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Generating Fibonacci Words via the Prefix--Suffix Duplication Operation

Jul 22, 2026

This work investigates the generation of Fibonacci words of the same parity—specifically, deriving \(F_{2n}\) from \(F_{2p}\) or \(F_{2n+1}\) from \(F_{2p+1}\)—using prefix-suffix duplication operations of length at most \(k\). Drawing on combinatorics on words and formal language operations, we confirm and strengthen Dumitran’s conjecture by establishing that \(k = 3\) is the optimal bound for such derivations. Furthermore, we present the first linear-time algorithm capable of constructing a complete derivation sequence in \(O(|F_{2n}|)\) or \(O(|F_{2n+1}|)\) time. These results theoretically establish the tightness of bounded duplication mechanisms for Fibonacci words and provide an efficient, constructive method for their generation.

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Complexity of Universality and Related Decision Problems for Unary Two-Dimensional Automata

Jun 30, 2026

This study investigates the computational complexity of fundamental decision problems—universality, equivalence, inclusion, and disjointness—for unary two-dimensional automata. Focusing on deterministic and nondeterministic variants with restricted movement directions (specifically, three-way and two-way models), the work provides the first complete characterization of the complexity landscape across these automaton classes. By leveraging automata-theoretic techniques, complexity analysis, and careful reductions, the paper establishes tight upper and lower bounds for multiple problems. Notably, it proves that bounded universality is coNP-complete and precisely classifies several other problems within established complexity classes, including P, NL-hard, coNP-hard, and ELEMENTARY.

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Primitive Two-Dimensional Words and Iterated Pedal Triangles via Symbolic Coding

Apr 29, 2026

This work uncovers a profound connection between two-dimensional primitive words and iterated pedal triangles. By introducing a four-symbol finite coding to symbolize the pedal map and employing branched orbit constructions, the authors establish a bijection between $2 \times n$ two-dimensional primitive words over a binary alphabet and the $n$-th order first self-similar pedal triangles. This study constitutes the first precise bridge between combinatorics on words and geometric dynamical systems, rigorously demonstrating an exact correspondence in cardinality between these two classes of objects. In doing so, it unifies symbolic dynamics, combinatorics, and geometric iteration within a coherent framework, offering a novel paradigm for interdisciplinary research at the intersection of discrete mathematics and dynamical geometry.

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Generating Fibonacci Words via the Prefix--Suffix Duplication Operation

Jul 22, 2026

This work investigates the generation of Fibonacci words of the same parity—specifically, deriving \(F_{2n}\) from \(F_{2p}\) or \(F_{2n+1}\) from \(F_{2p+1}\)—using prefix-suffix duplication operations of length at most \(k\). Drawing on combinatorics on words and formal language operations, we confirm and strengthen Dumitran’s conjecture by establishing that \(k = 3\) is the optimal bound for such derivations. Furthermore, we present the first linear-time algorithm capable of constructing a complete derivation sequence in \(O(|F_{2n}|)\) or \(O(|F_{2n+1}|)\) time. These results theoretically establish the tightness of bounded duplication mechanisms for Fibonacci words and provide an efficient, constructive method for their generation.

0 citationsRead paper

Complexity of Universality and Related Decision Problems for Unary Two-Dimensional Automata

Jun 30, 2026

This study investigates the computational complexity of fundamental decision problems—universality, equivalence, inclusion, and disjointness—for unary two-dimensional automata. Focusing on deterministic and nondeterministic variants with restricted movement directions (specifically, three-way and two-way models), the work provides the first complete characterization of the complexity landscape across these automaton classes. By leveraging automata-theoretic techniques, complexity analysis, and careful reductions, the paper establishes tight upper and lower bounds for multiple problems. Notably, it proves that bounded universality is coNP-complete and precisely classifies several other problems within established complexity classes, including P, NL-hard, coNP-hard, and ELEMENTARY.

0 citationsRead paper

Primitive Two-Dimensional Words and Iterated Pedal Triangles via Symbolic Coding

Apr 29, 2026

This work uncovers a profound connection between two-dimensional primitive words and iterated pedal triangles. By introducing a four-symbol finite coding to symbolize the pedal map and employing branched orbit constructions, the authors establish a bijection between $2 \times n$ two-dimensional primitive words over a binary alphabet and the $n$-th order first self-similar pedal triangles. This study constitutes the first precise bridge between combinatorics on words and geometric dynamical systems, rigorously demonstrating an exact correspondence in cardinality between these two classes of objects. In doing so, it unifies symbolic dynamics, combinatorics, and geometric iteration within a coherent framework, offering a novel paradigm for interdisciplinary research at the intersection of discrete mathematics and dynamical geometry.

0 citationsRead paper