Hertrampf's locally definable acceptance types show that many complexity classes can be defined in terms of polynomial-time bounded NTMs with simple local conditions on the nodes of its computation tree, rather than global concepts like number of accepting paths, etc. We introduce a modification of Hertrampf's locally definable acceptance types which allows to get a larger number of characterizable complexity classes. Among others the newly characterizable classes are UP and Mod Z k P. It is shown how different types of oracle access, e.g., guarded access, can be characterized by this model. This sheds new light on the discussion on how to access unambiguous computation. We present simple functions that describe precisely objects of current research as the unambiguous oracle, alternation, and promise hierarchies. We exhibit the new class UAP which seems to be an unambiguous analogue of Wagner's P. UAP (and thus P) contains Few and is currently the smallest class known with this property.
