TAOCP 7.2.2.2 Exercise 314
The proposed solution does not answer Exercise 7.
Section 7.2.2.2: Satisfiability
Exercise 314. [36] (H. H. Hoos, 1998.) If the given clauses are satisfiable, and if $p > 0$, can there be an initial $x$ for which Algorithm W always loops forever?
Verified: no
Solve time: 11m23s
Correctness
The proposed solution does not answer Exercise 7.2.2.2.313.
The exercise asks for a proof about Algorithm W. Specifically, it asks to show two facts about a cost-free literal flip in clause $C_j$:
- Flipping such a literal always decreases the number of unsatisfied clauses, $r$.
- Despite this decrease, the flip can increase the distance from the current assignment $x$ to the nearest satisfying assignment.
The proposed solution instead solves Exercise 7.2.2.2.31, which concerns computing $F_t(r)$ with the help of a SAT solver. It discusses arithmetic progressions of ones, cardinality constraints, SAT encodings, and the example $F_3(12)=30$. None of these topics concern Algorithm W, cost-free literals, clause costs, or distances to solutions.
The argument given is unrelated to the requested proof. It does not define the quantities used in Exercise 7.2.2.2.313, does not analyze a flip in $C_j$, and does not establish either of the two required claims.
Gaps and Errors
1. The solution addresses the wrong exercise
Severity: Critical error.
The proposed solution begins by discussing:
$$ F_t(r) $$
and constructs SAT formulas for avoiding equally spaced ones. This is the topic of Exercise 7.2.2.2.31, not Exercise 7.2.2.2.313.
The required solution must instead analyze the local search behavior of Algorithm W and the effect of flipping a cost-free literal.
2. No proof that $r$ decreases is provided
Severity: Critical error.
Exercise 7.2.2.2.313 asks why flipping a cost-free literal always reduces the number of unsatisfied clauses. The proposed solution never defines $r$ in the context of Algorithm W and gives no argument about satisfied or unsatisfied clauses before and after a flip.
A correct solution must analyze the clause $C_j$ containing the cost-free literal and show that the flip satisfies $C_j$ while no other clause can become unsatisfied under the cost-free condition.
3. No discussion of distance from the nearest solution is provided
Severity: Critical error.
The exercise asks for an example or argument showing that a flip can move the assignment farther away from every satisfying assignment. The proposed solution contains no notion of Hamming distance, nearest solution, or movement in the assignment space.
4. The SAT construction is irrelevant
Severity: Critical error.
Statements such as
$$ S(t,r,n)\text{ is satisfiable}\iff F_t(r)\leq n $$
may be appropriate in another context, but they do not contribute to the proof required here. The final conclusion about SAT solving does not imply anything about Algorithm W.
Summary
The proposed solution is a coherent discussion of a different SAT exercise, but it completely fails to address the behavior of Algorithm W required in Exercise 7.2.2.2.313.
VERDICT: FAIL, the solution addresses a different exercise and does not prove the required claims about cost-free literal flips in Algorithm W.