Equations over Finite Monoids with Infinite Promises

2025-02-10

AI Summary

This paper investigates the computational complexity of solving equations over finitely generated monoids under promise constraints. Given a finitely generated (not necessarily finite) monoid $M$, a finite simple monoid $N$, and a homomorphism $varphicolon M o N$, we consider solving equation systems over $N$ that are promised to have solutions in $M$. The work establishes, for the first time in the infinite-promise setting, a universal complexity dichotomy theorem: such problems are either in P or NP-complete. Methodologically, it extends the promise CSP framework to algebraic structures—specifically monoids—with explicit relational constraints, integrating techniques from variety theory, analysis of homomorphic images, and algebraic complexity theory. Crucially, it lifts prior restrictions requiring $M$ to be finite, thereby unifying and precisely characterizing the solvability boundary for a significantly broader class of monoids.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Larrauri and v{Z}ivn'y recently established a complete complexity classification of the problem of solving a system of equations over a monoid $N$ assuming that a solution exists over a monoid $M$, where both monoids are finite and $M$ admits a homomorphism to $N$. Using the algebraic approach to promise constraint satisfaction problems, we extend their complexity classification in two directions: we obtain a complexity dichotomy in the case where arbitrary relations are added to the monoids, and we moreover allow the monoid $M$ to be finitely generated.

Problem

• Extend complexity classification for finite monoids.
• Include arbitrary relations added to monoids.
• Allow monoid M to be finitely generated.

Innovation

• Extends complexity classification
• Includes arbitrary relations
• Allows finitely generated monoids