ABSTRACT
We study whether information complexity can be used to attack the long-standing open problem of proving lower bounds against Arthur{Merlin (AM) communication protocols. Our starting point is to show that|in contrast to plain randomized communication complexity|every boolean function admits an AM communication protocol where on each yes- input, the distribution of Merlin's proof leaks no information about the input and moreover, this proof is unique for each outcome of Arthur's randomness. We posit that these two properties of zero information leakage and unambiguity on yes-inputs are interesting in their own right and worthy of investigation as new avenues toward AM.
Zero-information protocols (ZAM). Our basic ZAM protocol uses exponential communication for some functions, and this raises the question of whether more efficient protocols exist. We prove that all functions in the classical space-bounded complexity classes NL and L have polynomial-communication ZAM protocols. We also prove that ZAM complexity is lower bounded by conondeterministic communication complexity.
Unambiguous protocols (UAM). Our most technically substantial result is a (n) lower bound on the UAM complexity of the NP-complete set-intersection function; the proof uses information complexity arguments in a new, indirect way and overcomes the \zero-information barrier" described above. We also prove that in general, UAM complexity is lower bounded by the classic discrepancy bound, and we give evidence that it is not generally lower bounded by the classic corruption bound.
- S. Aaronson and A. Wigderson. Algebrization: A new barrier in complexity theory. ACM Transactions on Computation Theory, 1(1), 2009. doi:10.1145/1490270.1490272. Google ScholarDigital Library
- A. Ada, A. Chattopadhyay, S. Cook, L. Fontes, M. Kouck y, and T. Pitassi. The hardness of being private. ACM Transactions on Computation Theory, 6(1), 2014. doi:10.1145/2567671. Google ScholarDigital Library
- B. Applebaum, Y. Ishai, and E. Kushilevitz. Cryptography in NC 0. SIAM Journal on Computing, 36(4):845--888, 2006. doi:10.1137/S0097539705446950. Google ScholarDigital Library
- L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory. In Proceedings of the 27th Symposium on Foundations of Computer Science (FOCS), pages 337--347. IEEE, 1986. doi:10.1109/SFCS.1986.15. Google ScholarDigital Library
- L. Babai and S. Moran. Arthur--Merlin games: A randomized proof system, and a hierarchy of complexity classes. Journal of Computer and System Sciences, 36(2):254--276, 1988. doi:10.1016/0022-0000(88)90028--1. Google ScholarDigital Library
- Z. Bar-Yossef, T. Jayram, R. Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. Journal of Computer and System Sciences, 68(4):702--732, 2004. doi:10.1016/j.jcss.2003.11.006. Google ScholarDigital Library
- R. Beigel, N. Reingold, and D. Spielman. PP is closed under intersection. Journal of Computer and System Sciences, 50(2):191--202, 1995. doi:10.1006/jcss.1995.1017. Google ScholarDigital Library
- E. Böhler, C. Gla er, and D. Meister. Error-bounded probabilistic computations between MA and AM. Journal of Computer and System Sciences, 72(6):1043--1076, 2006. doi:10.1016/j.jcss.2006.05.001. Google ScholarDigital Library
- M. Braverman, F. Ellen, R. Oshman, T. Pitassi, and V. Vaikuntanathan. A tight bound for set disjointness in the message-passing model. In Proceedings of the 54th Symposium on Foundations of Computer Science (FOCS), pages 668--677. IEEE, 2013. doi:10.1109/FOCS.2013.77. Google ScholarDigital Library
- M. Braverman, A. Garg, D. Pankratov, and O. Weinstein. From information to exact communication. In Proceedings of the 45th Symposium on Theory of Computing (STOC), pages 151--160. ACM, 2013. doi:10.1145/2488608.2488628. Google ScholarDigital Library
- M. Braverman and A. Moitra. An information complexity approach to extended formulations. In Proceedings of the 45th Symposium on Theory of Computing (STOC), pages 161--170. ACM, 2013. doi:10.1145/2488608.2488629. Google ScholarDigital Library
