Revisiting the Equivalence Problem for Finite Multitape Automata

The decidability of determining equivalence of deterministic multitape automata (or transducers) was a longstanding open problem until it was resolved by Harju and Karhumaki in the early 1990s. Their proof of decidability yields a co-NP upper bound, but apparently not much more is known about the complexity of the problem. In this paper we give an alternative proof of decidability, which follows the basic strategy of Harju and Karhumaki but replaces their use of group theory with results on matrix algebras. From our proof we obtain a simple randomised algorithm for deciding equivalence of deterministic multitape automata, as well as automata with transition weights in the field of rational numbers. The algorithm involves only matrix exponentiation and runs in polynomial time for each fixed number of tapes. If the two input automata are inequivalent then the algorithm outputs a word on which they differ.

[1]  A. S. Amitsur,et al.  Minimal identities for algebras , 1950 .

[2]  B. H. Neumann,et al.  On Ordered Groups , 1949 .

[3]  David J. Saltman,et al.  Lectures on Division Algebras , 1999 .

[4]  Vikraman Arvind,et al.  Derandomizing the Isolation Lemma and Lower Bounds for Circuit Size , 2008, APPROX-RANDOM.

[5]  Wen-Guey Tzeng,et al.  A Polynomial-Time Algorithm for the Equivalence of Probabilistic Automata , 1992, SIAM J. Comput..

[6]  Samuel Eilenberg,et al.  Automata, languages, and machines. A , 1974, Pure and applied mathematics.

[7]  Marcel Paul Schützenberger,et al.  On the Definition of a Family of Automata , 1961, Inf. Control..

[8]  Paul M. Cohn,et al.  Further Algebra and Applications , 2003 .

[9]  Hoeteck Wee,et al.  More on noncommutative polynomial identity testing , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

[10]  Catriel Beeri.  An Improvement on Valiant's Decision Procedure for Equivalence of Deterministic Finite Turn Pushdown Machines , 1976, Theor. Comput. Sci..

[11]  Timothy V. Griffiths.  The unsolvability of the Equivalence Problem for Λ-Free nondeterministic generalized machines , 1968, JACM.

[12]  Jorge E. Mezei,et al.  Two-sided finite-state transductions (abbreviated version) , 1963, SWCT.

[13]  B. H. Neumann,et al.  On ordered division rings , 1949 .

[14]  Leslie G. Valiant.  The Equivalence Problem for Deterministic Finite-Turn Pushdown Automata , 1974, Inf. Control..

[15]  Malcolm Bird,et al.  The Equivalence Problem for Deterministic Two-Tape Automata , 1973, J. Comput. Syst. Sci..

[16]  Jacques Sakarovitch,et al.  Elements of Automata Theory: THE POWER OF ALGEBRA , 2009 .

[17]  Vijay V. Vazirani,et al.  Matching is as easy as matrix inversion , 1987, STOC.

[18]  Joël Ouaknine,et al.  On the Complexity of Equivalence and Minimisation for Q-weighted Automata , 2013, Log. Methods Comput. Sci..

[19]  Tero Harju,et al.  The Equivalence Problem of Multitape Finite Automata , 1991, Theor. Comput. Sci..

[20]  Sheila A. Greibach,et al.  A Polynomial Time Algorithm for Deciding the Equivalence Problem for 2-Tape Deterministic Finite State Acceptors , 1982, SIAM J. Comput..

[21]  Dana S. Scott,et al.  Finite Automata and Their Decision Problems , 1959, IBM J. Res. Dev..