## Sunday, February 21, 2016

### xor as linear inequalities

I came across a question about using an xor condition in a MIP model. Here is a summary of the answer.

Expressing $$z=x\cdot y$$ (or $$z=x\text{ and }y$$ ) where $$x,y,z \in \{0,1\}$$ as a set of linear constraints is well-known:

and:     \begin{align} &z \le x \\ &z \le y \\ &z \ge x+y-1 \end{align} $$\>\>\>\Longleftrightarrow\>\>\>$$
 $$z$$ $$x$$ $$y$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$1$$ $$0$$ $$1$$ $$0$$ $$1$$ $$1$$ $$1$$
The relation $$z=x\text{ xor }y$$ is slightly more complicated. The xor (exclusive-or) condition can also be written as $$z = x<>y$$, i.e. $$z$$ is 1 if $$x$$ and $$y$$ are different (and $$z$$ is zero if they are the same). This we can write as the following set of linear inequalities:
xor:     \begin{align} & z \le x+y\\ & z \ge x-y \\ & z \ge y-x \\ & z \le 2 -x - y \end{align} $$\>\>\>\Longleftrightarrow\>\>\>$$
 $$z$$ $$x$$ $$y$$ $$0$$ $$0$$ $$0$$ $$1$$ $$0$$ $$1$$ $$1$$ $$1$$ $$0$$ $$0$$ $$1$$ $$1$$
To be complete $$z=x\text{ or }y$$ can be written as:
or:     \begin{align} & z \le x+y\\ & z \ge x \\ & z \ge y \end{align} $$\>\>\>\Longleftrightarrow\>\>\>$$
 $$z$$ $$x$$ $$y$$ $$0$$ $$0$$ $$0$$ $$1$$ $$0$$ $$1$$ $$1$$ $$1$$ $$0$$ $$1$$ $$1$$ $$1$$
In all of this we assumed $$x,y$$ and $$z$$ are binary variables.

1. Thanks! You help me a lot!

2. Dear colleagues,
recently we understood how to represent any boolean function f(x) as system of linear inequalities in the following sense.

Let B^n={(x_1,x_2, ..., x_n): x_k={0,1}} be is a set of n-dimensional unit cube's vertices, i.e. the set of all 0-1 arguments of some boolean function f:B^n -> {0,1}. Then there exist a system of a number of linear inequalities of n+1 variables (x,y):
S(x,y)={a_i + l_i(x) + b_i*y >= 0: i={1:m}} such that for any x\in B^n, f(x)=y if and only if S(x,y) holds.
It is important that we do not need to assume that y is a discrete, 0-1, variable! For any 0-1 vector x, corresponding y value will be either 0, or 1.

And I can explicitly produce the system S(x,y) for any function f(x) given either by formula or in tabular form.
Can any body tell me any reference on this subject?
I can not believe that it a "fresh result" in math. programming ... But my rather intensive search over literature did not bring any similar assertion...

3. what about detecting equality or not (xor) between 2 consecutive integer variables? not only binary. thanks