eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-08-21
66:1
66:12
10.4230/LIPIcs.MFCS.2023.66
article
On the Complexity Dichotomy for the Satisfiability of Systems of Term Equations over Finite Algebras
Mayr, Peter
1
2
Department of Mathematics, University of Colorado Boulder, CO, USA
Institute for Algebra, Johannes Kepler UniversitΓ€t Linz, Austria
For a fixed finite algebra π, we consider the decision problem SysTerm(π): does a given system of term equations have a solution in π? This is equivalent to a constraint satisfaction problem (CSP) for a relational structure whose relations are the graphs of the basic operations of π. From the complexity dichotomy for CSP over fixed finite templates due to Bulatov [Bulatov, 2017] and Zhuk [Zhuk, 2017], it follows that SysTerm(π) for a finite algebra π is in P if π has a not necessarily idempotent Taylor polymorphism and is NP-complete otherwise. More explicitly, we show that for a finite algebra π in a congruence modular variety (e.g. for a quasigroup), SysTerm(π) is in P if the core of π is abelian and is NP-complete otherwise. Given π by the graphs of its basic operations, we show that this condition for tractability can be decided in quasi-polynomial time.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol272-mfcs2023/LIPIcs.MFCS.2023.66/LIPIcs.MFCS.2023.66.pdf
systems of equations
general algebras
constraint satisfaction