Abstract
This exposition provides a proof of the existence of polynomial-size monotone formula for Majority. The exposition follows the main principles of Valiant’s proof (J. Algorithms, 1984), but deviates from it in the actual implementation. Specifically, we show that, with high probability, a full ternary tree of depth \(2.71\log _2n\) computes the majority of n values when each leaf of the tree is assigned at random one of the n values.
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Notes
- 1.
- 2.
One direction is almost trivial, for the other direction see [5].
- 3.
Sorting networks may be viewed as Boolean circuits with bit-comparison gates (a.k.a comparators), where each comparator is a (2-bit) sorting device. Observe that a comparator can be implemented by a monotone circuit (i.e., \(\mathtt{comp}(x,y)=(\min (x,y),\max (x,y))=((x\wedge y),(x\vee y))\)), and that the middle bit of the sorted sequence equals the majority value of the original sequence.
- 4.
One way to see this is to define \(p=\mathsf{Pr}[Z_1=0]\).
- 5.
Suppose that i iterations are sufficient for increasing the bias from 1/2n to \(\delta _0\). Then, \((1.5-2\delta _0^2)^i \cdot (1/2n) \ge \delta _0\) holds, which solves to \(i \ge \log _{1.5-2\delta _0^2}(2\delta _0n)\). Hence, we may use the minimal such \(i\in \mathbb {N}\).
- 6.
Actually, we have \(\mathsf{Pr}[MAJ(x)\!=\!F_\ell (R(x))]>1-2^{-n-\varOmega (n)}\), because we can use \(\ell _3=(2.71-c_1)\cdot \log _2 n-O(1)=\varOmega (\log n)\), where \(c_1\approx 1/\log _2(1.5)\approx 1.70951129\). (Alternatively, note that the analysis of the last \(\ell _3\) iterations actually yields an error probability of \(3^{2^{\ell _3}-1}\cdot (0.1)^{2^{\ell _3}}<(0.3)^{2^{\ell _3}}<2^{-1.7n}\). Furthermore, 0.1 can be replaced by any positive constant.)
- 7.
This is surprising only if we forget that V takes four inputs rather than three.
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Acknowledgments
We thank to Alina Arbitman for her comments and suggestions regarding the original write-up.
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Goldreich, O. (2020). On (Valiant’s) Polynomial-Size Monotone Formula for Majority. In: Goldreich, O. (eds) Computational Complexity and Property Testing. Lecture Notes in Computer Science(), vol 12050. Springer, Cham. https://doi.org/10.1007/978-3-030-43662-9_3
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