TY - GEN
T1 - Trakhtenbrot’s Theorem in Coq
T2 - 10th International Joint Conference on Automated Reasoning, IJCAR 2020
AU - Kirst, Dominik
AU - Larchey-Wendling, Dominique
N1 - Publisher Copyright:
© 2020, Springer Nature Switzerland AG.
PY - 2020/1/1
Y1 - 2020/1/1
N2 - 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. All our results are mechanised in the framework of a growing Coq library of synthetic undecidability proofs.
AB - 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. All our results are mechanised in the framework of a growing Coq library of synthetic undecidability proofs.
UR - http://www.scopus.com/inward/record.url?scp=85088253168&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-51054-1_5
DO - 10.1007/978-3-030-51054-1_5
M3 - Conference contribution
AN - SCOPUS:85088253168
SN - 9783030510534
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 79
EP - 96
BT - Automated Reasoning - 10th International Joint Conference, IJCAR 2020, Proceedings
A2 - Peltier, Nicolas
A2 - Sofronie-Stokkermans, Viorica
PB - Springer
Y2 - 1 July 2020 through 4 July 2020
ER -