPIcs.ITP.2022.19
M3 - Conference contribution
AN - SCOPUS:85136312742
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 13th International Conference on Interactive Theorem Proving, ITP 2022
A2 - Andronick, June
A2 - de Moura, Leonardo
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 13th International Conference on Interactive Theorem Proving, ITP 2022
Y2 - 7 August 2022 through 10 August 2022
ER -