The equation solvability problem over supernilpotent algebras with Mal’cev term

Kompatscher, Michael

doi:10.1142/s0218196718500443

Cited by 11 publications

(9 citation statements)

References 18 publications

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“…For CEQV this was observed in [AM10]. For CSAT similar observation was made in [Kom18], [IK18] and [Aic19b], using the fact that every polynomial can be expressed as a 'sum' of absorbing polynomials.…”

Section: Circuit Satisfiability and Equivalencesupporting

confidence: 67%

CC-circuits and the expressive power of nilpotent algebras

Kompatscher¹

2019

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We show that CC-circuits of bounded depth have the same expressive power as polynomials over finite nilpotent algebras from congruence modular varieties. We use this result to phrase and discuss an algebraic version of Barrington, Straubing and Thérien's conjecture, which states that CC-circuits of bounded depth need exponential size to compute AND.Furthermore we investigate the complexity of deciding identities and solving equations in a fixed nilpotent algebra. Under the assumption that the conjecture is true, we obtain quasipolynomial algorithms for both problems. On the other hand, if AND is computable by uniform CC-circuits of bounded depth and polynomial size, we can construct a nilpotent algebra with coNP-complete, respectively NP-complete problem.

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Section: Circuit Satisfiability and Equivalencesupporting

confidence: 67%

CC-circuits and the expressive power of nilpotent algebras

Kompatscher¹

2019

Preprint

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“…In both cases it was shown that considered algorithm works in polynomial time but the degree of the polynomial came from application of Ramsey Theory and was really huge. Later it was independently shown in [24] and [16] that essentially the same algorithm works for supernilpotent algebras from congruence modular variety in polynomial time with the same huge degree of the polynomial. This results was improved by Aichinger in [1].…”

Section: Introductionmentioning

confidence: 99%

“…Several articles considering this new approach to solving equations have appeared e.g. [24], [17], [1], [21], [23], [18]. In this paper we will present the results in terms of Csat, however for clarity we mention, that all the algorithms and upper bounds presented here apply also to the original definition of the problem as polynomials can be represented by circuits expanding size of the representation only by constant factor.…”

Section: Introductionmentioning

confidence: 99%

Even faster algorithms for CSAT over~supernilpotent algebras

Kawałek¹,

Krzaczkowski²

2020

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In this paper two algorithms solving circuit satisfiability problem over supernilpotent algebras are presented. The first one is deterministic and is faster than fastest previous algorithm presented in [1]. The second one is probabilistic with linear time complexity. Application of the former algorithm to finite groups provides time complexity that is usually lower than in previously best [5] and application of the latter leads to corollary, that circuit satisfiability problem for group G is either tractable in probabilistic linear time if G is nilpotent or is NP-complete if G fails to be nilpotent. The results are obtained, by translating equations between polynomials over supernilpotent algebras to bounded degree polynomial equations over finite fields.

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“…The results in [IK18] link the existence of absorbing polynomials, commutator theoretical properties, and the complexity of CSAT(A) and CEQV(A) for algebras from congruence modular varieties: For so called supernilpotent algebras A, which do not have absorbing polynomials of arbitrary big arities, the problems are in P (for CSAT this was independently shown in [Kom18], and for CEQV already in [AM10]). On the other hand, very roughly speaking, non-nilpotent algebras allow to efficiently construct circuits, which express absorbing polynomials, and therefore (almost always) have hard CSAT and CEQV problems.…”

Section: Introductionmentioning

confidence: 99%

CSAT and CEQV for nilpotent Maltsev algebras of Fitting length > 2

Kompatscher¹

2021

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The circuit satisfaction problem CSAT(A) of an algebra A is the problem of deciding whether an equation over A (encoded by two circuits) has a solution or not. While solving systems of equations over finite algebras is either in P or NP-complete, no such dichotomy result is known for CSAT(A). In fact Idziak, Kawa lek and Krzaczkowski [IKK20] constructed examples of nilpotent Maltsev algebras A, for which, under the assumption of ETH and an open conjecture in circuit theory, CSAT(A) can be solved in quasipolynomial, but not polynomial time. The same is true for the circuit equivalence problem CEQV(A).In this paper we generalize their result to all nilpotent Maltsev algebras of Fitting length > 2. This not only advances the project of classifying the complexity of CSAT (and CEQV) for algebras from congruence modular varieties initiated in [IK18], but we also believe that the tools developed are of independent interest in the study of nilpotent algebras.

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The equation solvability problem over supernilpotent algebras with Mal’cev term

Cited by 11 publications

References 18 publications

CC-circuits and the expressive power of nilpotent algebras

CC-circuits and the expressive power of nilpotent algebras

Even faster algorithms for CSAT over~supernilpotent algebras

CSAT and CEQV for nilpotent Maltsev algebras of Fitting length > 2

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