Complexity classes defined via k-valued functions

A lot of complexity classes can be characterized by posing some global acceptance condition on the computation trees produced by nondeterministic polynomial time machines. If the acceptance condition can be performed by a tree automaton, we obtain the concept of locally definable acceptance types (U. Hertrampf, 1992). This concept can be varied in different ways: if the acceptance condition depends only on the leaves of the computation tree, we obtain the concept of leaf languages (D. Bovet et al., 1991); if moreover the leaf language has to be a regular set, we obtain associative acceptance types. A special case appears, if we just count the number α of accepting paths up to a fixed maximal value c (i.e. α=max(# accepting paths, c)) and then check, whether α belongs to a given subset A⊆{0,…,c-1}. This concept leads to complexity classes with finite acceptance types. We survey all these concepts and compare their power