Efficient equivalence checking technique for some classes of finite-state machines

Finite transducers, two-tape automata, and biautomata are related computational models descended from the concept of Finite-State Automaton. In these models an automaton controls two heads that read or write symbols on the tapes in the one-way mode. The computations of these three types of automata show many common features, and it is surprising that the methods for analyzing the behavior of automata developed for one of these models do not find suitable utilization in other models. The goal of this paper is to develop a uniform technique for building polynomial-time equivalence checking algorithms for some classes of automata (finite transducers, two-tape automata, biautomata, single-state pushdown automata) which exhibit certain features of the deterministic or unambiguous behavior. This new technique reduces the equivalence checking of automata to solvability checking of certain systems of equations over the semirings of languages or transductions. It turns out that such a checking can be performed by the variable elimination technique which relies on some combinatorial and algebraic properties of prefix-free regular languages. The main results obtained in this paper are as follows:
1. Using the algebraic approach a new algorithm for checking the equivalence of states of deterministic finite automata is constructed; time complexity of this algorithm is $O(n \log n)$.
2. A new class of prefix-free finite transducers is distinguished and it is shown that the developed algebraic approach provides the equivalence checking of transducers from this class in quadratic time (for real-time prefix-free transducers) and cubic (for prefix-free transducers with $\varepsilon$-transitions) relative to the sizes of analysed machines.
3. It is shown that the equivalence problem for deterministic two-tape finite automata can be reduced to the same problem for prefix-free finite transducers and solved in cubic time relative to the size of the analysed machines.
4. In the same way it is proved that the equivalence problem for deterministic finite biautomata can be solved in cubic time relative to the sizes of analysed machines.
5. By means of the developed approach an efficient equivalence checking algorithm for the class of simple grammars corresponding to deterministic single-state pushdown automata is constructed.