Decidable Membership Problems for Finite Recurrent Systems over Sets of Naturals

Abstract

A finite recurrent system over the power set of the natural numbers of dimension n is a pair composed of nn-ary functions over the power set of the natural numbers and an n-tuple of singleton sets of naturals. Every function is applied to the components of the tuple and computes a set of natural numbers, that might also be empty. The results are composed into another tuple, and the process is restarted. Thus, a finite recurrent system generates an infinite sequence of n-tuples of sets of natural numbers. The last component of a generated n-tuple is the output of one step, and the union of all outputs is the set defined by the system. We will consider only special finite recurrent systems: functions are built from the set operations union (∪), intersection (∩) and complementation (–) and the arithmetic operations addition (⊕) and multiplication (⊗). Sum and product of two sets of natural numbers are defined elementwise. We will study two types of membership problems: given a finite recurrent system and a natural number, does the set defined by the system contain the queried number, and does the output of a specified step contain the queried number? We will determine upper and lower bounds for such problems where we restrict the allowed operations to subsets of { ∩ , ∪ , − − , ⊕ , ⊗ }. We will show completeness results for the complexity classes NL, NP and PSPACE.