Phase Transitions in Decision Problems Over Odd-Sized Alphabets

TL;DR

This paper extends the understanding of phase transitions in decision problems to include those over odd-sized alphabets, showing that such phenomena are not limited to even-sized cases.

Contribution

It demonstrates that decision problems over odd-sized alphabets also exhibit phase transitions, filling a gap in the existing theory.

Findings

Phase transitions occur in decision problems over odd-sized alphabets.

The subset of problems with phase transitions is similar to the even-sized case.

Supports the ubiquity of phase transitions across different alphabet sizes.

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.