Abstract
Irit Dinur’s proof of the PCP theorem via gap amplification (J. ACM, Vol. 54 (3) and ECCC TR05-046) has an important extension to Assignment Testers (a.k.a PCPPs). This extension is based on a corresponding extension of the gap amplification theorem from PCPs to Assignment Testers (a.k.a PCPPs). Specifically, the latter extension states that the rejection probability of an Assignment Tester can be amplified by a constant factor, at the expense of increasing the output size of the Assignment Tester by a constant factor (while retaining the alphabet). We point out a gap in the proof of this extension, and show that this gap can be bridged.
We stress that the gap refers to the amplification of Assignment Testers, and the underlying issue does not arise in the case of standard PCPs. Furthermore, it seems that the issue also does not arise with respect to the applications in Dinur’s paper, but it may arise in other applications.
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References
Ben-Sasson, E., Goldreich, O., Harsham, P., Sudan, M., Vadhan, S.: Robust PCPs of proximity, shorter PCPs and applications to coding. SIAM J. Comput. 36(4), 889–974 (2006). Preliminary version in STOC 2004, pp. 120–134
Dinur, I.: The PCP theorem by gap amplification. In: ECCC TR05-046
Dinur, I.: The PCP theorem by gap amplification. J. ACM 54(3), 12-es (2007). Preliminary version in STOC 2006, pp. 241–250
Dinur, I., Reingold, O.: Assignment testers: towards combinatorial proofs of the PCP theorem. SIAM J. Comput. 36(4), 975–1024 (2006). Preliminary version in FOCS 2004, pp. 155–164
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Goldreich, O., Meir, O. (2020). Bridging a Small Gap in the Gap Amplification of Assignment Testers. In: Goldreich, O. (eds) Computational Complexity and Property Testing. Lecture Notes in Computer Science(), vol 12050. Springer, Cham. https://doi.org/10.1007/978-3-030-43662-9_2
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DOI: https://doi.org/10.1007/978-3-030-43662-9_2
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