Automatic forcing and genericity: On the diagonalization strength of finite automata

Abstract

Algorithmic and resource-bounded Baire category and corresponding genericity concepts introduced in computability theory and computational complexity theory, respectively, have become elegant and powerful tools in these settings. Here we introduce some new genericity notions based on extension functions computable by finite automata which are tailored for capturing diagonalizations over regular sets and functions. We show that the generic sets obtained either by the partial regular extension functions of any fixed constant length or by all total regular extension of constant length are just the sets with saturated (also called disjunctive) characteristic sequence a. Here a sequence α is saturated if every string occurs in α as a substring. We also show that these automatic generic sets are not regular but may be context free. Furthermore, we introduce stronger automatic genericity notions based on regular extension functions of nonconstant length and we show that the corresponding generic sets are bi-immune for the class of regular and context free languages. © Springer-Verlag Berlin Heidelberg 2003.