Zakharov V.

The unification problem for parameterized finite state machines consists in finding for two given automata such values of their parameters that these automata will compute the same transduction relations. This presents a quadratic time algorithm for computing the most general unifiers of parameterized deterministic finite state machines. This algorithm is based on the Martelli–Montanari unification algorithm for terms and the equivalence checking algorithm for deterministic finite state machines

We present an efficient algorithm for checking language equivalence of states in top-down deterministic finite tree automata (DFTAs). Unlike string automata, tree automata operate over hierarchical structures, posing unique challenges for algorithmic analysis. Our approach reduces the equivalence checking problem to that of checking the solvability of a system of language-theoretic equations which specify the behavior of a DFTA. By constructing such a system of equations and systematically manipulating with it through substitution and conflict detection rules, we develop a decision procedure that determines whether two states accept the same tree language. We formally prove the correctness and termination of the algorithm and establish its worst-case time complexity as O(n²) under the RAM (Random Access Machine) model of computation augmented with pointers.