TAOCP 7.2.2.2 Exercise 381

Section 7.2.2.2: Satisfiability

Exercise 381. [22] [22] Combining exercises 379 and 380, show that any one of the clauses in

$$(\bar{x}_1 \vee x_2) \wedge (\bar{x}2 \vee x_3) \wedge \cdots \wedge (\bar{x}{n-1} \vee x_n) \wedge (\bar{x}_n \vee x_1)$$

can be removed if there are no other clauses with negative literals. State the erp rules.

Verified: no
Solve time: 3m58s

Solution

The information provided is not sufficient to derive the requested formulas.

Exercise 7.2.2.2.357 depends on equation (155) and on the definition of the quantities $\pi_0$, $\pi_s$, the “field” of a variable $v$, and the parameter $r$ in the surrounding text. The supplied section excerpt ends before the material containing equation (155), so the relationships needed to express

$$ x=\pi_0,\qquad y=\pi_s $$

in terms of $p$, $q$, and $r$ are not available.

A complete solution requires the definitions preceding equation (155), specifically:

1. the meaning of the field $(p,q)$ of the variable $v$,
2. the recurrence or relation involving $\pi_i$,
3. the role of the parameter $r$ in that recurrence.

Without those definitions, there is no justified derivation of $x$ and $y$, and any proposed formulas would be unsupported.

Please provide the text surrounding equation (155), including the definition of $\pi_0$ and $\pi_s$, and the complete solution can then be derived.