Toward Probabilistic Checking against Non-Signaling Strategies with Constant Locality

Non-signaling strategies are a generalization of quantum strategies that have been studied in physics over the past three decades. Recently, they have found applications in theoretical computer science, including to proving inapproximability results for linear programming and to constructing protocols for delegating computation. A central tool for these applications is probabilistically checkable proof (PCPs) systems that are sound against non-signaling strategies. In this paper we show, assuming a certain geometrical hypothesis about noise robustness of non-signaling proofs (or, equivalently, about robustness to noise of solutions to the Sherali-Adams linear program), that a slight variant of the parallel repetition of the exponential-length constant-query PCP construction due to Arora et al. (JACM 1998) is sound against non-signaling strategies with constant locality. Our proof relies on the analysis of the linearity test and agreement test (also known as the direct product test) in the non-signaling setting.

[1]  Yael Tauman Kalai,et al.  On the Space Complexity of Linear Programming with Preprocessing , 2016, Electron. Colloquium Comput. Complex..

[2]  S. Popescu,et al.  Quantum nonlocality as an axiom , 1994 .

[3]  László Lovász,et al.  Interactive proofs and the hardness of approximating cliques , 1996, JACM.

[4]  Thomas Holenstein,et al.  Parallel repetition: simplifications and the no-signaling case , 2007, STOC '07.

[5]  Mihir Bellare,et al.  Linearity testing in characteristic two , 1996, IEEE Trans. Inf. Theory.

[6]  Tsuyoshi Ito,et al.  Polynomial-Space Approximation of No-Signaling Provers , 2009, ICALP.

[7]  Manuel Blum,et al.  Self-testing/correcting with applications to numerical problems , 1990, STOC '90.

[8]  David Steurer,et al.  Analytical approach to parallel repetition , 2013, STOC.

[9]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[10]  Thomas Holenstein.  Parallel Repetition: Simplification and the No-Signaling Case , 2009, Theory Comput..

[11]  Leonid A. Levin,et al.  Checking computations in polylogarithmic time , 1991, STOC '91.

[12]  Tsuyoshi Ito,et al.  Oracularization and Two-Prover One-Round Interactive Proofs against Nonlocal Strategies , 2008, 2009 24th Annual IEEE Conference on Computational Complexity.

[13]  Igor Shinkar,et al.  Probabilistic Checking against Non-Signaling Strategies from Linearity Testing , 2018, Electron. Colloquium Comput. Complex..

[14]  Dhiraj Holden,et al.  Non-signaling proofs with o(√ log n) provers are in PSPACE , 2020, STOC.

[15]  Irit Dinur,et al.  Exponentially Small Soundness for the Direct Product Z-test , 2017, Electron. Colloquium Comput. Complex..

[16]  Justin Holmgren,et al.  The parallel repetition of non-signaling games: counterexamples and dichotomy , 2019, STOC.

[17]  Tsuyoshi Ito,et al.  A Multi-prover Interactive Proof for NEXP Sound against Entangled Provers , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[18]  R. Raz,et al.  How to delegate computations: the power of no-signaling proofs , 2014, Electron. Colloquium Comput. Complex..

[19]  Silvio Micali,et al.  Computationally Sound Proofs , 2000, SIAM J. Comput..

[20]  Nathan Rosen.  Bell’s theorem and quantum mechanics , 1994 .

[21]  S. KarthikC.,et al.  Toward a General Direct Product Testing Theorem , 2020, ACM Trans. Comput. Theory.

[22]  Yael Tauman Kalai,et al.  Delegation for bounded space , 2013, STOC '13.

[23]  Omer Reingold,et al.  Assignment testers: towards a combinatorial proof of the PCP-theorem , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[24]  David Steurer,et al.  Direct Product Testing , 2014, 2014 IEEE 29th Conference on Computational Complexity (CCC).

[25]  Eli Ben-Sasson,et al.  Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding , 2004, SIAM J. Comput..

[26]  Ivan Damgård,et al.  Linear zero-knowledge—a note on efficient zero-knowledge proofs and arguments , 1997, STOC '97.

[27]  Andreas J. Winter,et al.  Parallel repetition and concentration for (sub-)no-signalling games via a flexible constrained de Finetti reduction , 2015, Chic. J. Theor. Comput. Sci..

[28]  Avi Wigderson,et al.  New direct-product testers and 2-query PCPs , 2009, STOC '09.

[29]  Madhu Sudan,et al.  Optimal Testing of Reed-Muller Codes , 2009, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[30]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems , 1990, SIAM J. Discret. Math..

[31]  Igor Shinkar,et al.  Testing Linearity against Non-signaling Strategies , 2018, Electron. Colloquium Comput. Complex..

[32]  Peter Rastall,et al.  Locality, Bell's theorem, and quantum mechanics , 1985 .

[33]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[34]  Mihir Bellare,et al.  Free Bits, PCPs, and Nonapproximability-Towards Tight Results , 1998, SIAM J. Comput..

[35]  R. Cramer,et al.  Linear Zero-Knowledgde. A Note on Efficient Zero-Knowledge Proofs and Arguments , 1996 .

[36]  Rafail Ostrovsky,et al.  Fast Verification of Any Remote Procedure Call: Short Witness-Indistinguishable One-Round Proofs for NP , 2000, ICALP.

[37]  Carsten Lund,et al.  Non-deterministic exponential time has two-prover interactive protocols , 2005, computational complexity.

[38]  Warren P. Adams,et al.  A hierarchy of relaxation between the continuous and convex hull representations , 1990 .

[39]  Ran Raz,et al.  A parallel repetition theorem , 1995, STOC '95.

[40]  Avi Wigderson,et al.  Derandomizing homomorphism testing in general groups , 2004, STOC '04.

[41]  Guy Kindler,et al.  Direct Sum Testing , 2015, SIAM J. Comput..

[42]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[43]  Mark Braverman,et al.  Small Value Parallel Repetition for General Games , 2014, STOC.

[44]  Oleg Verbitsky,et al.  Towards the parallel repetition conjecture , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[45]  Ronitt Rubinfeld,et al.  Non‐Abelian homomorphism testing, and distributions close to their self‐convolutions , 2008, Random Struct. Algorithms.