Complexity of Equations over Sets of Natural Numbers

Artur Jeż, Alexander Okhotin
2009 Theory of Computing Systems  
Systems of equations of the form X i = ϕ i (X 1 , . . . , Xn) (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 | m ∈ 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
more » ... 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.
doi:10.1007/s00224-009-9246-y fatcat:ehvbpe7jnfcudbq7ayegs3xuxm