Abstract
A Boolean function on n variables is called k-mixed if for any two different restrictions fixing the same set of k variables must induce different functions on the remaining n − k variables. In this paper, we give an explicit construction of an n − o(n)-mixed Boolean function whose circuit complexity over the basis U 2 is 5n + o(n). This shows that a lower bound method on the size of a U 2-circuit that uses the property of k-mixed, which gives the current best lower bound of 5n − o(n) on a U 2-circuit size (Iwama, Lachish, Morizumi and Raz [STOC ’01, MFCS ’02]), has reached the limit.
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References
Alon, N., Nathanson, M.B., Ruzsa, I.: The Polynomial Method and Restricted Sums of Congruence Classes. J. Number Theory 56, 404–417 (1996)
Andreev, A., Baskakov, J., Clementi, A., Rolim, J.: Small Pseudo-Random Sets yields Hard Functions: New Tight Explicit Lower Bounds for Branching Programs. In: Proc. 26th ICALP, pp. 179–189 (1999); Also in ECCC TR97-053 (1997)
Forster, C.C., Stockton, F.D.: Counting Responders in an Associative Memory. IEEE Trans. Comput. C-20(12), 1580–1583 (1971)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1978)
Iwama, K., Morizumi, H.: An Explicit Lower Bound of 5n − o(n) for Boolean Circuits. In: Proc. 27th MFCS, pp. 353–364 (2002)
Iwama, K., Lachish, O., Morizumi, H., Raz, R.: An Explicit Lower Bound of 5n − o(n) for Boolean Circuits (manuscript, 2005), (preliminary versions are [9] and [5]), http://www.wisdom.weizmann.ac.il/~ranraz/publications/
Jukna, S.P.: Entropy of Contact Circuits and Lower Bounds on Their Complexity. Theoret. Comput. Sci. 57, 113–129 (1988)
Klein, P., Paterson, M.S.: Asymptotically Optimal Circuit for a Storage Access Function. IEEE Trans. Comput. 29(8), 737–738 (1980)
Lachish, O., Raz, R.: Explicit Lower Bound of 4.5n − o(n) for Boolean Circuits. In: Proc. 33rd STOC, pp. 399–408 (2001)
Savický, P., Zák, S.: A Large Lower Bound for 1-Branching Programs. ECCC TR96-036 Rev.01 (1996)
Dias da Silva, J.A., Hamidoune, Y.O.: Cyclic Spaces for Grassmann Derivatives and Additive Theory. Bull. London. Math. Soc. 26, 140–146 (1994)
Simon, J., Szegedy, M.: A New Lower Bound Theorem for Read Only Once Branching Programs and its Applications. In: Advances in Computational Complexity Theory, DIMACS Series, vol. 13, pp. 183–193 (1993)
Wegener, I.: The Complexity of Boolean Functions, Willey-Teubner (1987)
Zwick, U.: A 4n Lower Bound on the Combinatorial 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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Amano, K., Tarui, J. (2008). A Well-Mixed Function with Circuit Complexity 5n ±o(n): Tightness of the Lachish-Raz-Type Bounds. In: Agrawal, M., Du, D., Duan, Z., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2008. Lecture Notes in Computer Science, vol 4978. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79228-4_30
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DOI: https://doi.org/10.1007/978-3-540-79228-4_30
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