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Uniform proofs of ACC representations

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Abstract

We give a uniform proof of the theorems of Yao and Beigel–Tarui representing ACC predicates as constant depth circuits with \(\hbox {MOD}_{m}\) gates and a symmetric gate. The proof is based on a relativized, generalized form of Toda’s theorem expressed in terms of closure properties of formulas under bounded universal, existential and modular counting quantifiers. This allows the main proofs to be expressed in terms of formula classes instead of Boolean circuits. The uniform version of the Beigel–Tarui theorem is then obtained automatically via the Furst–Saxe–Sipser and Paris–Wilkie translations. As a special case, we obtain a uniform version of Razborov and Smolensky’s representation of \(\hbox {AC}^{0}[p]\) circuits. The paper is partly expository, but is also motivated by the desire to recast Toda’s theorem, the Beigel–Tarui theorem, and their proofs into the language of bounded arithmetic. However, no knowledge of bounded arithmetic is needed.

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References

  1. Allender, E.: A note on the power of threshold circuits. In: Proceedings 30th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 580–584 (1989)

  2. Allender, E.: The permanent requires large uniform threshold circuits. Chic. J. Theor. Comput. Sci. (article 7) (1999)

  3. Allender, E., Gore, V.: A uniform circuit lower bound for the permanent. SIAM J. Comput. 23(5), 1026–1049 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Allender, E., Hertrampf, U.: Depth reduction for circuits of unbounded fan-in. Inf. Comput. 112, 217–238 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  6. Barrington, D.A.M.: Quasipolynomial size circuit classes. In: Proc. 7th Structure in Complexity Conference, pp. 86–93 (1992)

  7. Beigel, R., Tarui, J.: On ACC. Comput. Complex. 4, 350–366 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Beigel, R., Tarui, J., Toda, S.: On probabilistic ACC circuits with an exact-threshold output gate. In: Algorithms and Computation: Third International Symposium, ISAAC’92, Lecture Notes in Computer Science 650, pp. 420–429 (1992)

  9. Buss, S.R.: Bounded Arithmetic. Bibliopolis, Naples, Italy (1986). Revision of 1985 Princeton University. Ph.D. thesis

  10. Buss, S.R., Kołodziejczyk, L.A., Zdanowski, K.: Collapsing modular counting in bounded arithmetic and constant depth propositional proofs. Trans. AMS 367, 7517–7563 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, S., Papakonstantinou, P.A.: Depth-reduction for composites. In: Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS’16), pp. 99–108. IEEE Computer Society (2016)

  12. Fortnow, L.: A simple proof of Toda’s theorem. Theory Comput. 5, 135–140 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Furst, M., Saxe, J.B., Sipser, M.: Parity, circuits and the polynomial-time hierarchy. Math. Syst. Theory 17, 13–27 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  14. Green, F., Köbler, J., Regan, K.W., Schwentick, T., Torán, J.: The power of the middle bit of a #P function. J. Comput. Syst. Sci. 50, 456–467 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hansen, K.A., Koucký, M.: A new characterization of \(ACC^0\) and probabilistic \(CC^0\). Conput. Complex. 19, 211–234 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jeřábek, E.: Approximate counting in bounded arithmetic. J. Symb. Log. 72(3), 959–993 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jeřábek, E.: Approximate counting by hashing in bounded arithmetic. J. Symb. Log. 74(3), 829–860 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kannan, R., Venkateswaran, H., Vinay, V., Yao, A.C.: A circuit-based proof of Toda’s theorem. Inf. Comput. 104(2), 271–276 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. Krajíček, J.: Bounded Arithmetic. Propositional Calculus and Complexity Theory. Cambridge University Press, Heidelberg (1995)

    Book  MATH  Google Scholar 

  20. Lautemann, C.: BPP and the polynomial hierarchy. Inf. Process. Lett. 17(4), 215–217 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Maciel, A., Pitassi, T.: Towards Lower Bounds for Bounded-Depth Frege Proofs with Modular Connectives. In: Beame, P.W., Buss, S.R. (eds.) Proof complexity and feasible arithmetics, pp. 195–227. American Mathematical Society, Providence (1998)

    Google Scholar 

  22. Papadimitriou, C.H., Zachos, S.K.: Two remarks on the power of counting. In: Proc. 6th Gesellschaft für Informatik Conference on Theoretical Computer Science, Lecture Notes in Computer Science 145, pp. 269–276. Springer, Berlin (1983)

  23. Paris, J.B., Wilkie, A.J.: \(\varDelta _{0}\) sets and induction. In: W. Guzicki, W. Marek, A. Pelc, C. Rauszer (eds.) Open Days in Model Theory and Set Theory, pp. 237–248. Warsaw University, Warsaw (1981)

  24. Razborov, A.A.: Lower bounds on the size of bounded depth networks over a complete basis with logical addition. Matematicheskie Zametki 41, 598–607 (1987). English translation in Mathematical Notes of the Academy of Sciences of the USSR 41, 333-338 (1987)

  25. Ruzzo, W.L.: On uniform circuit complexity. J. Comput. Syst. Sci. 22, 365–383 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schöning, U.: Probabilistic complexity classes and lowness. J. Comput. Syst. Sci. 39, 84–100 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sipser, M.: A complexity theoretic approach to randomness. In: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pp. 330–335. ACM Press (1983)

  28. Smolensky, R.: Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In: Proceedings of the Nineteenth Annual ACM Symposium on the Theory of Computing, pp. 77–82. ACM Press (1987)

  29. Straubing, H., Thérien, D.: A note on \(MOD_p\)-\(MOD_m\) circuits. Theory Comp. Syst. 39(5), 699–706 (2006)

    Article  MATH  Google Scholar 

  30. Tarui, J.: Probabilistic polynomials, AC\({}^0\) functions and the polynomial-time hierarchy. Theor. Comput. Sci. 113, 167–183 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  31. Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM J. Comput. 20(5), 865–877 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  32. Toda, S., Ogiwara, M.: Counting classes are at least as hard as the polynomial-time hierarchy. In: Proc. 6th Structure in Complexity Theory Conference, pp. 2–12 (1991)

  33. Valiant, L.G., Vazirani, V.V.: NP is as easy as detecting unique solutions. Theor. Comput. Sci. 47, 85–93 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  34. Williams, R.: Non-uniform ACC circuit lower bounds. Shorter version appeared in 26th IEEE Conference on Computational Complexity (CCC). Journal of the ACM, 61, pp. 115–125 (article 2) (2014)

  35. Yao, A.C.C.: On ACC and threshold circuits. In: Proc. 31st IEEE Symp. on Foundations of Computer Science (FOCS), pp. 619–627 (1990)

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  1. Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093-0112, USA

    Sam Buss

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  1. Sam Buss
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Correspondence to Sam Buss.

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Supported in part by NSF Grants DMS-1101228 and CCR-1213151, Simons Foundation Grant 306202 and Skolkovo Institute of Science and Technology. Part of this work was performed during the Special Semester program in Complexity in St. Petersburg, Russia.

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Buss, S. Uniform proofs of ACC representations. Arch. Math. Logic 56, 639–669 (2017). https://doi.org/10.1007/s00153-017-0560-9

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