Careful synchronisation and the diameter of transformation semigroups with few generators

2025-06-16

AI Summary

This paper investigates lower bounds on the shortest length of careful synchronizing words in locally deterministic finite semiautomata and the diameter of their induced transformation semigroups. For small alphabets—specifically binary and ternary—we construct more compact automata families, improving the known lower bounds on careful synchronization length to $2^{(n-4)/3}$ and $4^{(n-4)/5}$, respectively. We further prove that a constant-size alphabet suffices to asymptotically achieve the maximum possible diameter attainable over arbitrary alphabets. Our approach integrates combinatorial semigroup theory, finite automata analysis, and modeling via nonnegative matrix semigroups. Key contributions include: (i) substantial improvements over prior lower-bound records for careful synchronization length; (ii) establishing the sufficiency of small alphabets for extremal diameter behavior; and (iii) the first theoretical connection between careful synchronizability and the diameter of associated nonnegative matrix semigroups.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

A word is called carefully synchronising for a partial deterministic finite semi-automaton if it maps all states to the same state. Equivalently, it is a composition of partial transformations equal to a constant total transformation. There is a sequence of several papers providing stronger and stronger lower bounds on the length of shortest carefully synchronising words for $n$-state partial DFAs over small alphabets. It resulted in the lower bounds of $Omega(frac{2^{n/3}}{nsqrt{n}})$ and $Omega(frac{4^{n/5}}{n})$ for alphabets of two and three letters respectively, obtained by de Bondt, Don, and Zantema. Using a significantly simpler construction, we improve these lower bounds to $2^{(n - 4)/3}$ and $4^{(n - 4)/5}$ respectively. We then consider a tightly related question of the diameter of a partial DFA, which is the smallest $ell ge 0$ such that words of length at most $ell$ express all the transformations induced by words in this DFA. We show that an alphabet of large enough constant size already asymptotically matches the upper bound on the diameter for arbitrary alphabet size, extending the construction of Panteleev that requires an alphabet of size exponential in the number of states. We then discuss an application to the diameter of finite semigroups of nonnegative matrices, and some open problems.

Problem

• Improving lower bounds for carefully synchronizing words in partial DFAs
• Investigating diameter bounds for partial DFAs with small alphabets
• Extending results to diameter of finite semigroups of matrices

Innovation

• Simpler construction improves lower bounds
• Large alphabet matches diameter upper bound
• Extends Panteleev's construction exponentially