A Polynomial Algorithm for Deciding Complete Reachability and Determining Quadratic Reaching Thresholds

Core Concepts

A polynomial-time algorithm is developed to decide whether a given deterministic finite automaton is completely reachable, solving an open problem. Additionally, a quadratic upper bound is proven on the length of the shortest words reaching any non-empty subset of states in a completely reachable automaton.

Abstract

The content discusses the problem of deciding the complete reachability of deterministic finite automata (DFAs) and determining bounds on the reaching thresholds for such automata.
Key highlights:

The concept of completely reachable automata, where every non-empty subset of states can be reached from the full set of states, is introduced.
The complexity of deciding whether a given DFA is completely reachable was an open problem, and the content presents a polynomial-time algorithm to solve this problem.
The algorithm is based on a new "complement-intersecting" technique for finding extending words that can reach larger subsets of states.
The content also proves a weak version of Don's conjecture for completely reachable automata, showing that any non-empty subset of size S in an n-state automaton can be reached with a word of length at most 2n(n-|S|).
This implies a quadratic upper bound on the reset threshold (length of the shortest reset words) for the class of completely reachable automata, improving upon the previously known cubic bound.
The impact of the size of group orbits in a completely reachable automaton on the upper bounds is discussed, and it is observed that the Černý conjecture holds for automata without permutational letters or with a maximum orbit size of at most ln(n).

Stats

Every synchronizing n-state automaton has a reset threshold of at most (n-1)^2, as conjectured by Černý.
The currently known best upper bound on the length of the shortest reset words is cubic in n.
For the class of completely reachable automata, a cubic bound of 7/48n^3 + O(n^2) was previously known.

Quotes

"Every non-empty subset of states is the image of the action of some word applied to the set of all states. Thus, every completely reachable automaton is synchronizing."
"The notion of completely reachable automata was first introduced in 2016 by Bondar and Volkov, who also asked about the complexity of the computational problem of deciding whether a given automaton is completely reachable."

Key Insights Distilled From

A polynomial algorithm deciding the complete reachability and quadratic reaching thresholds

by Robe... at arxiv.org 04-10-2024

https://arxiv.org/pdf/2208.05956.pdf

Deeper Inquiries

What are the implications of the polynomial-time algorithm for deciding complete reachability on the study of synchronizing automata and their applications

The polynomial-time algorithm for deciding complete reachability has significant implications for the study of synchronizing automata and their applications. Firstly, it provides a more efficient way to determine if an automaton is completely reachable, which is a fundamental property in automata theory. This algorithm allows researchers to quickly verify if an automaton can reach every non-empty subset of states, which is crucial in various applications of automata theory.
Moreover, the algorithm can aid in the study of synchronizing automata, particularly in relation to the Černý conjecture. By efficiently determining the complete reachability of automata, researchers can further investigate the properties of synchronizing automata and potentially make progress towards resolving long-standing conjectures in the field.
In terms of applications, the polynomial algorithm can be utilized in various fields where automata theory is applied, such as in testing reactive systems and synchronization of codes. The ability to efficiently decide complete reachability can enhance the development and analysis of systems that rely on automata-based models, leading to improvements in system design, verification, and optimization.

How can the techniques developed in this work be extended to address the Černý conjecture or other open problems in the theory of synchronizing automata

The techniques developed in this work can be extended to address the Černý conjecture and other open problems in the theory of synchronizing automata. One possible extension is to refine the algorithm to specifically target the Černý conjecture, which aims to find the shortest reset word for synchronizing automata. By incorporating additional constraints or optimizations into the algorithm, researchers may be able to tackle the Černý conjecture more directly and potentially improve the existing bounds on the length of reset words.
Furthermore, the complement-intersecting technique used in the algorithm could be adapted to explore other conjectures or problems in synchronizing automata theory. By applying similar principles to different scenarios or variations of the Černý conjecture, researchers may uncover new insights and solutions to related challenges in the field.
Overall, the techniques developed in this work lay a strong foundation for further research into synchronizing automata and related problems. By building upon the algorithm and methodologies presented, researchers can advance the understanding of automata theory and potentially make significant contributions to the resolution of open problems in the field.

Are there any connections between the size of group orbits in a completely reachable automaton and the structural properties or complexity-theoretic aspects of such automata

There are indeed connections between the size of group orbits in a completely reachable automaton and the structural properties or complexity-theoretic aspects of such automata. The size of group orbits plays a crucial role in determining the reachability and synchronization properties of automata. In completely reachable automata, the size of group orbits can impact the efficiency of finding properly extending words and determining the reset thresholds.
The structural properties of completely reachable automata, such as the presence of primitive permutation groups in their transition monoids, can be linked to the complexity of deciding complete reachability and related problems. Automata with specific structural properties, such as primitive permutation groups, may exhibit different reachability characteristics and synchronization behaviors, influencing the overall complexity of analyzing such automata.
Moreover, the size of group orbits can provide insights into the diversity and complexity of behaviors exhibited by completely reachable automata. By studying the relationships between group orbits, structural properties, and reachability properties, researchers can gain a deeper understanding of the computational and structural aspects of automata theory. This holistic approach can lead to new discoveries and advancements in the field of automata theory and its applications.