Attribute estimation and testing quasi-symmetry

Authors:
Krzysztof Majewski;Nicholas Pippenger
Affiliations:
Department of Informatics, University of Oslo, Boks 1072 Blindern, NO-0316 Oslo, Norway;Department of Mathematics, Harvey Mudd College, 1250 North Dartmouth Avenue, Claremont, CA 91711, USA
Venue:
Information Processing Letters
Year:
2009

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Abstract

A Boolean function is symmetric if it is invariant under all permutations of its arguments; it is quasi-symmetric if it is symmetric with respect to the arguments on which it actually depends. We present a test that accepts every quasi-symmetric function and, except with an error probability at most @d0, rejects every function that differs from every quasi-symmetric function on at least a fraction @e0 of the inputs. For a function of n arguments, the test probes the function at O((n/@e)log(n/@d)) inputs. Our quasi-symmetry test acquires information concerning the arguments on which the function actually depends. To do this, it employs a generalization of the property testing paradigm that we call attribute estimation. Like property testing, attribute estimation uses random sampling to obtain results that have only ''one-sided'' errors and that are close to accurate with high probability.