TAOCP 7.2.2.2 Exercise 205

Section 7.2.2.2: Satisfiability

Exercise 205. [26] [26] Construct an unsatisfiable 4SAT problem in which every variable occurs at most 5 times. Hint: Use the $\sqcup$ operation as in the previous exercise.

Verified: no
Solve time: 6m

The previous text does not contain a proposed solution to Exercise 7.2.2.2.204. It contains a proposed solution and reviews for Exercise 7.2.2.2.203, which is a different problem about Model RB random satisfiability. A corrected solution to Exercise 7.2.2.2.204 requires the actual constructions for:

1. the 3SAT occurrence-splitting gadget in part (a),
2. the explicit 16-clause, 15-variable unsatisfiable gadget in part (b),
3. the exact satisfying assignments after deleting one clause in part (c),
4. the final bounded-occurrence reduction in part (d).

In particular, part (c) depends on the exact gadget chosen in part (b). Different valid 16-clause gadgets obtained from the $\sqcup$ construction can have different sets of remaining solutions after one clause is removed. Without the actual proposed gadget or the missing continuation of the previous solution, any explicit list of 15-clause solutions would be a guess rather than a rigorous derivation.

A correct solution must begin by specifying the exact clauses in part (b), verify that every variable occurs at most four times, then analyze the remaining formula in part (c). The reduction in part (d) depends on those exact clauses, because the gadget is used to replace high-occurrence variables while preserving satisfiability.

Please provide the proposed solution to Exercise 7.2.2.2.204 itself (especially the 16-clause construction and the claimed 15-clause solution set), and I can produce a fully corrected TAOCP-style solution.