Verifying Proofs in Constant Depth

Abstract

In this paper we initiate the study of proof systems where verification of proofs proceeds by NC 0 circuits. We investigate the question which languages admit proof systems in this very restricted model. Formulated alternatively, we ask which languages can be enumerated by NC 0 functions. Our results show that the answer to this problem is not determined by the complexity of the language. On the one hand, we construct NC 0 proof systems for a variety of languages ranging from regular to NP complete. On the other hand, we show by combinatorial methods that even easy regular languages such as Exact-OR do not admit NC 0 proof systems. We also show that Majority does not admit NC 0 proof systems. Finally, we present a general construction of NC 0 proof systems for regular languages with strongly connected NFA's.

Small Depth Proof Systems

“…Equivalently, the proof system is computable by an AC 0 circuit where each gate has fanin (log n) O(1) ; we refer to the class of such circuits as poly log AC 0 circuits. By the result of [11], Exact-Or requires depth Ω(log log n), so (upto constant multiplicative factors) this is tight. Our proof technique also generalises to show that Maj has poly log AC 0 proof systems (Theorem 7).…”

“…The most intriguing question here, posed in [11], is to characterize the regular languages that have NC 0 proof systems. We state a conjecture for this characterization; the conjecture throws up more questions regarding decidability of some properties of regular languages.…”

“…If L has an NC 0 proof system (C n ) and h is a regular transduction computed by a strongly connected automaton M , the construction from [11] (Proposition 1 (3)) with the output w of C n as input will produce a word x ∈ L(M ). A small modification allows us output the transduction h(x) instead of x.…”

“…But an AC 0 computation can verify that a given assignment satisfies a Boolean formula. Nonetheless, one can look for verifiers even weaker than AC 0 ; this kind of study was initiated in [11] where NC 0 proof systems were investigated. In an NC 0 proof system, each output bit depends on just O(1) bits of the input, so to enumerate L as the range of an NC 0 function f , f must be able to do highly local corrections to the alleged proof while maintaining the global property that the output word belongs to L. Unlike with locally-decodable error-correcting codes, the correction here must be deterministic and always correct.…”

“…The work in [11] shows that languages of varying complexity (complete for NC 1 , P, NP) have uniform NC 0 proof systems, while the languages Exact-Or, Maj amongst others do not have even non-uniform NC 0 proof systems. It then focuses on regular languages, and shows that a large subclass of regular languages has uniform NC 0 proof systems.…”