Harmonic Analysis of Polynomial Threshold Functions

The analysis of linear threshold Boolean functions has recently attracted the attention of those interested in circuit complexity as well as of those interested in neural networks. Here a generalization of linear threshold functions is defined, namely, polynomial threshold functions, and its relation to the class of linear threshold functions is investigated.A Boolean function is polynomial threshold if it can be represented as a sign function of a polynomial that consists of a polynomial (in the number of variables) number of terms. The main result of this paper is showing that the class of polynomial threshold functions (which is called $PT_1 $) is strictly contained in the class of Boolean functions that can be computed by a depth 2, unbounded fan-in polynomial size circuit of linear threshold gates (which is called $LT_2 $).Harmonic analysis of Boolean functions is used to derive a necessary and sufficient condition for a function to be an S-threshold function for a given set S of monomials. This cond...

[1]  C. K. Chow,et al.  On the characterization of threshold functions , 1961, SWCT.

[2]  Robert O. Winder,et al.  Threshold logic , 1971, IEEE Spectrum.

[3]  Saburo Muroga,et al.  Threshold logic and its applications , 1971 .

[4]  J. Dillon.  Elementary Hadamard Difference Sets , 1974 .

[5]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[6]  David S. Johnson,et al.  The NP-Completeness Column: An Ongoing Guide , 1982, J. Algorithms.

[7]  Pavel Pudlák,et al.  Threshold circuits of bounded depth , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[8]  Roman Smolensky,et al.  Algebraic methods in the theory of lower bounds for Boolean circuit complexity , 1987, STOC.

[9]  Georg Schnitger,et al.  Parallel Computation with Threshold Functions , 1988, J. Comput. Syst. Sci..

[10]  Jehoshua Bruck.  Computing with networks of threshold elements , 1989 .

[11]  Andrew Chi-Chih Yao.  Circuits and local computation , 1989, STOC '89.

[12]  Jehoshua Bruck,et al.  Neural networks, error-correcting codes, and polynomials over the binary n -cube , 1989, IEEE Trans. Inf. Theory.

[13]  J. Reif,et al.  On Threshold Circuits and Polynomial Computation , 1992, SIAM J. Comput..

[14]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .