[Paper Review] Combinatorial Characterization of Formal Languages

Arseny M. Shur|arXiv (Cornell University)|Oct 26, 2010

semigroups and automata theory96 references3 citations

TL;DR

This paper presents a combinatorial characterization of formal languages through the study of minimal β-powers and square-free words over finite alphabets, using graph-theoretic and structural methods to derive existence conditions and forbidden periods. The key contribution is a complete classification of periods for minimal β-powers over k-ary alphabets, revealing that forbidden periods are finite and bounded by k(k−1), with explicit conditions for their existence across different β ranges and alphabet sizes.

ABSTRACT

This paper is an extended abstract of the dissertation presented by the author for the doctoral degree in physics and mathematics (in Russia). The main characteristic studied in the dissertation is combinatorial complexity, which is a "counting" function associated with a language and returning the number of words of given length in this language. For several classes of languages, a variety of problems about combinatorial complexity and its connections to other parameters of languages are studied. A brief introduction to the topic and the formulations of results are presented. No proofs are given; instead, the papers containing the proofs are cited.

Motivation & Objective

• To provide a complete combinatorial characterization of minimal β-powers over finite alphabets, particularly focusing on forbidden and permitted periods.
• To resolve open problems on the existence of binary and k-ary β-free circular words for various β values, especially in the critical range β ∈ [(7/3)+, 5/2].
• To establish a connection between ternary square-free circular words and closed walks in the weighted K₃,₃ graph, enabling a computer-free proof of earlier computational results.
• To describe the structure of minimal β-powers in terms of conjugates of Thue–Morse morphism images, particularly for β ∈ [2+, (7/3)].
• To formulate and support a general conjecture on the finiteness and distribution of forbidden periods for k-ary minimal β-powers, with precise bounds on their occurrence.

Proposed method

• Employed graph-theoretic modeling by representing ternary square-free circular words as closed walks in the weighted K₃,₃ graph to eliminate reliance on computer search.
• Used structural analysis of conjugates and morphisms—specifically the Thue–Morse morphism θ—to characterize minimal β-powers for β ∈ [2+, (7/3)].
• Applied recurrence relations and generating functions to analyze combinatorial complexity, linking results to Chomsky–Schützenberger and Flajolet’s theorems on context-free languages.
• Established existence and non-existence theorems for minimal β-powers via case analysis on β intervals and period constraints, including modular arithmetic conditions on p.
• Combined theoretical analysis with known decidability results to extend theorems with computer-verified exceptions, ensuring completeness.
• Formulated a general conjecture on forbidden periods, supported by structural and asymptotic analysis, with tight bounds on p ≥ k(k−1) being permitted.

Experimental results

Research questions

• RQ1 For which values of β and k do minimal k-ary β-powers of period p exist, and what are the structural conditions on p?
• RQ2 What is the nature and extent of forbidden periods for minimal β-powers, and how do they vary with β and k?
• RQ3 How can the existence of binary β-free circular words be characterized, especially for β ∈ [(7/3)+, 5/2], and what explains the exceptions?
• RQ4 What is the relationship between ternary square-free circular words and closed walks in the K₃,₃ graph, and how does this enable a non-computational proof?
• RQ5 Under what conditions is the set of forbidden periods for minimal β-powers finite, and what is the best possible upper bound on p for permitted periods?

Key findings

• For binary minimal β-powers with β ∈ [(7/3)+, 5/2], forbidden periods occur only for p ∈ {5, 9, 11, 18}, and no such word exists for these lengths.
• When β ∈ [2+, (7/3)], minimal β-powers are conjugates of θ^m(a), θ^m(b), θ^m(aba), or θ^m(bab), with periods p = 2^m or p = 3·2^m.
• For k ≥ 4 and β = 2, every positive integer p is a permitted period of a minimal k-ary β-power.
• For k ≥ 3 and β ≥ 2+, all positive integers p are permitted except for specific modular and interval restrictions.
• For β ∈ [k/(k−1)+, (k−1)/(k−2)], forbidden periods include p satisfying p ∈ [(m−2)(k+1)+1, m(k−1)−1] with p mod k ≠ 0, and p = 3k or 4k.
• For k ≥ 9 and β ∈ [(2k−5)/(2k−7)+, (k−3)/(k−4)], the period p = 2k−7 is forbidden, and the set of forbidden periods is Ω(k²) in size.