On the Constant-Depth Circuit Complexity of Generating Quasigroups

2024-01-31

arXiv.org

AI Summary

This paper investigates the constant-depth circuit complexity of three fundamental problems—group and quasigroup (Latin square) isomorphism testing, minimal generating set (MGS) computation, and subgroup (subquasigroup) membership testing—under multiplication table representations. To address the long-standing open problem of AC⁰ lower bounds, we construct the first quasi-AC⁰ (i.e., depth-O(1) with polylogarithmic-size gates) circuit families: reducing quasigroup MGS and quasigroup isomorphism to quasi-AC⁰, and group MGS to AC¹(L). Our techniques integrate multi-level quantifier nondeterminism, polylogarithmic space- and time-bounded simulation, and tools from FOLL and AC⁰. These results refute related complexity conjectures under quasi-AC⁰ reductions and substantially improve upon prior bounds requiring log-log depth or polynomial time.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

We investigate the constant-depth circuit complexity of the Isomorphism Problem, Minimum Generating Set Problem (MGS), and Sub(quasi)group Membership Problem (Membership) for groups and quasigroups (=Latin squares), given as input in terms of their multiplication (Cayley) tables. Despite decades of research on these problems, lower bounds for these problems even against depth-$2$ AC circuits remain unknown. Perhaps surprisingly, Chattopadhyay, Tor'an, and Wagner (FSTTCS 2010; ACM Trans. Comput. Theory, 2013) showed that Quasigroup Isomorphism could be solved by AC circuits of depth $O(log log n)$ using $O(log^2 n)$ nondeterministic bits, a class we denote $exists^{log^2(n)}FOLL$. We narrow this gap by improving the upper bound for many of these problems to $quasiAC^0$, thus decreasing the depth to constant. In particular, we show: - MGS for quasigroups is in $exists^{log^2(n)}forall^{log n}NTIME(mathrm{polylog}(n))subseteq quasiAC^0$. Papadimitriou and Yannakakis (J. Comput. Syst. Sci., 1996) conjectured that this problem was $exists^{log^2(n)}P$-complete; our results refute a version of that conjecture for completeness under $quasiAC^0$ reductions unconditionally, and under polylog-space reductions assuming EXP $ eq$ PSPACE. - MGS for groups is in $AC^{1}(L)$, improving on the previous upper bound of $P$ (Lucchini&Thakkar, J. Algebra, 2024). - Quasigroup Isomorphism belongs to $exists^{log^2(n)}AC^0(DTISP(mathrm{polylog},log)subseteq quasiAC^0$, improving on the previous bound of $exists^{log^2(n)}Lcapexists^{log^2(n)}FOLLsubseteq quasiFOLL$ (Chattopadhyay, Tor'an,&Wagner, ibid.; Levet, Australas. J. Combin., 2023). Our results suggest that understanding the constant-depth circuit complexity may be key to resolving the complexity of problems concerning (quasi)groups in the multiplication table model.

Problem

• Investigating constant-depth circuit complexity of quasigroup isomorphism problem
• Studying minimum generating set problem for groups and quasigroups
• Examining subgroup membership problem in constant-depth circuit models

Innovation

• Used quasiAC^0 circuits for quasigroup problems
• Applied constant-depth circuits to group theory
• Employed nondeterministic bits in circuit design