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Synchronization of strongly connected partial DFAs and prefix codes

Mikhail V. Berlinkov, Robert Ferens, Andrew Ryzhikov, Marek Szykuła

TL;DR

This work analyzes synchronization for strongly connected partial DFAs, tying them to complete DFAs via fixing, collecting, and induced automata to transfer core results. It proves that the rank conjecture for complete DFAs implies the same for partial DFAs and provides a framework to relate Černý-type bounds between models, including an algebraic approach via induced automata. For finite prefix codes, it derives a tight bound of O(n log^3 n) on the shortest reset word length of partial literal automata, by exploiting small-rank words and induced constructions. The study also shows that many questions about synchronization in partial DFAs reduce to the complete case, and establishes corresponding lower bounds for properly incomplete automata, illustrating the limits of improving these bounds without breakthroughs in the complete DFA setting. Overall, the paper unifies synchronization theory across partial and complete automata and highlights the practical interplay with prefix codes and their literal automata.

Abstract

We study synchronizing partial DFAs, which extend the classical concept of synchronizing complete DFAs and are a special case of synchronizing unambiguous NFAs. A partial DFA is called synchronizing if it has a word (called a \emph{reset word}) whose action brings a non-empty subset of states to a unique state and is undefined for all other states. The class of strongly connected partial automata is precisely the class of automata recognized prefix codes. While in the general case the problem of checking whether a partial DFA is synchronizing is PSPACE-complete, we show that in the strongly connected case this problem can be efficiently reduced to the same problem for a complete DFA. Using combinatorial, algebraic, and formal languages methods, we develop techniques that relate main synchronization problems for strongly connected partial DFAs to the same problems for complete DFAs. In particular, this includes the Černý and the rank conjectures, the problem of finding a reset word, and upper bounds on the length of the shortest reset words of literal automata of finite prefix codes. We conclude that solving fundamental synchronization problems is equally hard in both models, as an essential improvement of the results for one model implies an improvement for the other.

Synchronization of strongly connected partial DFAs and prefix codes

TL;DR

This work analyzes synchronization for strongly connected partial DFAs, tying them to complete DFAs via fixing, collecting, and induced automata to transfer core results. It proves that the rank conjecture for complete DFAs implies the same for partial DFAs and provides a framework to relate Černý-type bounds between models, including an algebraic approach via induced automata. For finite prefix codes, it derives a tight bound of O(n log^3 n) on the shortest reset word length of partial literal automata, by exploiting small-rank words and induced constructions. The study also shows that many questions about synchronization in partial DFAs reduce to the complete case, and establishes corresponding lower bounds for properly incomplete automata, illustrating the limits of improving these bounds without breakthroughs in the complete DFA setting. Overall, the paper unifies synchronization theory across partial and complete automata and highlights the practical interplay with prefix codes and their literal automata.

Abstract

We study synchronizing partial DFAs, which extend the classical concept of synchronizing complete DFAs and are a special case of synchronizing unambiguous NFAs. A partial DFA is called synchronizing if it has a word (called a \emph{reset word}) whose action brings a non-empty subset of states to a unique state and is undefined for all other states. The class of strongly connected partial automata is precisely the class of automata recognized prefix codes. While in the general case the problem of checking whether a partial DFA is synchronizing is PSPACE-complete, we show that in the strongly connected case this problem can be efficiently reduced to the same problem for a complete DFA. Using combinatorial, algebraic, and formal languages methods, we develop techniques that relate main synchronization problems for strongly connected partial DFAs to the same problems for complete DFAs. In particular, this includes the Černý and the rank conjectures, the problem of finding a reset word, and upper bounds on the length of the shortest reset words of literal automata of finite prefix codes. We conclude that solving fundamental synchronization problems is equally hard in both models, as an essential improvement of the results for one model implies an improvement for the other.

Paper Structure

This paper contains 18 sections, 20 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Observing a reactive system
  3. Synchronizing automata
  4. Synchronizing codes
  5. Automata for X*
  6. Carefully synchronizing DFAs
  7. Our contribution and organization of the paper
  8. Preliminaries
  9. Upper bounds
  10. Inseparability equivalence
  11. Fixing automaton
  12. Rank conjecture
  13. Collecting automaton
  14. Algorithmic issues
  15. The Cerny conjecture
  16. ...and 3 more sections

Key Result

Lemma 3.1

Let $\mathrsfs{A}=(Q,\Sigma,\delta)$ be a partial automaton, and let $S \subseteq Q$ be a subset such that $\kappa(S) \ge 2$. Then there is a word $w \in \Sigma^*$ of length at most $\kappa(Q)-\kappa(S)+1 \le n-|S|+1$ and such that $1 \le \kappa(\delta(S,w)) < \kappa(S)$.

Figures (2)

  • Figure 1: Left: a strongly connected partial 6-state binary automaton; right: the literal automaton of the prefix code $\{abaaa,abaab,abab,abba\}$.
  • Figure 2: Left: a collecting tree with root $[q3]=\{q_3,q_6\}$; right: the corresponding collecting automaton of the example from Fig. \ref{['fig:examples']}(left).

Theorems & Definitions (43)

  • Definition 3.1
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Definition 3.2: Fixing automaton
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Corollary 3.5
  • ...and 33 more