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We study how the complexity of modular circuits computing AND depends on the depth of the circuits and the prime factorization of the modulus they use. In particular our construction of subexponential circuits of depth 2 for AND helps us to classify (modulo Exponential Time Hypothesis) modular circuits with respect to the complexity of their satisfiability. We also study a precise correlation between this complexity and the sizes of modular circuits realizing AND. On the other hand showing that AND can be computed by a polynomial size probabilistic modular circuit of depth 2 (with O(log n) random bits) providing a probabilistic computational model that can not be derandomized. We apply our methods to determine (modulo ETH) the complexity of solving equations over groups of symmetries of regular polygons with an odd number of sides. These groups form a paradigm for some of the remaining cases in characterizing finite groups with respect to the complexity of their equation solving.

The study of the complexity of the equation satisfiability problem in finite groups had been initiated by Goldmann and Russell (2002) where they showed that this problem is in polynomial time 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 $G$ having a nilpotent normal subgroup $H$ with nilpotent factor $G/H$. This paper shows that such normal subgroup must exist in each finite group with equation satisfiability solvable in polynomial time, unless the Exponential Time Hypothesis fails.

In arXiv:1710.08163 a generalization of Boolean circuits to arbitrary finite algebras had been introduced and applied to sketch P versus NP-complete borderline for circuits satisfiability over algebras from congruence modular varieties. However the problem for nilpotent (which had not been shown to be NP-hard) but not supernilpotent algebras (which had been shown to be polynomial time) remained open. In this paper we provide a broad class of examples, lying in this grey area, and show that, under the Exponential Time Hypothesis and Strong Exponential Size Hypothesis (saying that Boolean circuits need exponentially many modular counting gates to produce boolean conjunctions of any arity), satisfiability over these algebras have intermediate complexity between $\Omega(2^{c\log^{h-1} n})$ and $O(2^{c\log^h n})$, where $h$ measures how much a nilpotent algebra fails to be supernilpotent. We also sketch how these examples could be used as paradigms to fill the nilpotent versus supernilpotent gap in general. Our examples are striking in view of the natural strong connections between circuits satisfiability and Constraint Satisfaction Problem for which the dichotomy had been shown by Bulatov and Zhuk.

Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras $\mathbf A$ there is a polynomial time algorithm that decides if an equation over $\mathbf A$ has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time., Comment: 50 pages

We investigate quantitative properties of BCI and BCK logics. The first part of the paper compares the number of formulas provable in BCI versus BCK logics. We consider formulas built on implication and a fixed set of $k$ variables. We investigate the proportion between the number of such formulas of a given length $n$ provable in BCI logic against the number of formulas of length $n$ provable in richer BCK logic. We examine an asymptotic behavior of this fraction when length $n$ of formulas tends to infinity. This limit gives a probability measure that randomly chosen BCK formula is also provable in BCI. We prove that this probability tends to zero as the number of variables tends to infinity. The second part of the paper is devoted to the number of lambda terms representing proofs of BCI and BCK logics. We build a proportion between number of such proofs of the same length $n$ and we investigate asymptotic behavior of this proportion when length of proofs tends to infinity. We demonstrate that with probability 0 a randomly chosen BCK proof is also a proof of a BCI formula.