TAOCP 7.2.2.2 Exercise 475
In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values.
Section 7.2.2.2: Satisfiability
Exercise 475. [**] [$M22$] Entitled, a Boolean function is called asymmetric if the identity is its only symmetry; it is totally asymmetric if it is asymmetric and has no antisymmetries.
a) If $f$ is totally asymmetric, how many functions are equivalent to $f$ under the operations of permuting variables, complementing variables, and/or complementing the function? b) According to (a) and 7.1.1–(95), the function $(x \vee y) \wedge (x \oplus z)$ is not totally asymmetric. What is its nontrivial symmetry? c) Prove that if $f$ is not asymmetric, it has an automorphism of prime order $p$. d) Show that $(u\overline{v}wx\overline{y})$ has a symmetry of the form $(uvwxy)(\bar{u}\bar{v}\bar{w}\bar{x}\bar{y})$. e) Make a similar statement if $f$ has a symmetry of the form $(uvwxy)(\bar{u}\bar{v}\bar{w}\bar{x}\bar{y})$. f) Conclude that, if $n \le 5$, the Boolean function $f(x_1, \ldots, x_n)$ is totally asymmetric if and only if no signed involution is a symmetry or antisymmetry of $f$. g) However, exhibit a counterexample to that statement when $n = 6$.
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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. ∎