Satisfiability Problems for Finite Groups

Authors Paweł M. Idziak , Piotr Kawałek , Jacek Krzaczkowski , Armin Weiß

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Author Details

Paweł M. Idziak
  • Jagiellonian University, Kraków, Poland
Piotr Kawałek
  • Jagiellonian University, Kraków, Poland
Jacek Krzaczkowski
  • Maria Curie-Sklodowska University, Lublin, Poland
Armin Weiß
  • Universität Stuttgart, FMI, Germany

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Paweł M. Idziak, Piotr Kawałek, Jacek Krzaczkowski, and Armin Weiß. Satisfiability Problems for Finite Groups. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 127:1-127:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Over twenty years ago, Goldmann and Russell initiated the study of the complexity of the equation satisfiability problem (PolSat and the NUDFA program satisfiability problem (ProgramSat) in finite groups. They showed that these problems are in 𝖯 for nilpotent groups while they are NP-complete for non-solvable groups. In this work we completely characterize finite groups for which the problem ProgramSat can be solved in randomized polynomial time under the assumptions of the Randomized Exponential Time Hypothesis and the Constant Degree Hypothesis. We also determine the complexity of PolSat for a wide class of finite groups. As a by-product, we obtain a classification for ListPolSat, a version of PolSat where each variable can be restricted to an arbitrary subset. Finally, we also prove unconditional algorithms for these problems in certain cases.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Complexity classes
  • Satisifiability
  • Solvable groups
  • ProgramSat
  • PolSat
  • Exponential Time Hypothesis


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