ISSN 0032-9460, Problems of Information Transmission, 2013, Vol. 49, No. 3, pp. 276–291.
Pleiades Publishing, Inc., 2013.
Original Russian Text
M.N. Vyalyi, 2013, published in Problemy Peredachi Informatsii, 2013, Vol. 49, No. 3, pp. 86–104.
On Expressive Power
of Regular Realizability Problems
M. N. Vyalyi
Computing Center of the Russian Academy of Sciences, Moscow
Received July 13, 2012; in ﬁnal form, December 24, 2012
Abstract—A regular realizability (RR) problem is a problem of testing nonemptiness of inter-
section of some ﬁxed language (ﬁlter) with a regular language. We show that RR problems are
universal in the following sense. For any language L there exists an RR problem equivalent to L
under disjunctive reductions in nondeterministic log space. From this result, we derive existence
of complete problems under polynomial reductions for many complexity classes, including all
classes of the polynomial hierarchy.
A motivation for this work was to present a speciﬁc class of algorithmic problems that represents
complexities of all known complexity classes (there are hundreds of them presently known) in a
A typical algorithmic problem is the recognition problem for a language. But in the most
interesting cases an input is structured: it is a graph, a function description, etc. Our main goal is
to choose a structure of an input to satisfy two (somewhat contradictory) requirements: a speciﬁc
class of problems should be wide enough and it should be useful. The latter requirement reﬂects a
hope that analysis of a speciﬁc problem might be easier than the general case.
Here we consider regular realizability problems in this context. Let L be a language (hereafter
we refer to it as a ﬁlter). The regular realizability problem with this ﬁlter is the question about
realizability of regular properties on words in L. More exactly, the language RR(L) consists of
descriptions of regular languages R such that R ∩ L = ∅.
What are possible complexities of RR problems? In this paper we obtain a partial answer to
this question. It appears that RR problems are universal: for any other problem there exists an
equivalent RR problem.
To make exact statements, we need to ﬁx a format for descriptions of regular languages and an
We represent a regular language R by a deterministic ﬁnite automaton (DFA) A recognizing
the language R (and denote this fact as R = L(A)). DFAs are described in a natural way by
transition tables. Details of the format used can be found in . Important features for this work
are: (i) each binary word w is a description of some DFA A(w), and (ii) testing membership for
a regular language L(w)=L(A(w)) can be done using deterministic log space. Thus, a formal
deﬁnition of the language RR(L) corresponding to the RR problem with a ﬁlter L is
w ∈ RR(L) ⇐⇒ L ∩ L(A(w)) = ∅.
Supported in part by the Russian Foundation for Basic Research, project no. 11-01-00398.