Checking generalized debates with small space and randomness

We introduce a model of probabilistic debate checking, where a silent resource-bounded verifier reads a dialogue about the membership of the string in the language under consideration between a prover and a refuter. Our model combines and generalizes the concepts of one-way interactive proof systems, games of incomplete information, and probabilistically checkable complete-information debate systems. We consider debates of partial and zero information, where the prover is prevented from seeing some or all of the messages of the refuter, as well as those of complete information. The classes of languages with debates checkable by verifiers operating under severe bounds on the memory and randomness are studied. We give full characterizations of versions of these classes corresponding to simultaneous bounds of O(1) space and O(1) random bits, and of logarithmic space and polynomial time. It turns out that constant-space verifiers, which can only check complete-information debates for regular languages deterministically, can check for membership in any language in P when allowed to use a constant number of random bits. Similar increases also occur for zero- and partial- information debates, from NSPACE(n) to PSPACE, and from E to EXPTIME, respectively. Adding logarithmic space to these constant-randomness verifiers does not change their power. When logspace debate checkers are restricted to run in polynomial time without a bound on the number of random bits, the class of debatable languages equals PSPACE for all debate types. We also present a result on the hardness of approximating the quantified max word problem for matrices that is a corollary of this characterization.