Circuit equivalence in 2-nilpotent algebras

Kawałek, Piotr; Kompatscher, Michael; Krzaczkowski, Jacek

doi:10.48550/arxiv.1909.12256

Cited by 2 publications

(2 citation statements)

References 12 publications

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“…However, this left an intriguing gap for nilpotent, but not supernilpotent Maltsev algebras (see Problem 2 in [IK18]). Although such algebras admit non-trivial absorbing polynomials of all arities, there are some known examples A, for which CSAT(A) and CEQV(A) are solvable in polynomial time (see [IKK18,KKK19]). An explanation for this seemingly contradictory phenomenon is, that these non-trivial absorbing polynomials cannot be efficiently represented by circuits.…”

Section: Introductionmentioning

confidence: 99%

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

Kompatscher¹

2021

Preprint

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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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Section: Introductionmentioning

confidence: 99%

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

Kompatscher¹

2021

Preprint

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“…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

Preprint

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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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Circuit equivalence in 2-nilpotent algebras

Cited by 2 publications

References 12 publications

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

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

Even faster algorithms for CSAT over~supernilpotent algebras

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