TAOCP 7.2.2.2 Exercise 379
The proposed solution does not address Exercise 7.
Section 7.2.2.2: Satisfiability
Exercise 379. ▶ [20] (Blocked self-subsumption.) Consider the clause $(a \vee b \vee c \vee d)$, and suppose that every clause containing $\bar{a}$ but not $b$ nor $\bar{c}$ also contains $d$. Show that we can then shorten the clause to $(b \vee c \vee d)$ without affecting satisfiability. Is an erp rule needed?
Verified: no
Solve time: 10m33s
Correctness
The proposed solution does not address Exercise 7.2.2.2.379. The exercise concerns blocked self-subsumption for a single clause $(a \vee b \vee c \vee d)$ and asks for a satisfiability-preserving shortening to $(b \vee c \vee d)$ under a specific condition on clauses containing $\bar{a}$. It also asks whether an erp rule is necessary.
The submitted solution instead discusses the anti-maximal-element axioms from Exercise 7.2.2.2.373. It introduces unrelated notation $x_{ij}$, unrelated preprocessing transformations, and an unrelated contradiction proof. None of this proves the claimed clause shortening, and it does not answer the question about erp.
For Exercise 379, the required argument is a self-subsumption argument. Let
$$ C=(a\vee b\vee c\vee d). $$
The shortened clause is
$$ C'=(b\vee c\vee d). $$
To prove satisfiability is preserved, one must show that any satisfying assignment of the original formula together with $C'$ also satisfies every occurrence of $C$. If $C'$ is false, then $b,c,d$ are all false. Under this assignment, $C$ can only be satisfied by $a=1$. Therefore the only possible way that replacing $C$ by $C'$ could change satisfiability is if there exists another clause that forces $a=0$ while remaining satisfiable with $b=c=d=0$.
The hypothesis rules this out. Any clause containing $\bar{a}$ but neither $b$ nor $\bar{c}$ must contain $d$. When $b=c=d=0$, such a clause contains $d$ as a false literal, and also contains neither $b$ nor $\bar{c}$, so its only possible satisfaction from the listed literals would have to come from some other literal. The presence of $d$ blocks the problematic self-subsumption case: any clause that could contradict $a$ also contains $d$, so adding the stronger requirement $b\vee c\vee d$ cannot remove all satisfying assignments.
The proposed solution contains none of this reasoning.
Gaps and Errors
- The solution addresses the wrong exercise. Critical error.
The entire argument concerns Exercise 373 rather than Exercise 379. It never mentions the clause $(a\vee b\vee c\vee d)$, the condition on clauses containing $\bar{a}$, the shortened clause $(b\vee c\vee d)$, or erp. The answer therefore does not solve the problem asked.
- No proof of satisfiability preservation is given. Critical error.
The main requirement is to prove that replacing $(a\vee b\vee c\vee d)$ by $(b\vee c\vee d)$ preserves satisfiability. The submitted solution gives no argument about assignments, clauses containing $\bar{a}$, or equivalence of the two clause sets.
- The erp question is unanswered. Critical error.
The exercise explicitly asks whether an erp rule is needed. The proposed solution gives no discussion of erp. The correct answer requires explaining that the shortening is an instance of blocked self-subsumption and does not require erp.
- The notation and claims are unrelated to the exercise. Critical error.
The introduction of variables $x_{ij}$, anti-maximal-element clauses, and preprocessing transformations is unrelated material. Even if that proof were correct for Exercise 373, it cannot contribute to a solution of Exercise 379.
Summary
The proposed solution is a response to a different TAOCP exercise and fails to address any of the required claims in Exercise 7.2.2.2.379. The central clause-shortening argument and the erp discussion are entirely missing.
VERDICT: FAIL, the solution does not address Exercise 7.2.2.2.379 and provides no proof of the required blocked self-subsumption rule.