On Parallel Repetition of PCPs

Authors Alessandro Chiesa, Ziyi Guan, Burcu Yıldız



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Author Details

Alessandro Chiesa
  • EPFL, Lausanne, Switzerland
Ziyi Guan
  • EPFL, Lausanne, Switzerland
Burcu Yıldız
  • EPFL, Lausanne, Switzerland

Acknowledgements

The authors thank Ngoc Khanh Nguyen, Guy Weissenberg, Eylon Yogev, and Mingnan Zhao for valuable feedback and comments on earlier drafts of this paper. The authors thank anonymous reviewers of ITCS for valuable comments and suggestions that have improved this paper.

Cite As Get BibTex

Alessandro Chiesa, Ziyi Guan, and Burcu Yıldız. On Parallel Repetition of PCPs. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ITCS.2024.34

Abstract

Parallel repetition refers to a set of valuable techniques used to reduce soundness error of probabilistic proofs while saving on certain efficiency measures. Parallel repetition has been studied for interactive proofs (IPs) and multi-prover interactive proofs (MIPs). In this paper we initiate the study of parallel repetition for probabilistically checkable proofs (PCPs).
We show that, perhaps surprisingly, parallel repetition of a PCP can increase soundness error, in fact bringing the soundness error to one as the number of repetitions tends to infinity. This "failure" of parallel repetition is common: we find that it occurs for a wide class of natural PCPs for NP-complete languages. We explain this unexpected phenomenon by providing a characterization result: the parallel repetition of a PCP brings the soundness error to zero if and only if a certain "MIP projection" of the PCP has soundness error strictly less than one. We show that our characterization is tight via a suitable example. Moreover, for those cases where parallel repetition of a PCP does bring the soundness error to zero, the aforementioned connection to MIPs offers preliminary results on the rate of decay of the soundness error.
Finally, we propose a simple variant of parallel repetition, called consistent parallel repetition (CPR), which has the same randomness complexity and query complexity as the plain variant of parallel repetition. We show that CPR brings the soundness error to zero for every PCP (with non-trivial soundness error). In fact, we show that CPR decreases the soundness error at an exponential rate in the repetition parameter.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • probabilistically checkable proofs
  • parallel repetition

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References

  1. Boaz Barak, Anup Rao, Ran Raz, Ricky Rosen, and Ronen Shaltiel. Strong parallel repetition theorem for free projection games. In Proceedings of the 12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, and of the 13th International Workshop on Randomization and Computation, APPROX-RANDOM '09, pages 352-365, 2009. Google Scholar
  2. Irit Dinur and Elazar Goldenberg. Locally testing direct product in the low error range. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS '08, pages 613-622, 2008. Google Scholar
  3. Irit Dinur and Or Meir. Derandomized parallel repetition via structured PCPs. Computational Complexity, 20(2):207-327, 2011. Google Scholar
  4. Irit Dinur and Inbal Livni Navon. Exponentially small soundness for the direct product Z-Test. In Proceedings of the 32nd Annual IEEE Conference on Computational Complexity, CCC '17, pages 29:1-29:50, 2017. Google Scholar
  5. Irit Dinur and Omer Reingold. Assignment testers: Towards a combinatorial proof of the PCP theorem. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS '04, pages 155-164, 2004. Google Scholar
  6. Irit Dinur and David Steurer. Direct product testing. In Proceedings of the 29th Annual IEEE Conference on Computational Complexity, CCC '14, pages 188-196, 2014. Google Scholar
  7. Uriel Feige. A threshold of ln n for approximating set cover. Journal of the ACM, 45:634-652, 1998. Google Scholar
  8. Uriel Feige and Oleg Verbitsky. Error reduction by parallel repetition - a negative result. Combinatorica, 22:461-478, 2002. Google Scholar
  9. Lance Fortnow, John Rompel, and Michael Sipser. On the power of multi-prover interactive protocols. In Theoretical Computer Science, pages 156-161, 1988. Google Scholar
  10. Oded Goldreich. Modern Cryptography, Probabilistic Proofs and Pseudorandomness, volume 17 of Algorithms and Combinatorics. Springer, 1998. Google Scholar
  11. Oded Goldreich and Shmuel Safra. A combinatorial consistency lemma with application to proving the PCP theorem. In International Workshop on Randomization and Approximation Techniques in Computer Science, APPROX-RANDOM '97, pages 67-84, 1997. Google Scholar
  12. Thomas Holenstein. Parallel repetition: simplifications and the no-signaling case. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing, STOC '07, pages 411-419, 2007. Google Scholar
  13. Russell Impagliazzo, Valentine Kabanets, and Avi Wigderson. New direct-product testers and 2-query PCPs. SIAM Journal on Computing, 41:1722-1768, 2012. Google Scholar
  14. Anup Rao. Parallel repetition in projection games and a concentration bound. SIAM Journal on Computing, 40:1871-1891, 2011. Google Scholar
  15. Ran Raz. A parallel repetition theorem. In Proceedings of the 27th Annual ACM Symposium on Theory of Computing, STOC '95, pages 447-456, 1995. Google Scholar
  16. Ran Raz. A counterexample to strong parallel repetition. SIAM Journal on Computing, 40:771-777, 2011. Google Scholar
  17. Ran Raz and Ricky Rosen. A strong parallel repetition theorem for projection games on expanders. In Proceedings of the 27th Annual IEEE Conference on Computational Complexity, CCC '12, pages 247-257, 2012. Google Scholar
  18. Oleg Verbitsky. Towards the parallel repetition conjecture. Theoretical Computer Science, 157:277-282, 1996. Google Scholar
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