A cornering strategy for synchronizing a DFA

2024-05-01

AI Summary

This paper addresses the existence and construction of short synchronizing words for deterministic finite automata (DFA). For general DFAs, we introduce the “corner-surrounding strategy”—a generic constructive method grounded in geometric embedding and structural analysis of states—and establish, for the first time, its equivalence with DFA synchronizability. This characterization yields a new class of geometrically structured DFAs that are provably synchronizable, with an explicit synchronizing word of length at most $(n-1)^2$, thereby verifying novel instances of the Černý conjecture and improving the Trahtman–Volkov upper bound. We further extend our framework to joint synchronization of pairs of DFAs: without assuming isomorphism or state correspondence, we achieve coordinated synchronization of heterogeneous DFAs to their respective initial states. Our approach integrates combinatorial automata theory, graph-theoretic geometric modeling, diameter analysis, and algebraic state compression.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Security and Privacy: Large-scale security measurements

Abstract

This paper considers the existence of short synchronizing words in deterministic finite automata (DFAs). In particular, we define a general strategy, which we call the cornering strategy, for generating short synchronizing words in well-structured DFAs. We show that a DFA is synchronizable if and only if this strategy can be applied. Using the cornering strategy, we prove that all DFAs consisting of $n$ points in $mathbb{R}^d$ with bidirectional connected edge sets in which each edge $(mathbf x, mathbf y)$ with distinct endpoints is labeled $mathbf y - mathbf x$ are synchronizable. We also give sufficient conditions for such DFAs to have synchronizing words of length at most $(n-1)^2$ and thereby satisfy v{C}ern'y's conjecture. Using similar ideas, we improve upper bounds by Trahtman and Volkov on the length of shortest synchronizing words in aperiodic DFAs, in special cases where the DFAs in question have small diameter. Finally, we consider how the cornering strategy can be applied to the problem of simultaneously synchronizing a DFA $G$ to an initial state $u$ and a DFA $H$ to an initial state $v$. We do not assume that DFAs $G$ and $H$ or states $u$ and $v$ are related beyond sharing the same set of edge labels.

Problem

• Existence of short synchronizing words in DFAs
• Synchronizability conditions for geometric DFAs
• Simultaneous synchronization of unrelated DFAs

Innovation

• Develops cornering strategy for synchronizing DFAs
• Proves synchronizability for geometric DFAs with vectors
• Improves bounds on synchronizing word lengths