Testing Monotonicity

We present a (randomized) test for monotonicity of Boolean functions. Namely, given the ability to query an unknown function at arguments of its choice, the test always accepts a monotone , and rejects with high probability if it is -far from being monotone (i.e., every monotone function differs from on more than an fraction of the domain). The complexity of the test is ! #" %$ . The analysis of our algorithm relates two natural combinatorial quantities that can be measured with respect to a Boolean function; one being global to the function and the other being local to it. We also consider the problem of testing monotonicity based only on random examples labeled by the function. We show an &' %( ) " %$ lower bound on the number of required examples, and provide a matching upper bound (via an algorithm). * Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, ISRAEL. E-mail: [email protected]. Work done while visiting LCS, MIT. + Laboratory for Computer Science, MIT, 545 Technology Sq., Cambridge, MA 02139. E-mail: , shafi,e lehman,danar @theory.lcs.mit.edu. . Supported by an ONR Science Scholar Fellowship at the Bunting Institute.