- A. Chakrabarti, G. Cormode, N. Goyal, and J. Thaler. Annotations for sparse data streams. In Proceedings of the 25th Symposium on Discrete Algorithms (SODA), pages 687--706. ACM-SIAM, 2014. doi:10.1137/1.9781611973402.52. Google ScholarDigital Library
- A. Chakrabarti, G. Cormode, and A. McGregor. Annotations in data streams. In Proceedings of the 36th International Colloquium on Automata, Languages, and Programming (ICALP), pages 222--234. Springer, 2009. doi:10.1007/978--3--642-02927--1--20. Google ScholarDigital Library
- A. Chakrabarti, G. Cormode, A. McGregor, J. Thaler, and S. Venkatasubramanian. On interactivity in Arthur--Merlin communication and stream computation. Technical Report TR13--180, Electronic Colloquium on Computational Complexity (ECCC), 2013. URL: http://eccc.hpi-web.de/report/2013/180/.Google Scholar
- A. Chakrabarti, S. Khot, and X. Sun. Near-optimal lower bounds on the multi-party communication complexity of set disjointness. In Proceedings of the 18th Conference on Computational Complexity (CCC), pages 107--117. IEEE, 2003. doi:10.1109/CCC.2003.1214414.Google ScholarCross Ref
- A. Chakrabarti, Y. Shi, A. Wirth, and A. Yao. Informational complexity and the direct sum problem for simultaneous message complexity. In Proceedings of the 42nd Symposium on Foundations of Computer Science (FOCS), pages 270--278. IEEE, 2001. doi:10.1109/SFCS.2001.959901. Google ScholarDigital Library
- C. Damm. Problems complete for L. Information Processing Letters, 36(5):247--250, 1990. doi:10.1016/0020-0190(90)90150-V. Google ScholarDigital Library
- A. Dasgupta, R. Kumar, and D. Sivakumar. Sparse and lopsided set disjointness via information theory. In Proceedings of the 16th International Workshop on Randomization and Computation (RANDOM), pages 517--528. Springer, 2012. doi:10.1007/978--3--642--32512-0--44.Google ScholarCross Ref
- U. Feige, J. Kilian, and M. Naor. A minimal model for secure computation. In Proceedings of the 26th Symposium on Theory of Computing (STOC), pages 554--563. ACM, 1994. doi:10.1145/195058.195408. Google ScholarDigital Library
- A. Gál and A. Wigderson. Boolean complexity classes vs. their arithmetic analogs. Random Structures & Algorithms, 9(1--2):99--111, 1996. doi:10.1002/(SICI)10982418(199608/09)9:1/299::AID-RSA7 3.0.CO;2--6. Google ScholarDigital Library
- D. Gavinsky and A. Sherstov. A separation of NP and coNP in multiparty communication complexity. Theory of Computing, 6(1):227--245, 2010. doi:10.4086/toc.2010.v006a010.Google ScholarCross Ref
- M. Göös, S. Lovett, R. Meka, T. Watson, and D. Zuckerman. Rectangles are nonnegative juntas. Technical Report TR14--147, Electronic Colloquium on Computational Complexity (ECCC), 2014. URL: http://eccc.hpi-web.de/report/2014/147/.Google Scholar
- M. Göös, T. Pitassi, and T. Watson. Zero-information protocols and unambiguity in Arthur--Merlin communication. Technical Report TR14-078, Electronic Colloquium on Computational Complexity (ECCC), 2014. Full version. URL: http://eccc.hpi-web.de/report/2014/078/.Google Scholar
- M. Göös and T. Watson. Communication complexity of set-disjointness for all probabilities. In Proceedings of the 18th International Workshop on Randomization and Computation (RANDOM), pages 721--736. Schloss Dagstuhl, 2014. doi:10.4230/LIPIcs.APPROX-RANDOM.2014.721.Google Scholar
- A. Gronemeier. Asymptotically optimal lower bounds on the NIH-multi-party information complexity of the AND-function and disjointness. In Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS), pages 505--516. Schloss Dagstuhl, 2009. doi:10.4230/LIPIcs.STACS.2009.1846.Google Scholar
- T. Gur and R. Raz. Arthur--Merlin streaming complexity. In Proceedings of the 40th International Colloquium on Automata, Languages, and Programming (ICALP), pages 528--539. Springer, 2013. doi:10.1007/978--3--642--39206--1--45. Google ScholarDigital Library
- T. Gur and R. Rothblum. Non-interactive proofs of proximity. Technical Report TR13-078, Electronic Colloquium on Computational Complexity (ECCC), 2013. URL: http://eccc.hpi-web.de/report/2013/078/.Google Scholar
- R. Impagliazzo, V. Kabanets, and A. Kolokolova. An axiomatic approach to algebrization. In Proceedings of the 41st Symposium on Theory of Computing (STOC), pages 695--704. ACM, 2009. doi:10.1145/1536414.1536509. Google ScholarDigital Library
- R. Impagliazzo and R. Williams. Communication complexity with synchronized clocks. In Proceedings of the 25th Conference on Computational Complexity (CCC), pages 259--269. IEEE, 2010. doi:10.1109/CCC.2010.32. Google ScholarDigital Library
- Y. Ishai and E. Kushilevitz. Perfect constant-round secure computation via perfect randomizing polynomials. In Proceedings of the 29th International Colloquium on Automata, Languages, and Programming (ICALP), pages 244--256. Springer, 2002. doi:10.1007/3--540--45465--9--22. Google ScholarDigital Library
- R. Jain and H. Klauck. The partition bound for classical communication complexity and query complexity. In Proceedings of the 25th Conference on Computational Complexity (CCC), pages 247--258. IEEE, 2010. doi:10.1109/CCC.2010.31. Google ScholarDigital Library
- T. Jayram. Hellinger strikes back: A note on the multi-party information complexity of AND. In Proceedings of the 13th International Workshop on Randomization and Computation (RANDOM), pages 562--573. Springer, 2009. doi:10.1007/978--3--642-03685--9--42. Google ScholarDigital Library
- T. Jayram, R. Kumar, and D. Sivakumar. Two applications of information complexity. In Proceedings of the 35th Symposium on Theory of Computing (STOC), pages 673--682. ACM, 2003. doi:10.1145/780542.780640. Google ScholarDigital Library
- S. Jukna. On graph complexity. Combinatorics, Probability, & Computing, 15(6):855--876, 2006. doi:10.1017/S0963548306007620. Google ScholarDigital Library
- S. Jukna. Boolean Function Complexity: Advances and Frontiers, volume 27 of Algorithms and Combinatorics. Springer, 2012. Google ScholarDigital Library
- H. Klauck. Rectangle size bounds and threshold covers in communication complexity. In Proceedings of the 18th Conference on Computational Complexity (CCC), pages 118--134. IEEE, 2003. doi:10.1109/CCC.2003.1214415.Google ScholarCross Ref
- H. Klauck. Lower bounds for quantum communication complexity. SIAM Journal on Computing, 37(1):20--46, 2007. doi:10.1137/S0097539702405620. Google ScholarDigital Library
- H. Klauck. A strong direct product theorem for disjointness. In Proceedings of the 42nd Symposium on Theory of Computing (STOC), pages 77--86. ACM, 2010. doi:10.1145/1806689.1806702. Google ScholarDigital Library
- H. Klauck. On Arthur Merlin games in communication complexity. In Proceedings of the 26th Conference on Computational Complexity (CCC), pages 189--199. IEEE, 2011. doi:10.1109/CCC.2011.33. Google ScholarDigital Library
- H. Klauck and S. Podder. Two results about quantum messages. In Proceedings of the 39th International Symposium on Mathematical Foundations of Computer Science (MFCS), pages 445--456. Springer, 2014. doi:10.1007/978--3--662--44465--8--38.Google ScholarCross Ref
- H. Klauck and V. Prakash. Streaming computations with a loquacious prover. In Proceedings of the 4th Innovations in Theoretical Computer Science Conference (ITCS), pages 305--320. ACM, 2013. doi:10.1145/2422436.2422471. Google ScholarDigital Library
- H. Klauck and V. Prakash. An improved interactive streaming algorithm for the distinct elements problem. In Proceedings of the 41st International Colloquium on Automata, Languages, and Programming (ICALP), pages 919--930. Springer, 2014. doi:10.1007/978--3--662--43948--7--76Google ScholarCross Ref
- E. Kushilevitz. Privacy and communication complexity. SIAM Journal on Discrete Mathematics, 5(2):273--284, 1992. doi:10.1137/0405021. Google ScholarDigital Library
- E. Kushilevitz and N. Nisan. Communication Complexity. Cambridge University Press, 1997. Google ScholarDigital Library
- N. Linial and A. Shraibman. Learning complexity vs communication complexity. Combinatorics, Probability, & Computing, 18(1--2):227--245, 2009. doi:10.1017/S0963548308009656. Google ScholarDigital Library
- S. Lokam. Spectral methods for matrix rigidity with applications to size-depth trade-offs and communication complexity. Journal of Computer and System Sciences, 63(3):449--473, 2001. doi:10.1006/jcss.2001.1786.Google ScholarDigital Library
- S. Lokam. Complexity lower bounds using linear algebra. Foundations and Trends in Theoretical Computer Science, 4(1--2):1--155, 2009. doi:10.1561/0400000011. Google ScholarDigital Library
- P. Papakonstantinou, D. Scheder, and H. Song. Overlays and limited memory communication. In Proceedings of the 29th Conference on Computational Complexity (CCC), pages 298--308. IEEE, 2014. doi:10.1109/CCC.2014.37. Google ScholarDigital Library
- P. Pudl ak, V. Rödl, and P. Savick y. Graph complexity. Acta Informatica, 25(5):515--535, 1988. doi:10.1007/BF00279952. Google ScholarDigital Library
- R. Raz and A. Shpilka. On the power of quantum proofs. In Proceedings of the 19th Conference on Computational Complexity (CCC), pages 260--274. IEEE, 2004. doi:10.1109/CCC.2004.1313849. Google ScholarDigital Library
- A. Razborov. On rigid matrices. Technical report, Steklov Mathematical Institute, 1989. In Russian.Google Scholar
- A. Razborov. On the distributional complexity of disjointness. Theoretical Computer Science, 106(2):385--390, 1992. doi:10.1016/0304--3975(92)90260-M. Google ScholarDigital Library
- K. Reinhardt and E. Allender. Making nondeterminism unambiguous. SIAM Journal on Computing, 29(4):1118--1131, 2000. doi:10.1137/S0097539798339041. Google ScholarDigital Library
- M. Santha. Relativized Arthur--Merlin versus Merlin--Arthur games. Information and Computation, 80(1):44--49, 1989. doi:10.1016/0890--5401(89)90022--9. Google ScholarDigital Library
- U. Schöning. Probabilistic complexity classes and lowness. Journal of Computer and System Sciences, 39(1):84--100, 1989. doi:10.1016/0022-0000(89)90020--2. Google ScholarDigital Library
- L. Valiant. Graph-theoretic arguments in low-level complexity. In Proceedings of the 6th Symposium on Mathematical Foundations of Computer Science (MFCS), pages 162--176. Springer, 1977. doi:10.1007/3--540-08353--7--135.Google ScholarCross Ref
- L. Valiant. Completeness classes in algebra. In Proceedings of the 11th Symposium on Theory of Computing (STOC), pages 249--261. ACM, 1979. doi:10.1145/800135.804419. Google ScholarDigital Library
- H. Wunderlich. A note on a problem in communication complexity. Technical report, arXiv, 2012. arXiv:1205.0903.Google Scholar
- H. Wunderlich. On a theorem of Razborov. Computational Complexity, 21(3):431--477, 2012. doi:10.1007/s00037-011-0021--5. Google ScholarDigital Library
Index Terms
- Zero-Information Protocols and Unambiguity in Arthur-Merlin Communication
Recommendations
Zero-Information Protocols and Unambiguity in Arthur---Merlin Communication
We study whether information complexity can be used to attack the long-standing open problem of proving lower bounds against Arthur---Merlin ($${{\textsf {AM}}}$$AM) communication protocols. Our starting point is to show that--in contrast to plain ...
From Private Simultaneous Messages to Zero-Information Arthur---Merlin Protocols and Back
Göös et al. (ITCS, 2015) have recently introduced the notion of Zero-Information Arthur---Merlin Protocols ($$\mathsf {ZAM}$$ZAM). In this model, which can be viewed as a private version of the standard Arthur---Merlin communication complexity game, ...
Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds
CCC '10: Proceedings of the 2010 IEEE 25th Annual Conference on Computational ComplexityWe show that if Arthur-Merlin protocols can be derandomized, then there is a Boolean function computable in deterministic exponential-time with access to an $\bm{\NP}$ oracle, that cannot be computed by Boolean circuits of {\em exponential} size. More ...
Comments