doi 10.1007/s00224-009-9246-y form xi paper system set arithmetization 319-342 natural numbers Complexity of Equations over Sets of Natural Numbers addition article equations operation problem articles least solution operations of union membership problem number conjunctive grammars consequences intersection Systems of equations of the form Xi=φi(X1,…,Xn) (1≤i≤n) are considered, in which the unknowns are sets of natural numbers. Expressions φi may contain the operations of union, intersection and elementwise addition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S+T=\{m+n\mid m\in S$\end{document} , n∈T}. A system with an EXPTIME-complete least solution is constructed in the paper through a complete arithmetization of EXPTIME-completeness. At the same time, it is established that least solutions of all such systems are in EXPTIME. The general membership problem for these equations is proved to be EXPTIME-complete. Among the consequences of the result is EXPTIME-completeness of the compressed membership problem for conjunctive grammars. system of equations https://scigraph.springernature.com/explorer/license/ general membership problem https://doi.org/10.1007/s00224-009-9246-y unknowns results 2022-08-04T16:58 true time EXPTIME XI grammar expression solution same time 2009-12-23 such systems complexity of equations Union complexity 2009-12-23 Pure Mathematics Okhotin Alexander Springer Nature Theory of Computing Systems 1432-4350 1433-0490 Institute of Computer Science, University of Wrocław, Wrocław, Poland Institute of Computer Science, University of Wrocław, Wrocław, Poland 2 pub.1003872386 dimensions_id Academy of Finland, Helsinki, Finland Academy of Finland, Helsinki, Finland Department of Mathematics, University of Turku, Turku, Finland Mathematical Sciences Jeż Artur Springer Nature - SN SciGraph project 48