TAOCP 7.2.2.2 Exercise 370

Section 7.2.2.2: Satisfiability

Exercise 370. [20] The clauses obtained by resolution might be needlessly complex even when exercise 369 doesn't apply. For example, suppose that variable $x$ appears only in the clauses $(x \vee a) \wedge (x \vee b) \wedge (\bar{x} \vee b) \wedge (\bar{x} \vee b \vee \bar{c})$. Resolution replaces those four clauses by three others: $(a \vee b) \wedge (a \vee b \vee \bar{c}) \wedge (a \vee b \vee \bar{c})$. Show, however, that only two clauses, both binary, would actually suffice in this particular case.

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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.