Computational Depth
نویسندگان
چکیده
We introduce Computational Depth, a measure for the amount of “nonrandom” or “useful” information in a string by considering the difference of various Kolmogorov complexity measures. We investigate three instantiations of Computational Depth: Basic Computational Depth, a clean notion capturing the spirit of Bennett’s Logical Depth. Time-t Computational Depth and the resulting concept of Shallow Sets, a generalization of sparse and random sets based on low depth properties of their characteristic sequences. We show that every computable set that is reducible to a shallow set has polynomial-size circuits. Distinguishing Computational Depth, measuring when strings are easier to recognize than to produce. We show that if a Boolean formula has a nonnegligible fraction of its satisfying assignments with low depth, then we can find a satisfying assignment efficiently. Research done while visiting the Center for Discrete Mathematics and Computer Science (DIMACS). Supported in part by DIMACS and Fundação Calouste Gulbenkian. This author is also partially supported by funds granted to LIACC through the Programa de Financiamento Plurianual, Fundação para a Ciência e Tecnologia and Programa POSI. Address: Artificial Intelligence and Computer Science Laboratory, R.Campo Alegre, 823, 4150-180 Porto, Portugal. Email: [email protected]. Web: http://www.ncc.up.pt/ ̃lfa. yAddress: 4 Independence Way, Princeton, NJ 08540 Email: [email protected]. Web: http://www.neci.nj.nec.com/homepages/fortnow. zAddress: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540. Email: [email protected]. Web: http://www.math.ias.edu/ ̃dieter. Torturing an uninformed witness cannot give information about the crime. Leonid Levin [10]
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تاریخ انتشار 2001