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.

💡 Research Summary

The paper introduces a new computational model called probabilistic debate checking. In this model a resource‑bounded verifier V reads a transcript of a dialogue between a prover P and a refuter R and decides whether a given input string x belongs to a language L. The verifier is limited in two ways: the amount of work‑tape space it can use and the number of random bits it may generate. The dialogue can be of three information types. In a complete‑information debate both parties see each other’s messages; in a partial‑information debate the prover sees only a subset of the refuter’s messages; in a zero‑information debate the prover sees none of the refuter’s messages. This framework simultaneously generalizes one‑way interactive proof systems, games of incomplete information, and previously studied probabilistically checkable debate systems.

The authors study the power of such verifiers under two natural resource regimes.

1. Constant‑space, constant‑randomness verifiers (O(1) space, O(1) random bits).
   Complete‑information: When V is allowed even a single random bit, its power jumps from recognizing only regular languages (the deterministic case) to recognizing every language in P. The proof constructs a P‑complete problem (e.g., circuit evaluation) as a debate where the refuter supplies a random challenge and the prover supplies a deterministic proof that can be checked with constant memory using the random challenge.
   Partial‑information: With the same constant resources the class of debatable languages expands from NSPACE(n) (the deterministic bound) to PSPACE. The prover’s lack of full visibility is compensated by the verifier’s random sampling, which enables it to simulate a PSPACE computation using a constant‑space, constant‑randomness protocol.
   Zero‑information: The class rises from E (deterministic exponential time) to EXPTIME. Again a handful of random bits suffice to let the verifier orchestrate an exponential‑time computation through a carefully designed zero‑information debate.

   Moreover, adding logarithmic space to a constant‑randomness verifier does not increase its power: the resulting classes remain P, PSPACE, and EXPTIME for the three information settings respectively.

2. Logarithmic‑space, polynomial‑time verifiers (O(log n) space, poly‑time, unrestricted randomness).
   In this regime the verifier can use any polynomial number of random bits. The main theorem shows that for all three debate types the class of languages that admit such a verifier is exactly PSPACE. The proof reduces any PSPACE computation to a debate in which the refuter supplies a nondeterministic choice sequence and the prover supplies a deterministic witness; the verifier, using only log n space, checks the consistency of the two streams by random sampling and a polynomial‑time simulation of the underlying PSPACE machine. Consequently, log‑space verifiers are already as powerful as full‑space interactive proof systems when randomness is unrestricted.

The paper also derives a corollary concerning the quantified max‑word problem for matrices. By encoding a PSPACE computation as a zero‑information debate and then translating the debate into a matrix product problem, the authors prove that approximating the maximal word length in such a quantified setting is PSPACE‑hard. This links the debate‑checking model to well‑studied problems in algebraic complexity and approximation algorithms.

Overall, the work maps out a detailed landscape of how tiny amounts of memory and randomness affect the ability to verify debates. It shows that even a constant number of random bits can dramatically increase the verifier’s power, turning a regular‑language checker into a full P‑machine, and that log‑space verifiers with unrestricted randomness capture PSPACE regardless of the information asymmetry. These results bridge gaps between interactive proof theory, game‑theoretic models of incomplete information, and probabilistically checkable proofs, and they suggest new directions for designing verification protocols that operate under severe hardware constraints.