Abstract
Although it is well known by a counting argument that relative to the full basis most Boolean functions need exponentially many operations, for explicit Boolean functions only linear lower bounds with small constant factors are known. For monotone networks (i.e., networks without negations) exponential lower bounds for explicit monotone Boolean functions have been proved. We describe the state of the art and give some arguments why techniques developed for the proof of lower bounds for monotone networks cannot easily be extended to Boolean networks with negations.
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References
Alon, N., Boppana, R.B.: The monotone circuit complexity of Boolean functions. Combinatorica 7, 1–22 (1987)
Amano, K., Maruoka, A.: The potential of the approximation method. SIAM J. Comput. 33, 433–447 (2004)
Amano, K., Maruoka, A.: A superpolynomial lower bound for a circuit computing the clique function with at most (1/6)loglogN negation gates. SIAM J. Comput. 35, 201–216 (2005)
Andreev, A.E.: On a method for obtaining lower bounds for the complexity of individual monotone functions. Soviet Math. Dokl. 31, 530–534 (1985)
Beals, R., Nishino, T., Tanaka, K.: On the complexity of negation-limited Boolean networks. SIAM J. Comput. 27, 1334–1347 (1998)
Berg, C., Ulfberg, S.: Symmetric approximation arguments for monotone lower bounds without sunflowers. Comput. Complex. 8, 1–20 (1999)
Berkowitz, S.J.: On some relationships between monotone and non-monotone circuit complexity, Tech. Report, Comput. Sci. Dept., Univ. of Toronto (1982)
Blum, N.: A Boolean function requiring 3n network size. TCS 28, 337–345 (1984)
Blum, N.: An Ω(n 4/3) lower bound on the monotone network complexity of the n th degree convolution. TCS 36, 59–69 (1985)
Brown, W.G.: On graphs that do not contain a Thompson graph. Canad. Math. Bull. 9, 281–285 (1966)
Chow, T.Y.: Almost-natural proofs. In: Proc. 49th FOCS, pp. 72–77 (2008)
Clote, P., Kranakis, E.: Boolean Functions and Computation Models. Springer, Heidelberg (2002)
Dunne, P.E.: The Complexity of Boolean Networks. Academic Press, London (1988)
Fischer, M.J.: The complexity of negation-limited networks - a brief survey. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 71–82. Springer, Heidelberg (1975)
Haken, A.: Counting bottlenecks to show monotone \(P \not= NP\). In: Proc. 36th FOCS, pp. 36–40 (1995)
Harnik, D., Raz, R.: Higher lower bounds on monotone size. In: Proc. 32nd STOC, pp. 191–201 (2000)
Iwama, K., Lachish, O., Morizumi, H., Raz, R.: An explicit lower bound of 5n − o(n) for Boolean circuits (manuscript, 2005)
Jukna, S.: Combinatorics of monotone computations. Combinatorica 19, 65–85 (1999)
Karchmer, M.: On proving lower bounds for circuit size. In: Proc. 8th Structure in Complexity Theory, pp. 112–118 (1993)
Kővári, T., Sós, V.T., Turán, P.: On a problem of K. Zarankiewicz. Colloq. Math. 3, 50–57 (1954)
Lamagna, E.A.: The complexity of monotone networks for certain bilinear forms, routing problems, sorting and merging. IEEE Trans. Comput. 28, 773–782 (1979)
Markov, A.A.: On the inversion complexity of a system of functions. J. ACM 5, 331–334 (1958)
Mehlhorn, K., Galil, Z.: Monotone switching circuits and Boolean matrix product. Computing 16, 99–111 (1976)
Mehlhorn, K.: Some remarks on Boolean sums. Acta Inform. 12, 371–375 (1979)
Neciporuk, E.I.: On a Boolean matrix. Systems Theory Res. 21, 236–239 (1971)
Paterson, M.S.: Complexity of monotone networks for Boolean matrix product. TCS 1, 13–20 (1975)
Paul, W.J.: A 2.5n lower bound on the combinational complexity of Boolean functions. SIAM J. Comput. 6, 427–443
Pippenger, N.: On another Boolean matrix. TCS 11, 49–56 (1980)
Pippenger, N., Valiant, L.G.: Shifting graphs and their applications. J. ACM 23, 423–432
Pratt, V.R.: The power of negative thinking in multiplying Boolean matrices. SIAM J. Comput. 4, 326–330 (1974)
Razborov, A.A.: Lower bounds on the monotone complexity of some Boolean functions. Soviet Math. Dokl. 31, 354–357 (1985)
Razborov, A.A.: A lower bound on the monotone network complexity of the logical permanent. Math. Notes Acad. Sci. USSR 37, 485–493 (1985)
Razborov, A.A.: On the method of approximation. In: Proc. 21st STOC, pp. 167–176 (1989)
Razborov, A.A., Rudich, S.: Natural proofs. JCSS 55, 24–35 (1997)
Savage, J.E.: Models of Computation: Exploring the Power of Computing. Addison-Wesley, Reading (1998)
Schnorr, C.P.: Zwei lineare untere Schranken für die Komplexität Boolescher Funktionen. Computing 13, 155–171 (1974)
Shannon, C.E.: The synthesis of two-terminal switching circuits. Bell Syst. Techn. J. 28, 59–98 (1949)
Simon, J., Tsai, S.-C.: On the bottleneck counting argument. TCS 237, 429–437 (2000)
Stockmeyer, L.: On the combinational complexity of certain symmetric Boolean functions. Math. Systems Theory 10, 323–336 (1977)
Tardos, É.: The gap between monotone and non-monotone circuit complexity is exponential. Combinatorica 8, 141–142 (1988)
Tiekenheinrich, J.: A 4n lower bound on the monotone Boolean complexity of a one output Boolean function. IPL 18, 201–202 (1984)
Valiant, L.G.: Graph-theoretic properties in computational complexity. JCSS 13, 278–285 (1976)
Valiant, L.G.: Negation is powerless for Boolean slice functions. SIAM J. Comput. 15, 531–535 (1986)
Wegener, I.: Switching functions whose monotone complexity is nearly quadratic. TCS 9, 83–97 (1979)
Wegener, I.: A new lower bound on the monotone network complexity of Boolean sums. Acta Informatica 13, 109–114 (1980)
Wegener, I.: Boolean functions whose monotone complexity is of size n 2/logn. TCS 21, 213–224 (1982)
Wegener, I.: The Complexity of Boolean Functions. Wiley-Teubner, Chichester (1987)
Weiß, J.: An n 3/2 lower bound on the monotone network complexity of the Boolean convolution. Information and Control 59, 184–188 (1983)
Widgerson, A.: The fusion method for lower bounds in circuit complexity. In: Combinatorics, Paul Erdős is eighty. Elsevier, Amsterdam (1993)
Zwick, U.: A 4n lower bound on the combinational complexity of certain symmetric Boolean functions over the basis of unate dyadic Boolean functions. SIAM J. Comput. 20, 499–505 (1991)
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Blum, N. (2009). On Negations in Boolean Networks. In: Albers, S., Alt, H., Näher, S. (eds) Efficient Algorithms. Lecture Notes in Computer Science, vol 5760. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03456-5_2
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DOI: https://doi.org/10.1007/978-3-642-03456-5_2
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