Most classical results in circuit complexity theory concern circuits over the Boolean domain. Besides their simplicity and the ease of comparing different languages, the actual architecture of computers is also an important motivating factor. On the other hand, by restricting attention to Boolean circuits, we lose sight of the much richer landscape of circuits over larger domains. Our goal is to bridge these two worlds: to use deep algebraic tools to obtain results in computational complexity theory, including circuit complexity, and to apply results from computational complexity to gain a better understanding of the structure of finite algebras. In this paper, we propose a unifying algebraic framework which we believe will help achieve this goal. Our work is inspired by branching programs and nonuniform deterministic automata introduced by Barrington, as well as by their generalization proposed by Idziak et al. We begin our investigation by studying the languages recognized by natural classes of algebraic structures. In particular, we characterize language classes recognized by circuits over simple algebras and over algebras from congruence modular varieties.

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, Kawalek and Krzaczkowski 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, but we also believe that the tools we developed are of independent interest in the study of nilpotent algebras.

Nonuniform Deterministic Finite Automata (NUDFA) over monoids were invented by Barrington to study boundaries of nonuniform constant-memory computation. Later, results on these automata helped to indentify interesting classes of groups for which equation satisfiability problem is solvable in (probabilistic) polynomial-time. Based on these results, we present a full characterization of groups, for which the identity checking problem has a probabilistic polynomial-time algorithm. We also go beyond groups, and propose how to generalise the notion of NUDFA to arbitrary finite algebraic structures. We study satisfiability of these automata in this more general setting. As a consequence, we present full description of finite algebras from congruence modular varieties for which testing circuit equivalence can be solved by a probabilistic polynomial-time procedure. In our proofs we use two computational complexity assumptions: randomized Expotential Time Hypothesis and Constant Degree Hypothesis.

The Constant Degree Hypothesis was introduced by Barrington et. al. [David A. Mix Barrington et al., 1990] to study some extensions of q-groups by nilpotent groups and the power of these groups in a computation model called NuDFA (non-uniform DFA). In its simplest formulation, it establishes exponential lower bounds for MOD_q∘MOD_m∘AND_d circuits computing AND of unbounded arity n (for constant integers d,m and a prime q). While it has been proved in some special cases (including d = 1), it remains wide open in its general form for over 30 years. In this paper we prove that the hypothesis holds when we restrict our attention to symmetric circuits with m being a prime. While we build upon techniques by Grolmusz and Tardos [Vince Grolmusz and Gábor Tardos, 2000], we have to prove a new symmetric version of their Degree Decreasing Lemma and use it to simplify circuits in a symmetry-preserving way. Moreover, to establish the result, we perform a careful analysis of automorphism groups of MOD_m∘AND_d subcircuits and study the periodic behaviour of the computed functions. Our methods also yield lower bounds when d is treated as a function of n. Finally, we present a construction of symmetric MOD_q∘MOD_m∘AND_d circuits that almost matches our lower bound and conclude that a symmetric function f can be computed by symmetric MOD_q∘MOD_p∘AND_d circuits of quasipolynomial size if and only if f has periods of polylogarithmic length of the form p^k q^𝓁.

The Constant Degree Hypothesis was introduced by Barrington et. al. (1990) to study some extensions of $q$-groups by nilpotent groups and the power of these groups in a certain computational model. In its simplest formulation, it establishes exponential lower bounds for $\mathrm{AND}_d \circ \mathrm{MOD}_m \circ \mathrm{MOD}_q$ circuits computing AND of unbounded arity $n$ (for constant integers $d,m$ and a prime $q$). While it has been proved in some special cases (including $d=1$), it remains wide open in its general form for over 30 years. In this paper we prove that the hypothesis holds when we restrict our attention to symmetric circuits with $m$ being a prime. While we build upon techniques by Grolmusz and Tardos (2000), we have to prove a new symmetric version of their Degree Decreasing Lemma and apply it in a highly non-trivial way. Moreover, to establish the result we perform a careful analysis of automorphism groups of $\mathrm{AND} \circ \mathrm{MOD}_m$ subcircuits and study the periodic behaviour of the computed functions. Finally, our methods also yield lower bounds when $d$ is treated as a function of $n$.

Abstract 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.

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. In particular we use the superlinear lower bound from [10] to check satisfiability of CC0 circuits in probabilistic 2O(n/ε(n)) time, where ε is some extremely slowly increasing function. Moreover we show that AND can be computed by a polynomial size modular circuit of depth 2 (with O(log n) random bits) providing a probabilistic computational model that can not be derandomized.

In [15] 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 two algorithms solving circuit satisfiability problem over supernilpotent algebras are presented. The first one is deterministic and is faster than fastest previous algorithm presented by Aichinger. 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 (given by Földvári) 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.

By a result of Horváth the equation solvability problem over finite nilpotent groups and rings is in P. We generalize his result, showing that the equation solvability over every finite supernilpotent Mal'cev algebra is in P. We also give an example of a nilpotent, but not supernilpotent Mal'cev algebra, whose identity checking problem is coNP-complete.