A Discrete Approximation and Communication Complexity Approach to the Superposition Problem
The superposition (or composition) problem is a problem of representation of a function f by a superposition of “simpler” (in a different meanings) set Ω of functions. In terms of circuits theory this means a possibility of computing f by a finite circuit with 1 fan-out gates Ω of functions.
Using a discrete approximation and communication approach to this problem we present an explicit continuous function f from Deny class, that can not be represented by a superposition of a lower degree functions of the same class on the first level of the superposition and arbitrary Lipshitz functions on the rest levels. The construction of the function f is based on particular Pointer function g (which belongs to the uniform AC0) with linear one-way communication complexity.
KeywordsBoolean Function Communication Complexity Discrete Approximation Discrete Function Arbitrary Continuous Function
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