TAOCP 7.2.2.2 Exercise 373

Section 7.2.2.2: Satisfiability

Exercise 373. [35] The answer to exercise 228 defines $2m + \sum_{j=1}^{m}(j-1)^2 \approx m^2/3$ clauses in $m^2$ variables that suffice to refute the anti-maximal-element axioms of (99)–(101). Algorithm L needs exponential time to show this; so does Davis-Putnam-Robinson R; and experiments show that they are bad news for Algorithm C too. Show, however, that preprocessing with transformations 1–4 will rapidly prove them unsatisfiable.

Verified: no
Solve time: 3m46s

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.