# Derandomized Parallel Repetition Theorems for Free Games

## Abstract

Raz’s parallel repetition theorem (SIAM J Comput 27(3):763–803, 1998) together with improvements of Holenstein (STOC, pp 411–419, 2007) shows that for any two-prover one-round game with value at most $${1- \epsilon}$$ (for $${\epsilon \leq 1/2}$$), the value of the game repeated n times in parallel on independent inputs is at most $${(1- \epsilon)^{\Omega(\frac{\epsilon^2 n}{\ell})}}$$, where is the answer length of the game. For free games (which are games in which the inputs to the two players are uniform and independent), the constant 2 can be replaced with 1 by a result of Barak et al. (APPROX-RANDOM, pp 352–365, 2009). Consequently, $${n=O(\frac{t \ell}{\epsilon})}$$ repetitions suffice to reduce the value of a free game from $${1- \epsilon}$$ to $${(1- \epsilon)^t}$$, and denoting the input length of the game by m, it follows that $${nm=O(\frac{t \ell m}{\epsilon})}$$ random bits can be used to prepare n independent inputs for the parallel repetition game.

In this paper, we prove a derandomized version of the parallel repetition theorem for free games and show that O(t(m)) random bits can be used to generate correlated inputs, such that the value of the parallel repetition game on these inputs has the same behavior. That is, it is possible to reduce the value from $${1- \epsilon}$$ to $${(1- \epsilon)^t}$$ while only multiplying the randomness complexity by O(t) when m = O().

Our technique uses strong extractors to “derandomize” a lemma of Raz and can be also used to derandomize a parallel repetition theorem of Parnafes et al. (STOC, pp 363–372, 1997) for communication games in the special case that the game is free.

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## References

1. Boaz Barak, Moritz Hardt, Ishay Haviv, Anup Rao, Oded Regev & David Steurer (2008). Rounding Parallel Repetitions of Unique Games. In FOCS, 374–383.

2. Boaz Barak, Anup Rao, Ran Raz, Ricky Rosen & Ronen Shaltiel (2009). Strong Parallel Repetition Theorem for Free Projection Games. In APPROX-RANDOM, 352–365.

3. Mihir Bellare, Oded Goldreich, Shafi Goldwasser (1993) Randomness in Interactive Proofs. Computational Complexity 3: 319–354

4. Michael Ben-Or, Shafi Goldwasser, Joe Kilian & Avi Wigderson (1988). Multi-Prover Interactive Proofs: How to Remove Intractability Assumptions. In STOC, 113–131.

5. Uriel Feige (1991). On the Success Probability of the Two Provers in One-Round Proof Systems. In Structure in Complexity Theory Conference, 116–123.

6. Uriel Feige (1995). Error reduction by parallel repetition-the state of the art. Technical report, Weizmann Institute, Jerusalem, Israel.

7. Uriel Feige & Joe Kilian (1995). Impossibility results for recycling random bits in two-prover proof systems. In STOC, 457–468.

8. Uriel Feige, Oleg Verbitsky (2002) Error Reduction by Parallel Repetition - A Negative Result. Combinatorica 22(4): 461–478

9. Lance Fortnow, John Rompel & Michael Sipser (1990). Errata for On the Power of Multi-Prover Interactive Protocols. In Structure in Complexity Theory Conference, 318–319.

10. Oded Goldreich (1997). A Sample of Samplers - A Computational Perspective on Sampling (survey). Electronic Colloquium on Computational Complexity (ECCC) 4(20).

11. Oded Goldreich, Russell Impagliazzo, Leonid Levin, Venkatesan Ramarathanan & David Zuckerman (1990). Security Preserving Amplification of Hardness. In FOCS, 318–326.

12. Oded Goldreich, Noam Nisan & Avi Wigderson (1995). On Yao’s XOR-Lemma. Electronic Colloquium on Computational Complexity (ECCC) 2(50).

13. Venkatesan Guruswami, Christopher Umans & Salil P. Vadhan (2009). Unbalanced expanders and randomness extractors from Parvaresh–Vardy codes. J. ACM 56(4).

14. Thomas Holenstein (2007). Parallel repetition: simplifications and the no-signaling case. In STOC, 411–419.

15. Russell Impagliazzo (1995). Hard-Core Distributions for Somewhat Hard Problems. In FOCS, 538–545.

16. Russell Impagliazzo, Ragesh Jaiswal, Valentine Kabanets & Avi Wigderson (2008). Uniform direct product theorems: simplified, optimized, and derandomized. In STOC, 579–588.

17. Russell Impagliazzo & Avi Wigderson (1997). PBPP if E Requires Exponential Circuits: Derandomizing the XOR Lemma. In STOC, 220–229.

18. Eyal Kushilevitz & Noam Nisan (1997). Communication Complexity. Cambridge University Press.

19. Dror Lapidot, Adi Shamir (1995) A One-Round, Two-Prover, Zero-Knowledge Protocol for NP. Combinatorica 15(2): 204–214

20. Noam Nisan, Steven Rudich & Michael E. Saks (1999). Products and Help Bits in Decision Trees. SIAM J. Comput. 28(3), 1035–1050.

21. Noam Nisan, David Zuckerman (1996) Randomness is Linear in Space. J. Comput. Syst. Sci. 52(1): 43–52

22. Itzhak Parnafes, Ran Raz & Avi Wigderson (1997). Direct Product Results and the GCD Problem, in Old and New Communication Models. In STOC, 363–372.

23. Anup Rao (2008). Parallel repetition in projection games and a concentration bound. In STOC, 1–10.

24. Ran Raz (1998) A Parallel Repetition Theorem. SIAM J. Comput. 27(3): 763–803

25. Ran Raz (2008). A Counterexample to Strong Parallel Repetition. In FOCS, 369–373.

26. Ronen Shaltiel (2003) Towards proving strong direct product theorems. Computational Complexity 12(1-2): 1–22

27. Oleg Verbitsky (1994). Towards the Parallel Repetition Conjecture. In Structure in Complexity Theory Conference, 304–307.

28. Andrew C. Yao (1982). Theory and Applications of Trapdoor Functions (Extended Abstract). In FOCS, 80–91.

29. Andrew Chi-Chih Yao (1979). Some Complexity Questions Related to Distributive Computing. In STOC, 209–213.

30. David Zuckerman (1997) Randomness-optimal oblivious sampling. Random Struct. Algorithms 11(4): 345–367

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Correspondence to Ronen Shaltiel.

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Shaltiel, R. Derandomized Parallel Repetition Theorems for Free Games. comput. complex. 22, 565–594 (2013). https://doi.org/10.1007/s00037-013-0071-y

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### Keywords

• Parallel repetition
• 2-prover systems
• derandomization
• randomness extractors

• 68Q15
• 68Q17