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| Name | An average-case depth hierarchy theorem for Boolean - Li-Yang Tan |
| Video Uploader | Video From Institute for Advanced Study |
| Upload Date | This Video Uploaded At 04-04-2016 17:16:49 |
| Video Discription | Computer Science/Discrete Mathematics Seminar I Topic: An average-case depth hierarchy theorem for Boolean circuits I Speaker: Li-Yang Tan Affiliation: Toyota Technological Institute, Chicago Date: Monday, April 4 We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of AND, OR, and NOT gates. Our hierarchy theorem says that for every d≥2d≥2, there is an explicit nn-variable Boolean function ff, computed by a linear-size depth-dd formula, which is such that any depth-(d−1)(d−1) circuit that agrees with ff on (1/2+on(1))(1/2+on(1)) fraction of all inputs must have size exp(nΩ(1/d)).exp(nΩ(1/d)). This answers an open question posed by Håstad in his Ph.D. thesis. Our average-case depth hierarchy theorem implies that the polynomial hierarchy is infinite relative to a random oracle with probability 1, confirming a conjecture of Håstad, Cai, and Babai. We also use our result to show that there is no "approximate converse" to the results of Linial, Mansour, Nisan and Boppana on the total influence of constant-depth circuits, thus answering a question posed by O'Donnell, Kalai, and Hatami. A key ingredient in our proof is a notion of random projections which generalize random restrictions. Joint work with Ben Rossman and Rocco Servedio. Paper available at http://arxiv.org/abs/1504.03398 For more videos, visit http://video.ias.edu [0Vm2wL60Ncc] |
| Category | Education |
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