Decision problems for language equations

Abstract

Equations with formal languages as unknowns using all Boolean operations and concatenation are studied. Their main properties, such as solution existence and uniqueness, are characterized by first-order formulae. It is shown that testing solution existence is Π 1 -complete, while solution uniqueness and existence of a least and of a greatest solution are all Π 2 -complete problems. The families of languages defined by components of unique, least and greatest solutions of such systems are shown to coincide with the classes of recursive, recursively enumerable and co-recursively enumerable sets, respectively.

Representing Hyper-arithmetical Sets by Equations over Sets of Integers

“…For language equations with continuous operations, testing solution existence is a Π 0 1 -complete decision problem [22], and it remains Π 0 1 -complete already in the case of a unary alphabet, concatenation as the only operation and regular constants [10], that is, for equations over sets of natural numbers with addition only. For the same formalisms, solution uniqueness is Π 0 2 -complete.…”

“…Just like the recursive sets are the natural upper bound for equations with continuous operations [22], and this upper bound is reached by ultimately simple specimens of such equations [9,10,16], the hyper-arithmetical sets, which might have looked as a very rough upper bound, have been found representable by equations with the simplest sets of erasing operations. In addition, these simple equations can be regarded as a basic arithmetical object representing an important variant of formal arithmetic.…”

“…In particular, for equations with concatenation and Boolean operations, it was shown by Okhotin [20,22] that the families of languages representable by their unique, least and greatest solutions are exactly the recursive, the recursively enumerable (r.e.) and the co-recursively enumerable (co-r.e.)…”

“…The recursive upper bound on unique solutions [22] is applicable only to equations with continuous operations on languages. Most of the basic language-theoretic operations, such as concatenation, Kleene star, all Boolean operations, non-erasing homomorphisms, etc., are indeed continuous, and thus subject to the above methods.…”

“…The last question considered in the paper is the complexity of testing whether a given system of equations has a solution: in Sect. 6, this problem is proved to be Σ 1 1 -complete in the analytical hierarchy (vs. Π 0 1 -complete for language equations with continuous operations [9,22]). …”