30th Annual Symposium on Foundations of Computer Science (1989)
Research Triangle Park, NC, USA
Oct. 30, 1989 to Nov. 1, 1989
N. Linial , IBM Almaden Res. Center, San Jose, CA, USA
Boolean functions in AC/sup O/ are studied using the harmonic analysis of the cube. The main result is that an AC/sup O/ Boolean function has almost all of its power spectrum on the low-order coefficients. This result implies the following properties of functions in AC/sup O/: functions in AC/sup O/ have low average sensitivity; they can be approximated well be a real polynomial of low degree; they cannot be pseudorandom function generators and their correlation with any polylog-wide independent probability distribution is small. An O(n/sup polylog(/ /sup sup)/ /sup (n)/)-time algorithm for learning functions in AC/sup O/ is obtained. The algorithm observed the behavior of an AC/sup O/ function on O(n/sup polylog/ /sup (n)/) randomly chosen inputs and derives a good approximation for the Fourier transform of the function. This allows it to predict with high probability the value of the function on other randomly chosen inputs.
Boolean functions, Fourier transform, learnability
N. Nisan, N. Linial and Y. Mansour, "Constant depth circuits, Fourier transform, and learnability," 30th Annual Symposium on Foundations of Computer Science(FOCS), Research Triangle Park, NC, USA, 1989, pp. 574-579.