## Abstract

We present a (randomized) test for monotonicity of Boolean functions. Namely, given the ability to query an unknown function f : (0,1)^{n} → (0,1) at arguments of its choice, the test always accepts a monotone f, and rejects f with high probability if it is ∈-far from being monotone (i.e., every monotone function differs from f on more than an ∈ fraction of the domain). The complexity of the test is O(n/∈). 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. A key ingredient is the use of a switching (or sorting) operator on functions.

Original language | English |
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Pages (from-to) | 301-337 |

Number of pages | 37 |

Journal | Combinatorica |

Volume | 20 |

Issue number | 3 |

DOIs | |

State | Published - 2000 |

Externally published | Yes |

### Bibliographical note

Funding Information:* A preliminary (and weaker) version of th is work appeared in [25] † Work done wh ile visiting LCS, MIT. ‡Supported in part by DARPA grant DABT63-96-C-0018 an in part by a Guastella fellowsh ip. § Th is work was done wh ile visiting LCS, MIT, and was supported by an ONR Science Sch olar Fellowsh ip at th e Bunting Institute.