On the Smallest Synchronizing Terms of Finite Tree Automata

Abstract

This paper deals with properties of synchronizing terms for finite tree automata, which is a generalization of the synchronization principle of deterministic finite string automata (DFA) and such terms correspond to a connected subgraph, where a state in the root is always the same regardless of states of subtrees attached to it. We ask, what is the maximum height of the smallest synchronizing term of a deterministic bottom-up tree automaton (DFTA) with n states, which naturally leads to two types of synchronizing terms, called weak and strong, that depend on whether a variable, i.e., a placeholder for a subtree, must be present in at least one leaf or all of them. We prove that the maximum height in the case of weak synchronization has a theoretical upper bound ${\text {sl}}(n) + n - 1$, where ${\text {sl}}(n)$ is the maximum length of the shortest synchronizing string of an n-state DFAs. For strong synchronization, we prove exponential bounds. We provide a theoretical upper bound of $2^n-n-1$ for the height and two constructions of automata approaching it. One achieves the height of $\varTheta (2^{n-\sqrt{n}})$ with an alphabet of linear size, and the other achieves $2^{n-1}-1$ with an alphabet of quadratic size.

The authors acknowledge the support of the OP VVV MEYS funded project CZ.02.1.01/0.0/0.0/16_019/0000765 “Research Center for Informatics” and the Grant Agency of the Czech Technical University in Prague, grant No. SGS20/208/OHK3/3T/18. V. Blažej was supported by the Engineering and Physical Sciences Research Council [grant number EP/V044621/1].