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CSAT and CEQV for Nilpotent Maltsev Algebras of Fitting Length> 2

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CSAT and CEQV for Nilpotent Maltsev Algebras of Fitting Length> 2 CSAT AND CEQV FOR NILPOTENT MALTSEV ALGEBRAS OF FITTING LENGTH > 2 MICHAEL KOMPATSCHER Abstract. 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. 1. Introduction Solving systems of equations over a given algebraic structure is one of the most fundamental problems in mathematics. In its various forms, it was one of the driving forces in developing new methods and entire new fields in algebra (e.g. Gaussian elimination, Galois theory, and algebraic geometry, to only mention a few). A big interest nowadays often lies in studying the problem from a computational perspective. And while solving equations over infinite algebras can be undecidable (as it was for instance famously shown for Diophantine equations [Mat70]), it can always be done in non-deterministic polynomial time in finite algebras. So for finite algebras, one of the main tasks is to distinguish algebras, for which solving equations is tractable (in P), hard (NP-complete), or possibly of some intermediate complexity. Solving systems of polynomial equations over finite algebras is closely connected to the con- arXiv:2105.00689v1 [cs.CC] 3 May 2021 straint satisfaction problems (CSP) of finite relational structures: in fact, it is quite straight- forward to see that solving such systems over an algebra A = (A, f1,...,fn) can be modeled as the CSP, whose constraint relations are the function graphs of the operations f1,...,fn and all constants (e.g. observed in [LZ07]). Therefore the CSP dichotomy theorem, which was independently shown by Bulatov [Bul17] and Zhuk [Zhu17, Zhu20], implies that also solving systems of equations over A is either in P or NP-complete. Date: May 4, 2021. This paper was supported by grant No 18-20123S of the Czech Science Foundation (GACR),ˇ INTER- EXCELLENCE project LTAUSA19070 of the Czech Ministry of Education MSMT,ˇ and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 714532). The paper reflects only the authors’ views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein. 1 2 MICHAEL KOMPATSCHER But interestingly, if we restrict the input to a bounded number of equations over A, not much is known. In fact, already the base case, in which we we restrict the input to one polyno- mial equation is far from being understood. In the literature this problem is usually referred to as the polynomial satisfiability problem (or equation solvability problem) PolSAT(A). Complete classifications of PolSAT(A) exists for some classes algebras, such as rings [Hor11], or lattices [Sch04]. The problem is also quite well understood for finite groups (see [GR02, F¨ol17, FH20, Wei20]), despite the situation not being completely resolved for solvable groups of Fitting length 2 (Problem 1 in [IKKW20]). However, classifying the complexity of PolSAT(A) for arbitrary algebras A seems to be far out of reach at the moment. One of the main hurdles is, that the complexity of PolSAT(A) is sensitive to changes of the language: In [HS12] it was shown that PolSAT(A4, ·) is in P for the alternating group (A4, ·), but, if we also allow expressions using commutator brackets [x,y]= x−1y−1xy in the input, the problem becomes NP-complete. To resolve this problem, Idziak and Krzaczkowski suggested in [IK18] to look at the cir- cuit satisfaction problem CSAT(A) instead, in which the input equation is not encoded by polynomials, but by circuits over A. Then, unlike for PolSAT, the complexity is stable under changes of the signature of A, which allows for the use of methods from universal algebra. The same paper also introduced the circuit equivalence problem CEQV(A), which asks if two circuits over A are equivalent, and initiated a systematic study of CSAT and CEQV for algebras from congruence modular varieties. This class of algebras, encompasses many alge- bras of interest in algebra and computer science (such as lattices, Boolean algebras, groups, quasigroups, rings, fields, vector spaces, modules), while still exhibiting some nice regularity properties. In many cases, the computational hardness of CSAT(A) (and CEQV(A)) is then intimately connected to existence of circuits that allow to express conjunctions. For example, in a finite F p−1 p−1 p−1 field ( p, +, ·) the polynomial tn(x1,...,xn) = x1 · x2 · · · xn can be interpreted as the conjunctions of n variables, as it is equal to 0, if there is an xi = 0, and equal to 1 else. Equations of the form tn(x1 − y1,...,xn − yn) = 1 can then straightforwardly be used to encode the p-colouring problem, thus CSAT(Fp, +, ·) is NP-complete. However, if we remove the multiplication · from the signature (and therefore the ability to form conjunctions), we are left with the problem of solving a linear equation over Fp, which is clearly in P. In the context of universal algebra, polynomials with similar properties as tn are called (non constant) absorbing polynomials. The results in [IK18] link the existence of absorbing polynomials, commutator theoreti- cal 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 poly- nomials 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 polyno- mials, and therefore (almost always) have hard CSAT and CEQV problems. 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. CSAT AND CEQV FOR NILPOTENT MALTSEV ALGEBRAS OF FITTING LENGTH > 2 3 In fact, a conjecture of Barrington, Straubing and Th´erien about so called CC-circuits [BST90] was shown in [Kom19] to be equivalent to absorbing polynomials in nilpotent algebras having the following lower bound on their size: Conjecture 1 ( [BST90, Kom19]). For every finite nilpotent Maltsev algebra A, there are constants c > 1, 0 < d ≤ 1 (that depend on A), such that any circuit Cn(x1,...,xn) that is d not constant and absorbing, has size Ω(cn ). Under Conjecture 1 we get following conditional result, which indicated that the problems might be easy for nilpotent Maltsev algebras: Theorem 2 ( [Kom19]). Let A be a finite nilpotent Maltsev algebra. Under the assumption d of Conjecture 1, CSAT(A) and CEQV(A) can be solved in quasipolynomial time O(clog(n) ) (where c> 1, d ≥ 1 depend on A). In a recent paper [IKK20] Idziak, Kawa lek and Krzaczkowski contrasted this theorem with examples of nilpotent Maltsev algebras D[p1,...,pk], obtained as an extensions of Abelian 1 − groups, in which there are indeed non constant n-ary absorbing polynomials of size cn k 1 with some additional nice properties. Under the exponential time hypothesis, this allowed them to prove also quasipolynomial lower bounds for the complexity of CSAT(D[p1,...,pk]) and CEQV(D[p1,...,pk]), which indicates, together with Theorem 2, that both CSAT(D[p1,...,pk]) and CEQV(D[p1,...,pk]) have intermediate complexities. We are going to generalize this result to all nilpotent Maltsev algebra of Fitting length k ≥ 2 (where the Fitting length, in some sense, measures how far the algebra is from being supernilpotent). Theorem 3. Let A be a nilpotent Maltsev algebra of Fitting length k ≥ 2. Then, under the exponential time hypothesis (ETH), the deterministic complexity of CSAT(A) and CEQV(A) k−1 has a lower bound of O(clog(n) ), for some c> 1. Theorem 3 together with Theorem 2 then implies the following: Corollary 4. Let A be a nilpotent Maltsev algebra of Fitting length k > 2. Then, under the ETH and Conjecture 1, CSAT(A) and CEQV(A) can be solved in quasipolynomial, but not polynomial time. So then, in particular • CSAT(A) is NP-intermediate, • CEQV(A) is coNP-intermediate. By Corollary 4, nilpotent Maltsev algebras of Fitting length k = 2 remain essentially the only algebras from congruence modular varieties, for which we do not have a clear picture about the complexity of CSAT(A) and CEQV(A). The tools we use to prove Theorem 3 involve the study of wreath product representation of nilpotent algebras, which might have some further applications in understanding algorithmic and structural aspects of nilpotent algebras.
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