Unambiguous polynomial hierarchies and exponential size

The classes NC/sup k/ and AC/sup k/ are defined by computational devices of polynomial size, i.e. by devices using a polynomially bounded number of gates or processors. We consider the case of exponential size, which results in classes between P and PSPACE. In this way, we get new characterizations of P and UP. The resulting relations of nondeterminism, unambiguity, and determinism to several types of simultaneous write access to a shared memory perfectly resemble the polynomial case. A new phenomenon is the equivalence of concurrent read, exclusive read, and owner read for arbitrary types of write access in the case of exponential size. In the exponential case, circuits of bounded depth characterize the polynomial hierarchy. Using the notion of an unambiguous circuit, we give a uniform framework to relate the various types of unambiguous polynomial hierarchies and to explain their differences.<>