Complexity of Equations over Sets of Natural Numbers

Authors:
Artur Jeż;Alexander Okhotin
Affiliations:
University of Wrocław, Institute of Computer Science, Wrocław, Poland;University of Turku, Department of Mathematics, Turku, Finland and Academy of Finland, Helsinki, Finland
Venue:
Theory of Computing Systems
Year:
2011

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Abstract

Systems of equations of the form X i =φ i (X 1,…,X n ) (1≤ i ≤ n) are considered, in which the unknowns are sets of natural numbers. Expressions φ i may contain the operations of union, intersection and elementwise addition $S+T=\{m+n\mid m\in S$, 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.