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# A Lower Bound on Circuit Complexity of Vector Function in U2

• Evgeny Demenkov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7353)

## Abstract

In 1973, Lamagna and Savage proved the following result. If f j : {0,1}n → {0,1} for 1 ≤ j ≤ m depends on at least two variables and if for i ≠ j, f i  ≠ f j and $$f_i \neq \bar{f_j}$$, then for any binary basis Ω,
$$C_{\Omega}(f_1, \ldots f_m) \geq \min\limits_j C_{\Omega}(f_j) + m -1,$$
where C Ω(f) is the minimal size of a circuit computing f in the basis Ω.

The main purpose of this paper is to give a better lower bound for the following case. Let f : {0,1}n → {0,1} and f i  = f ⊕ x i for 1 ≤ i ≤ n. Assume that f is not a constant after any three substitutions x i  = c i for different variables. Then

$$C_{U_2}(f_1, \ldots f_n) \geq \min\limits_{i \neq j, c_i, c_j } C_{U_2}(f\mid_{x_i = c_i, x_j = c_j})+2n-O(1),$$
where U 2 = B 2 ∖ { ⊕ , ≡ }. This implies a 7n lower bound on the circuit complexity over U 2 of f 1, …, f n if f has circuit complexity at least 5n.

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## References

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Shannon, C.E.: The synthesis of two-terminal switching circuits. Bell System Technical Journal 28, 59–98 (1949)
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Blum, N.: A Boolean function requiring 3n network size. Theoretical Computer Science 28, 337–345 (1984)
3. 3.
Demenkov, E., Kulikov, A.S.: An Elementary Proof of a 3no(n) Lower Bound on the Circuit Complexity of Affine Dispersers. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 256–265. Springer, Heidelberg (2011)
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Iwama, K., Morizumi, H.: An Explicit Lower Bound of 5no(n) for Boolean Circuits. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 353–364. Springer, Heidelberg (2002)
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Lamagna, E., Savage, J.: On the logical complexity of symmetric switching functions in monotone and complete bases. Discrete Optimization, Technical report, Brown University, Rhode Island (1973)Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 2012

## Authors and Affiliations

• Evgeny Demenkov
• 1
1. 1.St. Petersburg State UniversityRussia