Abstract
In this paper we initiate the study of proof systems where verification of proofs proceeds by \(\protect{\ensuremath{\mathsf{NC}}}^{0}\) 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 \(\protect{\ensuremath{\mathsf{NC}}}^{0}\) 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 \(\protect{\ensuremath{\mathsf{NC}}}^{0}\) proof systems for a variety of languages ranging from regular to \(\protect{\ensuremath{\mathsf{NP}}}\)-complete. On the other hand, we show by combinatorial methods that even easy regular languages such as Exact-OR do not admit \(\protect{\ensuremath{\mathsf{NC}}}^{0}\) proof systems. We also present a general construction of \(\protect{\ensuremath{\mathsf{NC}}}^{0}\) proof systems for regular languages with strongly connected NFA’s.
Research supported by a DAAD/DST grant, DFG grant VO 630/6-2, and by grant N. 20517 from the John Templeton Foundation.
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Beyersdorff, O. et al. (2011). Verifying Proofs in Constant Depth. In: Murlak, F., Sankowski, P. (eds) Mathematical Foundations of Computer Science 2011. MFCS 2011. Lecture Notes in Computer Science, vol 6907. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22993-0_11
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