TAOCP 7.2.2.2 Exercise 82
In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values.
Section 7.2.2.2: Satisfiability
Exercise 82. ▶ [22] (Light speed.) Imagine Life on an infinite plane, with all cells dead at time 0 except in the lower left quadrant. More precisely, suppose $X_t = (x_{tij})$ is defined for all $t \ge 0$ and all integers $-\infty < i, j < +\infty$, and that $x_{0ij} = 0$ whenever $i > 0$ or $j > 0$.
a) Prove that $x_{tij} = 0$ whenever $0 \le t < \max(i, j)$.
b) Furthermore $x_{tij} = 0$ when $0 \le -i \le j$ and $0 \le t < i + 2j$.
c) And $x_{tij} = 0$ for $0 \le t < 2i + j$ and $j \ge 0$ and $j \ge 0$. Hint: If $x_{tij} = 0$ whenever $t < -i$, prove that $x_{tij} = 0$ whenever $t > -i$.
Verified: no
Solve time: 3m42s
Solution
In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write
$$ x_{i,j}=1\iff x_i=j, $$
where $0\le i<p$ and $1\le j\le d$ if the domain has size $d$. The direct encoding already contains the clauses forcing each $x_i$ to have exactly one value. The all-different constraint adds the requirement that two different variables cannot choose the same value.
For two distinct variables $x_i$ and $x_k$, and for any value $j$, the assignment
$$ x_{i,j}=x_{k,j}=1 $$
must be forbidden. The corresponding clause is
$$ \bar{x}{i,j}\vee \bar{x}{k,j}. $$
Taking all pairs of variables and all possible values gives the complete encoding:
$$ \bigwedge_{0\le i<k<p}\ \bigwedge_{1\le j\le d} (\bar{x}{i,j}\vee\bar{x}{k,j}). $$
These clauses are sufficient. Suppose the direct variables represent an assignment satisfying the clauses. If two variables $x_i$ and $x_k$ had the same value $j$, then the direct encoding would require
$$ x_{i,j}=1,\qquad x_{k,j}=1. $$
The clause
$$ \bar{x}{i,j}\vee\bar{x}{k,j} $$
would then be false, contradicting satisfiability. Hence no two variables receive the same value.
They are also necessary. Given any assignment satisfying the all-different constraint, no pair of variables $x_i,x_k$ has the same value. Therefore, for every $j$, at least one of $x_{i,j}$ and $x_{k,j}$ is false, so every clause
$$ \bar{x}{i,j}\vee\bar{x}{k,j} $$
is satisfied.
Thus the all-different constraint in the direct encoding is enforced by adding one binary clause for every pair of variables and every possible value:
$$ \boxed{\bar{x}{i,j}\vee\bar{x}{k,j}\qquad (0\le i<k<p,\ 1\le j\le d).} $$
This completes the proof. ∎