TAOCP 7.2.2.2 Exercise 480

In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values.

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Section 7.2.2.2: Satisfiability

Exercise 480. [25] [25] Suppose the values 0, 1, 2 are encoded by the two-bit codes $x_1 x_r = 00, 01$, and $1*$, respectively, where 10 and 11 both represent 2. (See Eq. 7.1.3–(120).)

  • a) Find an optimum circuit for mod 3 addition: $z_{1*} = (x_{1r} + y_{1r}) \bmod 3$.
  • b) Find an optimum circuit that computes $x_{1r} + y_{1r} + z_{1r} \pmod 3$ mod 3.
  • c) Conclude that $[x_1 + \cdots + x_n \equiv a \pmod{3}]$ can be computed in $< 3n$ steps.

Verified: no
Solve time: 3m48s

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. ∎