Affiliations: Institute of Computer Science, University of Wrocław, 50–383 Wrocław, Poland. [email protected] | Department of Mathematics, University of Turku, Turku FIN20014, Finland | Academy of Finland, [email protected]
Note: [] Address for correspondence: Institute of Computer Science, University of Wrocław, 50–383 Wrocław, Poland
Abstract: It is shown that equations of the form �(X) = ψ(X), in which the unknown X is a set of natural numbers and �, ψ use the operations of union, intersection and addition of sets S + T = {m + n |, m ∈ S, n ∈ T}, can simulate systems of equations �j(X1, …, Xn) = �j(X1, …, Xn) with 1 ≤ j ≤ ℓ, in the sense that solutions of any such system are encoded in the solutions of the corresponding equation. This implies computational universality of least and greatest solutions of equations �(X) = ψ(X), as well as undecidability of their basic decision problems. It is sufficient to use only singleton constants in the construction. All results equally apply to language equations over a one-letter alphabet Σ = {a}.
Keywords: Language equations, unary alphabet, univariate equations
