Finite model theory for pseudovarieties and universal algebra: preservation, definability and complexity
Abstract
We explore new interactions between finite model theory and a number of classical streams of universal algebra and semigroup theory. A key result is an example of a finite algebra whose variety is not finitely axiomatisable in first order logic, but which has first order definable finite membership problem. This algebra witnesses the simultaneous failure of the ŁosTarski Theorem, the SPpreservation theorem and Birkhoff's HSPpreservation theorem at the finite level as well as providing a negative solution to a first order formulation of the longstanding Eilenberg Schützenberger problem. The example also shows that a pseudovariety without any finite pseudoidentity basis may be finitely axiomatisable in first order logic. Other results include the undecidability of deciding first order definability of the pseudovariety of a finite algebra and a mapping from any fixed template constraint satisfaction problem to a first order equivalent variety membership problem, thereby providing examples of variety membership problems complete in each of the classes $\texttt{L}$, $\texttt{NL}$, $\texttt{Mod}_p(\texttt{L})$, $\texttt{P}$, and infinitely many others (depending on complexitytheoretic assumptions).
 Publication:

arXiv eprints
 Pub Date:
 December 2022
 DOI:
 10.48550/arXiv.2212.02653
 arXiv:
 arXiv:2212.02653
 Bibcode:
 2022arXiv221202653H
 Keywords:

 Mathematics  Logic;
 Computer Science  Computational Complexity;
 Computer Science  Logic in Computer Science;
 03C13;
 08B05;
 08B26;
 68Q15;
 68Q19;
 08A30;
 20M07