63-74 ϕj such systems set system of equations basic decision problems 2008-01-01 2022-08-04T17:14 false decision problem recursive sets family operations of union constants https://scigraph.springernature.com/explorer/license/ unique solution chapters On the Computational Completeness of Equations over Sets of Natural Numbers Systems of equations of the form ϕj(X1, ..., Xn) = ψj(X1, ..., Xn) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1 \leqslant j \leqslant m$\end{document} are considered, in which the unknowns Xi are sets of natural numbers, while the expressions ϕj,ψj may contain singleton constants and the operations of union (possibly replaced by intersection) and pairwise addition . 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. addition pairwise addition chapter computational completeness operation system arithmetical hierarchy problem 2008-01-01 unknowns Xi solution https://doi.org/10.1007/978-3-540-70583-3_6 completeness form natural numbers Union XI number equations hierarchy family of sets Halldórsson Magnús M. Igor Walukiewicz Goldberg Leslie Ann Department of Mathematics, University of Turku, Finland Department of Mathematics, University of Turku, Finland Academy of Finland 978-3-540-70582-6 978-3-540-70583-3 Automata, Languages and Programming Mathematical Sciences Springer Nature - SN SciGraph project pub.1009467480 dimensions_id Applied Mathematics Okhotin Alexander Institute of Computer Science, University of Wrocław, Poland Institute of Computer Science, University of Wrocław, Poland Ingólfsdóttir Anna 10.1007/978-3-540-70583-3_6 doi Jeż Artur Aceto Luca Ivan Damgård Springer Nature