## Abstract

We show that a boolean valued function over n variables, where each variable ranges in an arbitrary probability space, can be tested for the property of depending on only J of them using a number of queries that depends only polynomially on J and the approximation parameter ε. We present several tests that require a number of queries that is polynomial in J and linear in ε^{-1}. We show a non-adaptive test that has one-sided error, an adaptive version of it that requires fewer queries, and a non-adaptive two-sided version of the test that requires the least number of queries. We also show a two-sided non-adaptive test that applies to functions over n boolean variables, and has a more compact analysis. We then provide a lower bound of Ω̃(J) on the number of queries required for the non-adaptive testing of the above property; a lower bound of Ω(log(J+1)) for adaptive algorithms naturally follows from this. In establishing this lower bound we also prove a result about random walks on the group Z^{q}_{2} that may be interesting in its own right. We show that for some t(q)=Õ(q ^{2}), the distributions of the random walk at times t and t+2 are close to each other, independently of the step distribution of the walk. We also discuss related questions. In Particular, when given in advance a known J-junta function h, we show how to test a function for the property of being identical to h up to a permutation of the variables, in a number of queries that is polynomial in J and ε^{-1}.

Original language | English |
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Pages (from-to) | 753-787 |

Number of pages | 35 |

Journal | Journal of Computer and System Sciences |

Volume | 68 |

Issue number | 4 |

DOIs | |

State | Published - Jun 2004 |

Externally published | Yes |

### Bibliographical note

Funding Information:·Corresponding author. E-mail address: eldar@cs.technion.ac.il (E. Fischer). 1Research supported by a Technion VPR fund Dent Charitable Trust, non-military research fund, and by a joint Haifa University–Technion research fund. 2Research supported by the Israel Science Foundation (Grant 32/00-1). 3Research supported by an Israeli Science Foundation grant and a United States–Israel Binational Science Foundation grant. 4Research supported by the Israel Science Foundation (Grant 039-7165).

## Keywords

- Boolean functions
- Discrete Fourier Analysis
- Juntas
- Property testing