Logspace verifiers, NC, and NP

Abstract

We explore the connection between public-coin interactive proof systems with logspace verifiers and $\mathcal{N}\mathcal{C}$using two different approaches. In the first approach, we describe an interactive proof system for accepting any language in $\mathcal{N}\mathcal{C}$after a logspace reduction, where the verifier is logspace-bounded and the protocol requires polylog time. These results are proved by describing $\mathcal{N}\mathcal{C}$computations as computations over arithmetic circuits using maximum and average gates, and then translating the arithmetic circuits into interactive proof systems in a natural way. In the second approach, we give a characterization of $\mathcal{N}\mathcal{C}$in terms of interactive proof systems where the verifier is logspace-bounded and runs in polylog time. The equivalent interactive proof systems work with error-correcting encodings of inputs, using the polylogarithmically checkable codes introduced in the context of transparent proofs.

We also characterize $\mathcal{N}\mathcal{C}$and $\mathcal{P}\mathcal{S}\mathcal{P}\mathcal{A}\mathcal{C}\mathcal{E}$via public-coin interactive proof systems where the verifier is logspace-bounded, but has restricted access to auxiliary storage.

Work done while the author was with the Institute of Mathematical Sciences, Madras 600 113, India.