ISSN 0032-9460, Problems of Information Transmission, 2011, Vol. 47, No. 4, pp. 342–352.
Pleiades Publishing, Inc., 2011.
Original Russian Text
M.N. Vyalyi, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 4, pp. 43–54.
On Regular Realizability Problems
M. N. Vyalyi
Computing Center of the Russian Academy of Sciences, Moscow
Received February 8, 2011; in ﬁnal form, June 16, 2011
Abstract—We consider the class of regular realizability problems. Any of such problems is
speciﬁed by some language (ﬁlter) and consists in verifying that the intersection of a given reg-
ular language and the ﬁlter is nonempty. The main question is diversity of the computational
complexity of such problems. We show that any regular realizability problem with an inﬁnite
ﬁlter is hard for a class of problems decidable in logarithmic space with respect to logarith-
mic reductions. We give examples of NP-complete and PSPACE-complete regular realizability
By a regular realizability problem we mean verifying that the intersection of a ﬁxed language
(ﬁlter) and a given regular language is nonempty. Thus, regular realizability problems form a vast
family of problems which is parametrized by arbitrary languages (ﬁlters).
Interest to such parametric families of algorithmic problems is due to the possibility of de-
scribing computational complexity classes in terms of complete problems belonging to a chosen
family. Therefore, the main problem is how diverse the complexity of regular realizability problems
It turns out that regular realizability problems are closely related to generalized nondetermin-
ism models [1, 2]. In Section 3 we prove that for each generalized nondeterminism class there
exists a natural regular realizability problem which is complete in this class. It follows from the
results of [1, 2] that complete regular realizability problems exist for such complexity classes as
LOG, NLOG, NP, PSPACE, EXP, and also the class of enumerable languages. For the NP and
PSPACE classes, we present simpler proofs of completeness of the corresponding problems (see
Note also that in  there is given an example of a regular realizability problem to which the well-
known Skolem’s problem on zeros in linear recurrence sequences can be reduced, whose algorithmic
decidability is still an open problem. It is also shown in  that this problem is equivalent to a
natural extension of Skolem’s problem to orbits of linear maps.
Breadth and diversity of obtained examples suggest that regular realizability problems should
be of use in computational complexity theory.
The present paper is a ﬁrst step in studying general properties of regular realizability problems.
Namely, we show that regular realizability problems with inﬁnite ﬁlters are not simpler than the
class of problems decidable in logarithmic space (see Section 4).
Supported in part by the Russian Foundation for Basic Research, project no. 11-01-00398.