On the IO-complexity and approximation languages

The approach to complexity theory called IO-complexity requires only that conditions meet infinitely often. A connection is established between the IO-complexity theory and the notion of approximation solutions. IO time and space bounds for deterministic Turing machines are defined. Particular attention is paid to what it means for a machine to accept a language with resource bound f(n) and density d(n). Formal definitions for the notion of finding an approximate solution for a hard problem are given. Given a language L that can be approximated by another language, the properties of Turing machines capable of accepting the other language are studied. An interpretation is given of the IO-complexity classes in terms of classes of languages that have approximate solutions. It is shown that the recursive languages of an IO-complexity class with resource bound f(n) and density d(n) are precisely those languages L that can be approximated by f(n) bounded machines agreeing with L on input w with probability at least d(w).