Symmetric Functions Capture General Functions

Abstract

We show that the set of all functions is equivalent to the set of all symmetric functions up to deterministic time complexity. In particular, for any function f , there is an equivalent symmetric function f sym such that f can be computed form f sym and vice-versa (modulo an extra deterministic linear time computation). For f over finite fields, f sym is (necessarily) over an extension field. This reduction is optimal in size of the extension field. For polynomial functions, the degree of f sym is not optimal. We present another reduction that has optimal degree "blowup" but is worse in the other parameters.