Abstract

Combinatorial property testing, initiated by Rubinfeld and Sudan [15] and formally dened by Goldreich, Goldwasser and Ron in [12], deals with the following relaxation of decision problems: Given a xed property P and an input f , distinguish between the case that f satises P , and the case that no input that diers from f in less than some xed fraction of the places satises P . An (; q)-test for P is a randomized algorithm that queries at most q places of an input x and distinguishes with probability 2/3 between the case that f has the property and the case that at least an -fraction of the places of f need to be changed in order for it to have the property. Here we concentrate on labeled, d-dimensional grids, where the grid is viewed as a partially ordered set (poset) in the standard way (i.e as a product order of total orders). The main result here presents an (; poly(1=))-test for every property of 0/1 labeled, d-dimensional grids that is characterized by a nite collection of forbidden induced posets. Such properties include the `monotonicity' property studied in [7, 6], other more complicated forbidden chain patterns, and general forbidden poset patterns. We also present a (less ecient) test for such properties of labeled grids with larger xed size alphabets. All the above tests have in addition a 1-sided error probability. Another result is a test for any bipartite graph property that is characterized by a nite set of forbidden induced subgraphs. Our test for such properties requires signicantly less queries than the previously known algorithm. Both collections above are variants of properties that are dened by certain rst order formulae with no quantier alternation over the syntax containing the grid order relations (and some add...

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