Theory of Computing Systems

, Volume 48, Issue 2, pp 319–342

# Complexity of Equations over Sets of Natural Numbers

Article

## 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 $$S+T=\{m+n\mid m\in S$$ , nT}. 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.

Language equations Integer expressions Conjunctive grammars Computational complexity

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