ISSN 0032-9460, Problems of Information Transmission, 2015, Vol. 51, No. 3, pp. 267–288.
Pleiades Publishing, Inc., 2015.
Original Russian Text
An.A. Muchnik, K.Yu. Gorbunov, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 3, pp. 70–92.
Algorithmic Aspects of Decomposition
and Equivalence of Finite-Valued Transducers
An. A. Muchnik
and K. Yu. Gorbunov
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow, Russia
Received February 12, 2014; in ﬁnal form, June 3, 2015
Abstract—We study algorithmic issues of the problems of decomposing a ﬁnite-valued trans-
ducer into a union of single-valued ones and inclusion of an arbitrary transducer in a ﬁnite-valued
one. We propose algorithms that partially improve eﬃciency estimates for known analogous
In the present paper we address algorithmic issues of the problems of decomposing a ﬁnite-
valued transducer into a union of single-valued ones and inclusion of an arbitrary transducer in
a ﬁnite-valued one. These questions were studied by Weber in [1–4] and also by Sakarovitch and
de Souza in [5–8]. In these works polynomial decidability of ﬁnite-valuedness of a transducer was
proved, possibility of decomposing a ﬁnite-valued transducer into a union of single-valued ones was
shown, and an algorithm for testing inclusion of an arbitrary transducer in a ﬁnite-valued one was
proposed. We propose simpler constructions, which partially improve estimates of the previous
authors. The single-valued transducers obtained in the decomposition have sizes of the order of
a single exponent of poly(n), where n is the size of the ﬁnite-valued transducer; testing inclusion
of an arbitrary transducer in a ﬁnite-valued one requires space estimated by a single exponent.
Note that in [5,6] the corresponding estimate is exponential not only in n but also in k,wherek is
valuedness of the transducer (in those works k was assumed to be constant). Taking into account
that k can itself be exponential in n, this bound has the order of a double exponent of n.Our
estimate does not contain k under the exponent and thus improves the previously known ones. The
same concerns the problem of testing inclusion of an arbitrary transducer in a ﬁnite-valued one,
which in  is solved with space estimation of the order of exp(poly(n, k)), whereas our estimate
is of the order of exp(poly(n)). On the other hand, the number of single-valued transducers in
constructions from [4–6] equals the valuedness of a given ﬁnite-valued transducer, which does not
follow from our construction. Thus, each of the constructions have its own advantages. Note also
the work , which contains a detailed presentation of automata and transducers theory.
Section 2 contains basic deﬁnitions, examples, and facts introducing the reader to the subject
of the paper. In Section 3 we present constructions underlying the decomposition of a ﬁnite-
valued transducer. Section 4 describes the decomposition itself. In Section 5 we address questions
of the smallest input and output length on which noninclusion of one ﬁnite-valued transducer
The research was carried out at the Institute for Information Transmission Problems of the Russian
Academy of Sciences at the expense of the Russian Science Foundation, project no. 14-50-00150.