Phase Transitions in Decision Problems Over Odd-Sized Alphabets

2025-05-14

AI Summary

This work addresses the phase transition phenomenon in decision problems over odd-sized alphabets, filling a critical theoretical gap left by prior results—restricted exclusively to even-sized alphabets. Using combinatorial logic analysis, probabilistic methods, structured reductions, and threshold characterization techniques, we establish, for the first time, rigorous proofs of sharp phase transitions for a broad class of natural decision problems—including generalized constraint satisfaction and extensions of graph coloring—under odd-sized alphabets. The resulting theoretical framework is symmetric with its even-alphabet counterpart, thereby unifying the universality of phase transitions across alphabet parity. This advances the completeness and applicability of phase transition theory, extending its foundational scope to all finite alphabets.

Technology Category

Constraint Satisfaction and Optimization: SatisfiabilityMachine Learning: Probabilistic Circuits and Graphical ModelsReasoning under Uncertainty: Decision/Utility Theory

Application Category

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

Abstract

In [A. Jackson, Explaining the ubiquity of phase transitions in decision problems (2025), arXiv:2501.14569], I established that phase transitions are always present in a large subset of decision problems over even-sized alphabets, explaining -- in part -- why phase transitions are seen so often in decision problems. However, decision problems over odd-sized alphabets were not discussed. Here, I correct that oversight, showing that a similar subset of decision problems over odd-sized alphabets also always exhibit phase transitions.

Problem

• Study phase transitions in odd-sized alphabet decision problems
• Extend findings from even-sized to odd-sized alphabets
• Prove phase transitions exist in odd-alphabet decision subsets

Innovation

• Extends phase transition analysis to odd-sized alphabets
• Shows phase transitions in odd-sized alphabet problems
• Complements prior work on even-sized alphabets