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Abstract: We show that there are functions computable by linear size boolean circuits of depth k that require super-polynomial size perceptrons of depth k \Gamma 1, for k ! logn=(6loglogn). This result implies the existence of an oracle A such that S p;A k 6` PP S p;A k\Gamma2 and in particular this oracle separates the levels in the PP PH hierarchy. Using the same ideas, we show a lower bound for another function, which makes it possible to strengthen the oracle separation to D p;A k 6`... (Update)
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...depends on circuit lower bounds. This approach has since then been employed fruitfully a number of times, see e.g. Has88, Ko89, Gre91, BU97] to name only very few examples. However, all the times the actual construction of the oracle requires a diagonalization where one...
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BibTeX entry: (Update)
C. Berg and S. Ulfberg. A lower bound for perceptrons and an oracle separation of the PP PH hierarchy. In Proceedings 12th Conference on Computational Complexity, pages 165--172. IEEE Computer Society Press, 1997. http://citeseer.ist.psu.edu/berg97lower.html More
@inproceedings{ berg97lower,
author = "Christer Berg and Staffan Ulfberg",
title = "A Lower Bound for Perceptrons and an Oracle Separation of the {PP} {PH} Hierarchy",
booktitle = "{IEEE} Conference on Computational Complexity",
pages = "165-172",
year = "1997",
url = "citeseer.ist.psu.edu/berg97lower.html" }
Citations (may not include all citations):84 Mathematical Systems Theory (context) - Furst, Saxe et al. - 1984 ACM
66 Pp is closed under intersection - Beigel, Reingold et al. - 1995 ACM DBLP
48 Borel sets and circuit complexity (context) - Sipser - 1983 ACM DBLP
23 and the polynomial hierarchy (context) - Beigel, PP - 1994
19 Probabilistic polynomial time is closed under parity reductr.. - Beigel, Hemachandra et al. - 1991
18 Information Processing Letters - Green, separating et al. - 1991
16 Relativized polynomial time hierarchies having exactly k lev.. (context) - Ko - 1989 ACM DBLP
15 Almost optimal lower bounds for small depth circuits - Hstad - 1989 ACM DBLP
15 the power of small-depth threshold circuits - Hstad, Goldmann - 1991
6 Separating PH from PP by relativization (context) - Fu - 1992
4 Separating the polynomial-time hierarchy by oracles (context) - Yao - 1985 ACM
3 Computational Limitations for Small-Depth Circuits (context) - Hstad - 1987 ACM
1 A lower bound for monotone perceptrons - Green - 1995
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