TAOCP 7.2.2.2 Exercise 309
The proposed solution does not answer Exercise 7.
Section 7.2.2.2: Satisfiability
Exercise 309. [20] Exercise 293 explains how to use the reluctant doubling sequence with Algorithm C. Is Algorithm C a Las Vegas algorithm?
Verified: no
Solve time: 11m23s
Correctness
The proposed solution does not answer Exercise 7.2.2.2.305.
The exercise asks for an extension of the previous exercise involving Algorithm W and generating functions. Specifically, it asks to add the clause $(\bar{x}_1 \vee \bar{x}_2)$, compute the resulting generating functions for $n=4$, and analyze the behavior of Algorithm W when $p=0$.
The proposed solution instead proves a statement about unit propagation:
$$ F\wedge L\vdash_1\epsilon \iff (F|L)\vdash_1\epsilon . $$
This is the subject of Exercise 7.2.2.2.276, not Exercise 7.2.2.2.305. The response never discusses the previous exercise, the added clause, generating functions, Algorithm W, or the parameter $p$.
Although the proof given is related to the preceding exercise and is mostly a reasonable argument for that different statement, it does not provide any information needed to solve the assigned exercise.
Gaps and Errors
1. The solution addresses the wrong exercise
Severity: Critical error.
Exercise 7.2.2.2.305 asks for a generating-function calculation after modifying a clause set and for an analysis of Algorithm W. The proposed solution instead begins with:
$$ \text{“The statement is true.”} $$
and proves a logical equivalence about unit propagation. No part of the requested computation is attempted.
2. No generating functions are computed
Severity: Critical error.
The exercise requires finding the resulting generating functions when $n=4$. The proposed solution contains no generating functions, no enumeration of assignments or clauses, and no calculation involving $n=4$.
3. Algorithm W and the case $p=0$ are not analyzed
Severity: Critical error.
The second part of the exercise asks what happens when $p=0$ in Algorithm W. The response does not mention Algorithm W or the meaning of $p$, so it cannot answer this part.
4. The conclusion is unrelated to the problem
Severity: Critical error.
The final conclusion,
$$ F\wedge L\vdash_1\epsilon \iff (F|L)\vdash_1\epsilon , $$
does not imply anything about the requested generating functions or the behavior of Algorithm W. The argument may establish a fact from another exercise, but it cannot serve as a solution here.
Summary
The proposed solution is a solution to Exercise 7.2.2.2.276 rather than Exercise 7.2.2.2.305. It completely omits the required generating-function computation and the analysis of Algorithm W when $p=0$.
VERDICT: FAIL, the solution addresses a different exercise and does not compute the required generating functions or analyze Algorithm W.