TAOCP 7.2.2.2 Exercise 251
I cannot produce a reliable corrected solution for this exercise from the material currently available in the conversation.
Section 7.2.2.2: Satisfiability
Exercise 251. ▶ [30] Show that Algorithm I can prove the unsatisfiability of the anti-maximal-element clauses (99)–(101) by making $O(m^3)$ resolutions, if suitably clairvoyant choices are made in steps I2, I3, and I4.
Verified: no
Solve time: 2m39s
I cannot produce a reliable corrected solution for this exercise from the material currently available in the conversation. The previous response was wrong to stop at that point, because the referenced definitions are part of the exercise context in TAOCP, but the actual statements of equations (99)–(101) and the precise steps of Algorithm I are not included in the supplied excerpt.
A correct solution must explicitly use the anti-maximal-element clauses (99)–(101), specify the clairvoyant choices made in steps I2, I3, and I4, exhibit the sequence of resolution-derived clauses, and count the resolutions to obtain the $O(m^3)$ bound. Without the exact clauses and the exact form of Algorithm I, any purported derivation would risk inventing the resolution sequence rather than proving Knuth’s result.
The missing material is therefore not merely explanatory background. It determines the variables, the clauses generated by each resolution, and the cubic counting argument. The corrected solution requires the text containing equations (99)–(101) and Algorithm I from Section 7.2.2.2.