On the Computational Completeness of Equations over Sets of Natural Numbers

Abstract

Systems of equations of the form ϕ j (X 1 , . . . , X n ) = ψ j (X 1 , . . . , X n ) with 1 j m are considered, in which the unknowns X i are sets of natural numbers, while the expressions ϕ j , ψ j may contain singleton constants and the operations of union and pairwise addition S + T = { m + n | m ∈ S, n ∈ T }. It is shown that the family of sets representable by unique (least, greatest) solutions of such systems is exactly the family of recursive (r.e., co-r.e., respectively) sets of numbers. Basic decision problems for these systems are located in the arithmetical hierarchy. The same results are established for equations with addition and intersection.

Nondeterministic State Complexity of Positional Addition

“…The present paper continues this study by investigating another operation, which has recently been used by Jeż and Okhotin [7,8] in the study of language equations. This is the operation of addition of strings in base-k positional notation.…”

On Language Equations with One-sided Concatenation

“…A systematic study of the hardness of decision problems for such language equations was carried out by Okhotin [26,27,28,29], who also characterized recursive and recursively enumerable sets by solutions of these equations. Recently these computational completeness results were extended to language equations over a one-letter alphabet by Jeż and Okhotin [16].…”

“…A systematic study of the hardness of decision problems for such language equations was carried out by Okhotin [26,27,28,29], who also characterized recursive and recursively enumerable sets by solutions of these equations. Recently these computational completeness results were extended to language equations over a one-letter alphabet by Jeż and Okhotin [16].A surprising proof of the computational universality of very simple language equations of the form LX = XL, where L ⊆ {a, b} * is a finite constant language, was given by Kunc [18]. Later, Jeż and Okhotin [17] and Lehtinen and Okhotin [20] demonstrated that already systems of two equations {XXK = XXL, XM = N }, with regular constants K, L, M, N ⊆ a * , possess a full range of undecidable problems, and can represent an encoding of any recursive (r.e., co-r.e.)…”

“…This topic was further studied in the monographs on algebraic automata theory by Salomaa [34] and Conway [12]. There has been a renewed interest in the topic over the last two decades, with the state-of-the-art as of 2007 presented in a survey by Kunc [19], and with more research on various aspects of language equations appearing in the last few years [13,16,17,20,22,30].As an example, consider the equation X = AX ∪ B, where A, B are fixed formal languages. It is well-known, that this equation has A * B as a solution.…”

Complex Algebras of Arithmetic

“…A conjunctive grammar is a context-free grammar with an explicit intersection operation [13]. This section largely draws together work by Okhotin [14], Jeż and Okhotin [7], and Pinus and Vazhenin [15]. We have the following undecidability results.…”