Generation of small complexity classes using logical formulas, schemes and operations of bounded concatenation

Abstract

The FO and FOM classes are small classes of recursive functions that characterize parallel computing with constant running time and a polynomial number of processors. The article provides an overview of the results describing the basic properties of the FO and FOM classes. Various ways of defining these classes and proving the equivalence of these definitions are given. The position of FO and FOM in the hierarchy of complexity classes is characterized. The results on the affiliation of various functions to the FO and FOM classes are presented. Superposition bases in the FO and FOM classes are described and a complete proof of the result on the superposition basis in the FOM class, consisting only of simple, mainly arithmetic functions, is given.