Cornering Robots to Synchronize a DFA
arxiv(2024)
摘要
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 ℝ^d with bidirectional connected edge sets in which each edge
( x, y) is labeled y - 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 Černý's conjecture. Using similar
ideas, we generalise a result of Ananichev and Volkov
from monotonic automata to a wider class of
DFAs admitting well-behaved partial orders. 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 edge labels.
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