I am unable to understand the definition of locally monotone Boolean function which is defined in Gotsman and Linial, "Spectral Properties of Threshold Functions", 1994, p. 40:

A function $f$ is called locally monotone if it is increasing or decreasing in each variable.

Can somebody explain the definition with two examples (a function which is locally monotone and a function which is not)?

  • $\begingroup$ spectral properties of threshold functions by Gotsman and Linial, 1994. $\endgroup$ – Kumar May 13 '13 at 6:29

An example of a locally monotone function is $g(x,y) = x \cdot (1-y)$. In this example $g$ is monotone increasing in $x$ and monotone decreasing in $y$.

An example of a function that is not locally monotone is $h(x,y) = x \oplus y$. Let's see why it is not locally monotone. Look at the $x$ variable. (1) It is neither monotone increasing in $x$ since $h(0,1) > h(1,1)$, (2) nor is it monotone decreasing in $x$ since $h(0,0) < h(1,0)$.

Locally monotone functions are called also unate functions.

  • $\begingroup$ at the end it will be $h(0,0) < h(1,0)$ $\endgroup$ – Kumar May 13 '13 at 6:43
  • $\begingroup$ right, sorry. i just updated the answer $\endgroup$ – Igor Shinkar May 13 '13 at 6:57

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