Complexity of Modular Circuits

TL;DR

This paper investigates the complexity of modular circuits computing AND, revealing new subexponential and probabilistic circuit constructions, and explores implications for solving equations over symmetry groups, linking circuit complexity with algebraic problems.

Contribution

It introduces subexponential depth-2 modular circuits for AND, analyzes their complexity, and connects these findings to the complexity of solving equations over certain symmetry groups.

Findings

Subexponential depth-2 circuits for AND are constructed.

Probabilistic polynomial-size depth-2 circuits for AND are shown to exist.

Complexity classifications for equations over symmetry groups are derived.

Abstract

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.