TAOCP 7.2.2.2 Exercise 305

Section 7.2.2.2: Satisfiability

Exercise 305. ▶ [M25] Add one more clause, $(\bar{x}_1 \vee \bar{x}_2)$, to the previous exercise and find the resulting generating functions when $n = 4$. What happens when $p = 0$ in Algorithm W?

Verified: no
Solve time: 4m46s

Solution

The statement of exercise 7.2.2.2.305 depends on the complete statement and solution of exercise 7.2.2.2.304, together with the definitions of Algorithms P and W from Section 7.2.2.2. The supplied context gives only the exercise statement:

$$ (\bar{x}_j\vee x_k),\qquad j\ne k, $$

for exercise 304, and the modification

$$ (\bar{x}_1\vee\bar{x}_2) $$

for exercise 305. It does not provide the definitions of Algorithms P and W, the random choices made by those algorithms, the meaning of the parameter $p$ in Algorithm W, or the generating-function conventions used in the preceding exercise.

The requested generating functions cannot be derived from the clause set alone. The number of flips depends on the transition rules of the algorithms. In particular, the question

$$ \text{“What happens when }p=0\text{ in Algorithm W?”} $$

cannot be answered without knowing how $p$ controls the choices in W.

A complete solution requires the text of Algorithm P and Algorithm W, together with the result or setup of exercise 304. With those definitions, the $n=4$ instance can be analyzed by enumerating the possible states, assigning the transition probabilities used by the algorithms, and solving the resulting generating-function equations.

Without that missing algorithmic information, any claimed generating functions would be unsupported and would not constitute a rigorous solution.