More on Computing Boolean Functions by Sparse Real Polynomials and Related Types of Threshold Circuits

In this paper we investigate the computational power of threshold{ AND circuits versus threshold{XOR circuits. Starting from the observation that small weight threshold{AND circuits can be simulated by small weight threshold{XOR circuits we pose the question whether a Supported by the grant A1019602 of the Academy of Sciences of the Czech Republic. A major part of the results were obtained while the author was supported by the Humboldt{ Fellowship at the University Dortmund. 1 similar simulation exists for small size unbounded weight circuits. The answer to this question, and this is the main result of this paper, is no. We present a function with small threshold{AND circuits for which all threshold{XOR circuits have exponentially many nodes. This answers a fundamental question on separating subsets of the hypercube by hy-perplanes induced by sparse real polynomials: Is it generally better to choose domain f1; ?1g n or are there functions having more compact polynomials over f0; 1g n ? We prove our result by a new lower bound argument which, we hope, contributes to a better understanding of the computational limitations of small depth threshold circuits. As a further result we show that there are AC 0;3 {functions which do not have small threshold{AND circuits. This completes the picture obtained from a series of results 1]]4]]7]]14] on computing AC 0 {functions by small depth threshold circuits. The result can be generalized to any level of the AC 0 {hierarchy, and, thus, illustrates the power of alternation with respect to unbounded weight threshold operations.