Kunz languages for numerical semigroups are context sensitive

TL;DR

The paper investigates the computational complexity of numerical semigroups by encoding them as Kunz languages, linking Kunz depth to the Chomsky hierarchy. It shows that the Kunz language of depth $2$ is regular, while depths $q\ge 3$ yield context-sensitive languages that are not regular, with $K_q$ not context-free for $q\ge 5$ and a high-level LBA construction illustrating context-sensitivity for $q=3$. These results imply increasing difficulty for language-encoded questions about numerical semigroups as depth grows and motivate exploring intermediate languages between context-free and context-sensitive. The work establishes a concrete bridge between semigroup theory and formal language theory, with potential implications for complexity considerations and Wilf-type conjectures in restricted depth classes.

Abstract

There is a one-to-one and onto correspondence between the class of numerical semigroups of depth $n$, where $n$ is an integer, and a certain language over the alphabet $\{1,\ldots,n\}$ which we call a Kunz language of depth $n$. The Kunz language associated with the numerical semigroups of depth $2$ is the regular language $\{1,2\}^*2\{1,2\}^*$. We prove that Kunz languages associated with numerical semigroups of larger depth are context-sensitive but not regular.

Figures (2)

• Figure 1: dfa recognizing $\{1,2\}^*2\{1,2\}^*$.
• Figure 2: lba that accepts $K_3$.

Theorems & Definitions (18)

• Definition 1
• Example 2
• Definition 3
• Example 4
• Lemma 5
• proof
• Definition 6
• proof
• Proposition 10
• proof