Testing of matrix properties

Combinatorial property testing deals with the following relaxation of decision problems: Given a fixed property P and an input f, distinguish between the case that f satisfies P, and the case that no input that differs from f in less than some fixed fraction of the places satisfies P. An (ε, 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 ε-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 is an (ε,poly(1/ε))-test for every property of 0/1 labeled, d-dimensional grids that is characterized by a finite collection of forbidden induced posets. Such properties include the 'monotonicity' property and many other properti es. A (less efficient) test for such properties with larger fixed size alphabets is also presented. Another result is a more efficient test than was previously known for a collection of bipartite graph properties. Both collections above are variants of properties that are defined by certain first order formulae with no quantifier alternation over the syntax containing the grid order relations (and some additional relations for the bipartite graph properties). We also show that with one quantifier alternation, a certain property can be defined, for which no test with query complexity of O(n^1/10) exists. The above results identify new classes of efficiently testable properties.