Category:Symmetric function
Contents
Definition
A symmetric Boolean function is a Boolean function whose value does not depend on the permutation of its input bits, i.e., it depends only on the number of ones in the input.
From the definition it follows that there are 2n+1 symmetric n-ary Boolean functions. It implies that instead of the truth table, traditionally used to represent Boolean functions, one may use a more compact representation for an n-variable symmetric Boolean function: the (n + 1)-vector, whose i-th entry (i = 0, ..., n) is the value of the function on an input vector with i ones.
Properties
- A restricted Boltzman machine with [math]n[/math] input units and [math]n^2+1[/math] hidden units can represent any symmetric Boolean function. [1]
Examples
- Threshold functions: their value is 1 on input vectors with k or more ones for a fixed k
- Exact-value functions: their value is 1 on input vectors with k ones for a fixed k
- Counting functions : their value is 1 on input vectors with the number of ones congruent to k mod m for fixed k, m
- Parity functions: their value is 1 if the input vector has odd number of ones.
References
- ↑ Theorem 7 in James Martens, Arkadev Chattopadhyay, Toniann Pitassi, Richard Zemel, On the Representational Efficiency of Restricted Boltzmann Machines, NIPS 2013
Pages in category "Symmetric function"
The following 4 pages are in this category, out of 4 total.
