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Complexity of Equations over Sets of Natural Numbers

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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.

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Authors and Affiliations

  1. Institute of Computer Science, University of Wrocław, Wrocław, Poland

    Artur Jeż

  2. Department of Mathematics, University of Turku, Turku, Finland

    Alexander Okhotin

  3. Academy of Finland, Helsinki, Finland

    Alexander Okhotin

Authors
  1. Artur Jeż
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  2. Alexander Okhotin
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Correspondence to Alexander Okhotin.

Additional information

A preliminary version of this paper was presented at STACS 2008 conference held in Bordeaux, France on 21–23 February, 2008.

A. Jeż was supported by MNiSW grant N206 259035 2008–2010.

A. Okhotin was supported by the Academy of Finland under grant 118540.

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Jeż, A., Okhotin, A. Complexity of Equations over Sets of Natural Numbers. Theory Comput Syst 48, 319–342 (2011). https://doi.org/10.1007/s00224-009-9246-y

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