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2013•ACM Transactions on Computation TheoryRequires access

Verifying proofs in constant depth

Olaf Beyersdorff, Samir Datta, Andreas Krebs, Meena Mahajan, Gido Scharfenberger-Fabian, Karteek Sreenivasaiah, Michael E. Thomas, Heribert Vollmer

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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.

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What this paper is about

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.

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Available 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.

Key concepts: Mathematical proof, Proof complexity, Combinatorial proof, Analytic proof, Constant (computer programming), Regular language, Construct (python library), Computer-assisted proof

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