Negation-Limited Formulas

Authors Siyao Guo, Ilan Komargodski

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Siyao Guo
Ilan Komargodski

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Siyao Guo and Ilan Komargodski. Negation-Limited Formulas. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 850-866, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


We give an efficient structural decomposition theorem for formulas that depends on their negation complexity and demonstrate its power with the following applications. We prove that every formula that contains t negation gates can be shrunk using a random restriction to a formula of size O(t) with the shrinkage exponent of monotone formulas. As a result, the shrinkage exponent of formulas that contain a constant number of negation gates is equal to the shrinkage exponent of monotone formulas. We give an efficient transformation of formulas with t negation gates to circuits with log(t) negation gates. This transformation provides a generic way to cast results for negation-limited circuits to the setting of negation-limited formulas. For example, using a result of Rossman (CCC'15), we obtain an average-case lower bound for formulas of polynomial-size on n variables with n^{1/2-epsilon} negations. In addition, we prove a lower bound on the number of negations required to compute one-way permutations by polynomial-size formulas.
  • Negation complexity
  • De Morgan formulas
  • Shrinkage


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  1. Miklós Ajtai, János Komlós, and Endre Szemerédi. An o(n log n) sorting network. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, STOC, pages 1-9, 1983. Google Scholar
  2. Kazuyuki Amano and Akira Maruoka. A superpolynomial lower bound for a circuit computing the clique function with at most (1/6)log log n negation gates. SIAM J. Comput., 35(1):201-216, 2005. Google Scholar
  3. Benny Applebaum, Yuval Ishai, and Eyal Kushilevitz. Cryptography in NC⁰. SIAM J. Comput., 36(4):845-888, 2006. Google Scholar
  4. Robert Beals, Tetsuro Nishino, and Keisuke Tanaka. On the complexity of negation-limited boolean networks. SIAM J. Comput., 27(5):1334-1347, 1998. Google Scholar
  5. Eric Blais, Clément L. Canonne, Igor Carboni Oliveira, Rocco A. Servedio, and Li-Yang Tan. Learning circuits with few negations. To appear in Proceedings of the 19th International Workshop on Randomization and Computation, RANDOM, 2015. Available at URL:
  6. Nader H. Bshouty and Christino Tamon. On the Fourier spectrum of monotone functions. J. ACM, 43(4):747-770, 1996. Google Scholar
  7. Ruiwen Chen, Valentine Kabanets, Antonina Kolokolova, Ronen Shaltiel, and David Zuckerman. Mining circuit lower bound proofs for meta-algorithms. In Proceedings of the 29th Conference on Computational Complexity, CCC, pages 262-273, 2014. Google Scholar
  8. Ruiwen Chen, Valentine Kabanets, and Nitin Saurabh. An improved deterministic #SAT algorithm for small De Morgan formulas. In Proceedings of the 39th International Symposium on Mathematical Foundations of Computer Science, MFCS, pages 165-176, 2014. Google Scholar
  9. Moshe Dubiner and Uri Zwick. How do read-once formulae shrink? Combinatorics, Probability & Computing, 3:455-469, 1994. Google Scholar
  10. Michael. J. Fischer. The complexity of negation-limited networks-a brief survey. Automata Theory and Formal Languages, 33:71-82, 1975. Google Scholar
  11. Oded Goldreich and Rani Izsak. Monotone circuits: One-way functions versus pseudorandom generators. Theory of Computing, 8(1):231-238, 2012. Google Scholar
  12. Mika Göös and Toniann Pitassi. Communication lower bounds via critical block sensitivity. In Proceedings of the 46th Annual Symposium on Theory of Computing, STOC, pages 847-856, 2014. Google Scholar
  13. Siyao Guo, Tal Malkin, Igor Carboni Oliveira, and Alon Rosen. The power of negations in cryptography. In Proceedings of the 12th Theory of Cryptography Conference, TCC, pages 36-65, 2015. Google Scholar
  14. Danny Harnik and Ran Raz. Higher lower bounds on monotone size. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, STOC, pages 378-387, 2000. Google Scholar
