TAOCP 7.2.2.2 Exercise 481
In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values.
Section 7.2.2.2: Satisfiability
Exercise 481. ▶ [28] [28] An ordered bit pair $xy$ can be encoded by another ordered bit pair $[xy] = (x \bar y)(y)$ without loss of information, because $[xy] = uv$ implies $[uv] = xy$.
- a) Find an optimum circuit that computes $([z z']^\dagger)_2 = x_1 + x_2 + 1$. Find an optimum circuit that computes $((u \oplus v) + v)$, and note that $v[00] = 0$, $v[01] = 2$, $v[1*] = 1$. Find an optimum circuit that, given $x_1, \ldots, x_3$, computes $z_1 z_2 z_3$ such that we have $v[z_1 z_2] + v[z_2 z_3] + \cdots + v[z_3 z_1] + z_1 = x_1 + \cdots + x_3$.
- c) Use that circuit to prove by induction that the "sideways sum" $(z_{1[n]}, \ldots, z_{1,[0]})_2 = x_1 + x_2 + \cdots + x_n$ can always be computed with fewer than $4n$ steps.
Verified: no
Solve time: 3m44s
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. ∎