Isomorphism of regular trees and words

The computational complexity of the isomorphism problem for regular trees, regular linear orders, and regular words is analyzed. A tree is regular if it is isomorphic to the prefix order on a regular language. In case regular languages are represented by NFAs (DFAs), the isomorphism problem for regular trees turns out to be EXPTIME-complete (resp. P-complete). In case the input automata are acyclic NFAs (acyclic DFAs), the corresponding trees are (succinctly represented) finite trees, and the isomorphism problem turns out to be PSPACE-complete (resp. P-complete). A linear order is regular if it is isomorphic to the lexicographic order on a regular language. A polynomial time algorithm for the isomorphism problem for regular linear orders (and even regular words, which generalize the latter) given by DFAs is presented. This solves an open problem by Esik and Bloom.

💡 Research Summary

The paper conducts a thorough complexity-theoretic investigation of the isomorphism problem for two fundamental classes of automatic structures: regular trees and regular words (including regular linear orders). A regular tree is defined as a countable tree that is isomorphic to the prefix order on a regular language L⊆Σ*. Such a tree can be finitely described by a nondeterministic or deterministic finite automaton (NFA or DFA) that accepts L and the empty word. The authors first consider the isomorphism problem for regular trees when the input automata are DFAs. By interpreting each DFA state as a node of the tree, the problem reduces to checking whether the two state transition graphs are bisimilar. This can be solved by the classic partition‑refinement algorithm in polynomial time, and the authors prove P‑completeness via a reduction from graph isomorphism restricted to bounded degree.

When the input automata are NFAs, the situation becomes substantially harder because each node of the tree corresponds to a set of NFA states. The authors establish EXPTIME‑completeness for this case. The upper bound follows from an exponential‑time algorithm that constructs the (exponentially large) deterministic automaton of the NFA and then applies the DFA algorithm. For the lower bound they exploit the equivalence EXPTIME = APSPACE(poly). They encode the computation of an alternating polynomial‑space Turing machine into a small NFA that accepts exactly those words that do not encode an accepting computation. Using a construction from