Efficient Equivalence Checking Technique for Some Classes of Finite-State Machines

Automatic Control and Computer Sciences (AC&CS), Switzerland. 2021. Vol. 55. No. 7. P. 670–701.

Zakharov V.

Translator: Zakharov V.

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,
and 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 .
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 time (for prefix-free transducers with -transitions) relative
to the size of analyzed machines. 3. It is shown that the equivalence problem for deterministic twotape
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 analyzed 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 size
of analyzed 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.

Keywords: проблема эквивалентности в моделях вычислений, finite state transducer, конечный автомат-преобразователь, контекстно-свободная грамматика, context-free grammar, multi-tape automaton, многоленточный автомат, biautomaton, equivalence checking problem, биавтомат

Publication based on the results of: Process models: checking conformance with the observed behavior, automated discovery and behavioral analysis (2021)