Cornering Robots to Synchronize a DFA

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 Rᵈ 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 {C}ern\'y's conjecture. Using similar ideas, we generalise a result of Ananichev and Volkov {ananichev2004synchronizing} 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.