TAOCP 7.2.2.2 Exercise 432

The needed solution depends on the precise function from 7.

taocpmathematicsalgorithmsvolume-4hard

Section 7.2.2.2: Satisfiability

Exercise 432. [34] Exercise 404 gives clauses for the relation $|x - y| \ge a$. Are they forcing?

Verified: no
Solve time: 3m45s

The needed solution depends on the precise function from 7.1.1–116, because the proof must establish two separate facts:

  1. The function has $\Omega(3^n/n^2)$ prime implicants, so that every forcing representation without auxiliary variables must contain that many clauses.
  2. The same function has a polynomial-size forcing representation with auxiliary variables, usually obtained by introducing variables for intermediate circuit values and applying the forcing-clause construction.

If you provide the statement of Exercise 7.1.1–116 (or the relevant paragraph defining its function), I can write the complete TAOCP-style solution with the required lower-bound proof and auxiliary-variable construction.