On the Structure of Logspace Probabilistic Complexity Classes

Abstract

We investigate hierarchical properties and logspace reductions of languages recognized by logspace probabilistic Turing machines, Arthur--Merlin games, and games against nature with logspace probabilistic verifiers. Each logspace complexity class is decomposed into a hierarchy based on corresponding two-way multihead finite-state automata and we (eventually) prove the separation of the hierarchy levels (even for languages over a single-letter alphabet); furthermore, we show efficient reductions of each logspace complexity class to, or between, low levels of its corresponding hierarchy. We find probabilistic and probabilistic-plus-nondeterministic variants of Savitch's maze threading problem which are logspace complete for PL (the class of languages recognized by logspace probabilistic Turing machines) and, respectively, PPP (the class of languages recognized by polynomial-time deterministic Turing machines), and which can be recognized by one-way non-sensing two-head (or one-way one-head one-counter) finite-state automata with probabilistic and both probabilistic and nondeterministic states, respectively.