Phase Transitions in Decision Problems Over Odd-Sized Alphabets

TL;DR

The paper extends the ubiquity of phase transitions in decision problems to odd-sized alphabets. It proves that any adequately-balanced language over an odd alphabet is preserving-P-isomorphic to a paddable not-anywhere-exponentially-unbalanced language over an even alphabet, which by prior results exhibits a phase transition; this property is transferred back through the preserving-P-isomorphism. The approach combines paddability, RoughP constructions, and preserving-P-isomorphisms (including alt-NAEU) to transport phase-transition behavior across alphabet sizes. This broadens the scope of phase-transition phenomena in computational problems and suggests avenues for application in areas like quantum-computation verification. The work also discusses potential relaxations and limitations, indicating future research directions toward further generalization.

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.

Figures (7)

• Figure 1: [From Ref. jackson2025explainingubiquityphasetransitions] A typical example of a phase transition. The defining feature of a phase transition -- in decision problems -- is the change in the accepting fraction as the parameter -- a real-valued polynomial time computable function of an instance of the problem -- approaches the threshold value (the vertical red line).
• Figure 2: Diagrams of the situation constructed in Lemma \ref{['lem:BasicIsomorpLemma']}, where $\mathbf{1}_{\mathcal{L}}: \Sigma^* \longrightarrow \{ 0, 1\}$ is the indicator function of $\mathcal{L}$, defined as, $\forall x \in \Sigma^*$, $\mathbf{1}_{\mathcal{L}}(x) = 1\text{ if } x \in \mathcal{L}0\text{ if } x \not \in \mathcal{L}$ and $\mathbf{1}_{\mathcal{H}}: \Sigma^* \longrightarrow \{ 0, 1\}$ is the indicator function of $\mathcal{H}$, defined as, $\forall x \in \Pi^*$, $\mathbf{1}_{\mathcal{H}}(x) = 1\text{ if } x \in \mathcal{H}0\text{ if } x \not \in \mathcal{H}$. The upper diagram depicts the construction of $\xi$ and $\xi^{-1}$, while the lower diagram depicts the functioning of $\xi$ and $\xi^{-1}$ to preserve membership of the respective languages. Lemma \ref{['lem:BasicIsomorpLemma']} is equivalent to saying that a $\mathcal{H} \subseteq \Pi^*$ exists for any $\mathcal{L} \subseteq \Sigma^*$ such that the lower diagram commutes.
• Figure 3: Diagram of the situation constructed in Lemma \ref{['lem:PIsoExists']}, where $\mathbf{1}_{\mathcal{L}}: \Sigma^* \longrightarrow \{ 0, 1\}$ is the indicator function of $\mathcal{L}$, as in Fig. \ref{['fig:Lemma3Diagram']}, and $\mathbf{1}_{\mathcal{H}}: \Sigma^* \longrightarrow \{ 0, 1\}$ is the indicator function of $\mathcal{H}$, as in Fig. \ref{['fig:Lemma3Diagram']}. Additionally, $i_2^\Sigma: \Sigma^* \times \Sigma^* \longrightarrow \Sigma^*$ is defined by, $\forall x, y \in \Sigma^*$, $i_2^\Sigma(x,y) = y$, and $i_2^\pi: \Pi^* \times \Pi^* \longrightarrow \Pi^*$ is defined similarly. Lemma \ref{['lem:PIsoExists']} implies this diagram commutes and all mappings -- except the indicator functions -- can be applied in polynomial time. Note the symmetry of the above diagram, along the center line, between $\Sigma^*$ and $\Pi^*$ and that the left-hand-side of the above diagram follows completely from the right-hand-side, given $\xi$ and $\xi^{-1}$.
• Figure 4: A diagram entirely entailed by the lower half, which follows from both $\mathcal{L}$ and $H$ being paddable and mutually reducible (implying the existence of $\phi$), and $\xi: \Sigma^* \longrightarrow \Pi^*$ being a preserving-P-isomorphism. Note the symmetry between the upper and lower halves. I additionally note that the two instance of $\{ 0, 1 \}$ can be merged into one, keeping all arrows into them the same, and the diagram still commutes (with the horizontal symmetry also maintained).
• Figure 5: Diagram used in the proof of Lemma \ref{['lem:RelationTheorem']}. Note that $H_{\mathcal{L}}$ is defined from $H_{\mathcal{H}}$ (which is defined from $\mathcal{H}$ by the method in Ref. farago2016roughly) by the commutativity of this diagram.

Theorems & Definitions (22)

• Theorem 1: Theorem 2 in Ref. jackson2025explainingubiquityphasetransitions
• Theorem 2
• Theorem 3
• proof
• proof : Proof of Theorem \ref{['lem:MainLemma']}
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof