Abstract
The Merlin-Arthur class of languages MA is included into Arthur-Merlin class AM, and into PP. For a standard transformation of a given MA protocol with Arthur’s message (= random string) of length a and Merlin’s message of length m to a PP machine, the latter needs O(ma) random bits. The same holds for simulating MA protocols by AM protocols: in the resulting AM protocol the length of Arthur’s message (= random string) is O(ma). And the same holds for simulating heuristic MA protocols by heuristic AM protocols as well. In the paper [A. Knop, Circuit Lower Bounds for Average-Case MA, CSR 2015] it was conjectured that, in the transformation of heuristic MA protocols to heuristic AM protocols, O(ma) can be replaced by a polynomial of a only. A similar question can be asked for normal MA and AM protocols, and for the simulation of MA protocols by PP machines. In the present paper we show that, relative to an oracle, both latter questions answer in the negative and Knop’s conjecture is false. Moreover, the same is true for simulation of MA protocols by AM protocols in which the error probability is not bounded away from 1/2, the so called PP\(\cdot \)NP protocols. The latter protocols generalize both AM protocols and PP machines .
The results are supported by Russian Science Foundation (20-11-20203).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alon, N., Feige, U., Wigderson, A., Zuckerman, D.: Derandomized graph products. Comput. Complex. 5(1), 60–75 (1995)
Aaronson, S., Wigderson, A.: Algebrization: a new barrier in complexity theory. ACM Trans. Comput. Theory 1(1), 2:1–2:54 (2009)
Babai, L.: Trading group theory for randomness. In: 17th STOC, pp. 421–429 (1985)
Baker, T., Gill, J., Solovay, R.: Relativization of P=?NP question. SIAM J. Comput. 4(4), 431–442 (1975)
Boppana, R.B., Håstad, J., Zachos, S.: Does co-np have short interactive proofs? Inf. Process. Lett. 25, 127–132 (1987)
Knop, A.: Circuit lower bounds for average-case MA. In: Beklemishev, L.D., Musatov, D.V. (eds.) CSR 2015. LNCS, vol. 9139, pp. 283–295. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-20297-6_18
Lautemann, C.: BPP and the polynomial hierarchy. Inf. Proc. Lett. 17(4), 215–217 (1983)
Sipser, M.: A complexity theoretic approach to randomness. In: Proceedings of the 15th ACM Symposium on the Theory of Computing, pp. 330–335. ACM, New York (1983)
Vereshchagin, N.: On the power of PP. In: Proceedings of 7th Annual IEEE Conference on Structure in Complexity Theory, Boston, MA, pp. 138–143 (1992)
Acknowledgements
The author is sincerely grateful to anonymous referees for helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Vereshchagin, N. (2022). How Much Randomness is Needed to Convert MA Protocols to AM Protocols?. In: Kulikov, A.S., Raskhodnikova, S. (eds) Computer Science – Theory and Applications. CSR 2022. Lecture Notes in Computer Science, vol 13296. Springer, Cham. https://doi.org/10.1007/978-3-031-09574-0_21
Download citation
DOI: https://doi.org/10.1007/978-3-031-09574-0_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-09573-3
Online ISBN: 978-3-031-09574-0
eBook Packages: Computer ScienceComputer Science (R0)