# 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.

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