Notes on extended equation solvability and identity checking for groups

Kompatscher, Michael

doi:10.1007/s10474-019-00924-7

Cited by 5 publications

(6 citation statements)

References 18 publications

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“…Equations in groups. An expression (also called a polynomial in [35,20,27]) over a group G is a word α over the alphabet G ∪ X ∪ X −1 where X is a set of variables. Here X −1 denotes a formal set of inverses of the variables.…”

Section: Contribution In This Work We Give a Negative Answer To This ...mentioning

confidence: 99%

“…Notice that in the literature EQN-SAT is also denoted by POL-SAT [35,20] or Eq [27], while EQN-ID is also referred to as POL-EQ (e.g. in [35,20,24]) or Id [27].…”

Section: Contribution In This Work We Give a Negative Answer To This ...mentioning

confidence: 99%

“…In this section we describe the reduction of C-Coloring to EQN-SAT(G) and EQN-ID(G) in the spirit of [14,27]. For this, we rely on the fact that G has some normal subgroups meeting some special requirements.…”

Section: Reducing C-coloring To Equationsmentioning

confidence: 99%

“…The construction for Lemma 17 resembles the ones in Lemmas 5 and 6 of [27]. However, while in [27] a minimal normal subgroup N of a quotient G/K is constructed such that r g with r g (x) = [x, g] is an automorphism of N (and N is abelian), in our case this is not enough since we need to apply commutator constructions to our analog of N in the spirit of the divide-and-conquer approach of Proposition 8. While our construction is rather easier than [27], it cannot cope with the case that…”

Section: Consequencesmentioning

confidence: 99%

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Hardness of equations over finite solvable groups under the exponential time hypothesis

Weiß

2020

Preprint

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Goldmann and Russell (2002) initiated the study of the complexity of the equation satisfiability problem in finite groups by showing that it is in P for nilpotent groups while it is NP-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups of Fitting length two. In this work, we present the first lower bounds for the equation satisfiability problem in finite solvable groups: under the assumption of the exponential time hypothesis, we show that it cannot be in P for any group of Fitting length at least four and for certain groups of Fitting length three. Moreover, the same hardness result applies to the equation identity problem. ACM Subject Classification Theory of computation → Problems, reductions and completeness Keywords and phrases equations in groups, solvable groups, exponential time hypothesisFunding Armin Weiß: Funded by DFG project DI 435/7-1.Acknowledgements I am grateful to Moses Ganardi for bringing my attention to the AND-weakness conjecture and pointing out the relation to the exponential time hypothesis. I am also thankful to David A. Mix Barrington for an interesting email exchange concerning the AND-weakness conjecture and the idea to include steps of the lower central series in Proposition 8 to get a more refined upper bound. Finally, I want to thank Caroline Mattes and Jan Philipp Wächter for many helpful discussions.

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Section: Contribution In This Work We Give a Negative Answer To This ...mentioning

confidence: 99%

“…Notice that in the literature EQN-SAT is also denoted by POL-SAT [35,20] or Eq [27], while EQN-ID is also referred to as POL-EQ (e.g. in [35,20,24]) or Id [27].…”

Section: Contribution In This Work We Give a Negative Answer To This ...mentioning

confidence: 99%

Section: Reducing C-coloring To Equationsmentioning

confidence: 99%

Section: Consequencesmentioning

confidence: 99%

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Hardness of equations over finite solvable groups under the exponential time hypothesis

Weiß

2020

Preprint

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“…problems and co-NP-complete PolEqv problems [9] [14]. Roughly speaking this results from the fact that some operations can be written in a much more concise ways using commutators than just the group operations alone.…”

Section: Introductionmentioning

confidence: 99%

Circuit equivalence in 2-nilpotent algebras

Kawałek,

Kompatscher,

Krzaczkowski

2019

Preprint

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The circuit equivalence problem of a finite algebra A is the computational problem of deciding whether two circuits over A define the same function or not. This problem not just generalises the equivalence problem for Boolean circuits, but is also of high interest in universal algebra, as it models the problems of checking identities in A. In this paper we discuss the complexity for algebras from congruence modular varieties. A partial classification was already given in [11], leaving essentially only a gap for nilpotent but not supernilpotent algebras. We start a systematic study of this open case, proving that the circuit equivalence problem is in P for 2-nilpotent such algebras.

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Equation Satisfiability in Solvable Groups

et al. 2022

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The study of the complexity of the equation satisfiability problem in finite groups had been initiated by Goldmann and Russell in (Inf. Comput. 178(1), 253–262, 2002) where they showed that this problem is in for nilpotent groups while it is -complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups G having a nilpotent normal subgroup H with nilpotent factor G/H. This paper shows that such a normal subgroup must exist in each finite group with equation satisfiability solvable in polynomial time, unless the Exponential Time Hypothesis fails.

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Notes on extended equation solvability and identity checking for groups

Cited by 5 publications

References 18 publications

Hardness of equations over finite solvable groups under the exponential time hypothesis

Hardness of equations over finite solvable groups under the exponential time hypothesis

Circuit equivalence in 2-nilpotent algebras

Equation Satisfiability in Solvable Groups

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