[Paper Review] State complexity of multiple catenation

Pascal Caron, Jean-Gabriel Luque+1 more|arXiv (Cornell University)|Jul 14, 2016

semigroups and automata theory9 references3 citations

TL;DR

This paper improves the state complexity of multiple catenation by reducing the alphabet size of witnesses from $2\alpha - 1$ to $\alpha + 1$ using Brzozowski's deterministic finite automata (DFAs). It introduces a recursive formula for computing the upper bound and provides computational and analytical evidence for a conjecture that $\alpha$-letter alphabets may suffice, validating it for $\alpha = 2$ and $\alpha = 3$. The key contribution is a significant reduction in alphabet size with minimal impact on state complexity bounds.

ABSTRACT

We improve some results relative to the state complexity of the multiple catenation described by Gao and Yu. In particular we nearly divide by 2 the size of the alphabet needed for witnesses. We also give some refinements to the algebraic expression of the state complexity, which is especially complex with this operation. We obtain these results by using peculiar DFAs defined by Brzozowski.

Motivation & Objective

• To reduce the alphabet size required for state complexity witnesses in multiple catenation of regular languages.
• To provide a recursive formula for computing the state complexity of sequential catenation of $\alpha$ regular languages.
• To validate the use of Brzozowski's DFAs as a powerful tool for constructing minimal-state witnesses.
• To investigate the possibility of achieving optimal alphabet size $\alpha$ for $\alpha$-language catenation.

Proposed method

• Constructs a family of $\alpha$ DFAs over an $\alpha + 1$-letter alphabet to serve as witnesses for multiple catenation.
• Applies Brzozowski's construction of minimal DFAs from reverse automata to generate state-minimal witnesses.
• Uses recursive state transition analysis to derive a computable expression for the state complexity of $\alpha$-fold catenation.
• Employs a detailed state equivalence proof via distinguishing words to verify minimality of the constructed automaton.
• Introduces a combinatorial framework based on compositions to express the state complexity bound more efficiently than prior algebraic formulas.
• Validates conjectures computationally up to $\alpha = 6$ or $7$, and proves them for $\alpha = 2$ and $\alpha = 3$.

Experimental results

Research questions

• RQ1 Can the alphabet size of witnesses for multiple catenation be reduced below $2\alpha - 1$?
• RQ2 Is there a recursive formula that simplifies the computation of state complexity for $\alpha$-fold catenation?
• RQ3 Can Brzozowski's DFAs be used to construct optimal witnesses for multiple catenation with $\alpha$-letter alphabets?
• RQ4 Is the conjecture that $\alpha$-letter alphabets suffice for $\alpha$-fold catenation true for all $\alpha \geq 2$?
• RQ5 What is the minimal alphabet size required to achieve the maximal state complexity in multiple catenation?

Key findings

• The paper reduces the alphabet size for witnesses of multiple catenation from $2\alpha - 1$ to $\alpha + 1$, nearly halving the required alphabet size.
• A recursive formula is provided for computing the state complexity of $\alpha$-fold catenation, enabling efficient computation.
• The constructed $\alpha$-DFA family over $\alpha + 1$ letters is proven to be a minimal witness for the maximal state complexity.
• For $\alpha = 2$ and $\alpha = 3$, the authors prove that $\alpha$-letter alphabets suffice, supporting the conjecture of optimality.
• The state complexity bound is expressed via a combinatorial formula involving compositions, offering a more efficient alternative to Gao and Yu’s complex algebraic expressions.
• The minimality of the constructed automaton is confirmed via a distinguishing-word argument that proves all states are pairwise non-equivalent.