  15. Johan Håstad. One-way permutations in NC0. Inf. Process. Lett., 26(3):153-155, 1987. Google Scholar
  16. Johan Håstad. The shrinkage exponent of de Morgan formulas is 2. SIAM J. Comput., 27(1):48-64, 1998. Google Scholar
  17. Johan Håstad, Alexander A. Razborov, and Andrew Chi-Chih Yao. On the shrinkage exponent for read-once formulae. Theor. Comput. Sci., 141(1&2):269-282, 1995. Google Scholar
  18. Russell Impagliazzo and Valentine Kabanets. Fourier concentration from shrinkage. In Proceedings of the 29th Conference on Computational Complexity, CCC, pages 321-332, 2014. Google Scholar
  19. Russell Impagliazzo, Raghu Meka, and David Zuckerman. Pseudorandomness from shrinkage. In Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 111-119, 2012. Google Scholar
  20. Russell Impagliazzo and Noam Nisan. The effect of random restrictions on formula size. Random Struct. Algorithms, 4(2):121-134, 1993. Google Scholar
  21. Kazuo Iwama, Hiroki Morizumi, and Jun Tarui. Negation-limited complexity of parity and inverters. Algorithmica, 54(2):256-267, 2009. Google Scholar
  22. Stasys Jukna. Boolean Function Complexity - Advances and Frontiers, volume 27 of Algorithms and combinatorics. Springer, 2012. Google Scholar
  23. Ilan Komargodski, Ran Raz, and Avishay Tal. Improved average-case lower bounds for DeMorgan formula size. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 588-597, 2013. Google Scholar
  24. Andrey A. Markov. On the inversion complexity of a system of functions. J. ACM, 5(4):331-334, 1958. Google Scholar
  25. Hiroki Morizumi. Limiting negations in formulas. In Proceedings of the 36th International Colloquium on Automata, Languages and Programming, ICALP, pages 701-712, 2009. Google Scholar
  26. Moni Naor and Omer Reingold. Number-theoretic constructions of efficient pseudo-random functions. J. ACM, 51(2):231-262, 2004. Google Scholar
  27. Eduard I. Nechiporuk. On the complexity of schemes in some bases containing nontrivial elements with zero weights. Problemy Kibernetiki, 8:123-160, 1962. In Russian. Google Scholar
  28. Noam Nisan. Pseudorandom bits for constant depth circuits. Combinatorica, 11(1):63-70, 1991. Google Scholar
  29. Noam Nisan and Avi Wigderson. Hardness vs randomness. J. Comput. Syst. Sci., 49(2):149-167, 1994. Google Scholar
  30. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. Google Scholar
  31. Mike Paterson and Uri Zwick. Shrinkage of de Morgan formulae under restriction. Random Struct. Algorithms, 4(2):135-150, 1993. Google Scholar
  32. Ran Raz and Avi Wigderson. Probabilistic communication complexity of boolean relations (extended abstract). In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, FOCS, pages 562-567, 1989. Google Scholar
  33. Benjamin Rossman. Correlation bounds against monotone NC¹. In Proceedings of the 30th Conference on Computational Complexity, CCC, pages 392-411, 2015. Google Scholar
  34. Miklos Santha and Christopher B. Wilson. Polynomial size constant depth circuits with a limited number of negations. In Proceedings of the 8th Annual Symposium on Theoretical Aspects of Computer Science, STACS, pages 228-237, 1991. Google Scholar
  35. Bella A. Subbotovskaya. Realizations of linear function by formulas using +,⋅,-. Doklady Akademii Nauk SSSR, 136:3:553-555, 1961. In Russian. Google Scholar
  36. Shao Chin Sung and Keisuke Tanaka. Limiting negations in bounded-depth circuits: An extension of markov’s theorem. Inf. Process. Lett., 90(1):15-20, 2004. Google Scholar
  37. Avishay Tal. Shrinkage of De Morgan formulae by spectral techniques. In Proceedings of the 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 551-560, 2014. Google Scholar
  38. Michel Talagrand. How much are increasing sets positively correlated? Combinatorica, 16(2):243-258, 1996. Google Scholar
  39. Keisuke Tanaka, Tetsuro Nishino, and Robert Beals. Negation-limited circuit complexity of symmetric functions. Inf. Process. Lett., 59(5):273-279, 1996. Google Scholar
  40. Leslie G. Valiant. Short monotone formulae for the majority function. J. Algorithms, 5(3):363-366, 1984. Google Scholar