TAOCP 7.2.2.2 Exercise 270
The proposed solution does not answer Exercise 7.
Section 7.2.2.2: Satisfiability
Exercise 270. [25] [25] (On-the-fly subsumption.) The intermediate clauses that arise in step C7, immediately before resolving with a reason $R_l$, occasionally turn out to be equal to the shorter clause $R_l \setminus l$. In such cases we have an opportunity to strengthen that clause by deleting $l$ from it, thus making it potentially more useful in the future.
a) Construct an example where two clauses can each be subsumed in this way while resolving a single conflict. The subsumed clauses should both contain two literals assigned at the current level of the trail, as well as one literal from a lower level.
b) Show that it's easy to recognize such opportunities, and to strengthen such clauses efficiently, by modifying the steps of answer 263.
Verified: no
Solve time: 13m57s
Correctness
The proposed solution does not answer Exercise 7.2.2.2.266. The exercise asks how a random decision policy in a CDCL SAT solver changes step C6 of the CDCL algorithm. The response instead gives a proof about satisfiability of 7SAT formulas with bounded variable occurrences, which is the topic of Exercise 7.2.2.2.210.
The proposed argument is therefore unrelated to the requested problem. It never discusses CDCL, variable activity, decision heuristics, random choices, or the modification of step C6. Since the question asks for a change to an algorithmic step, the solution must describe how the deterministic maximum-activity choice is replaced or augmented by a randomized choice with probability $p$.
The mathematical argument presented is internally coherent as a solution to a different exercise. The Local Lemma calculation is not the issue; the response fails because it addresses the wrong exercise entirely.
Gaps and Errors
1. The response solves the wrong exercise
Severity: Critical error.
Exercise 7.2.2.2.266 concerns CDCL solver behavior:
- step C6,
- choosing a decision variable,
- maximum activity selection,
- random decisions with probability $p$.
The proposed solution instead proves:
$$ \text{Every 7SAT formula with at most 13 occurrences of each variable is satisfiable.} $$
This is unrelated to the requested task.
2. No analysis of step C6 is provided
Severity: Critical error.
The required answer should explain the modification of C6. The original step C6 chooses a branching variable according to the activity heuristic. A randomized policy would alter C6 so that the solver performs a random choice with probability $p$ and uses the activity heuristic with probability $1-p$.
The proposed solution contains no reference to the CDCL decision procedure and therefore does not establish the requested algorithmic change.
3. The conclusion does not follow from the exercise statement
Severity: Critical error.
The final statement,
$$ \boxed{\text{Every 7SAT formula with at most 13 occurrences of each variable is satisfiable.}} $$
does not answer what the exercise asks. Even if correct, it provides no information about how random decisions affect CDCL step C6.
Summary
The proposed solution is a mathematically reasonable proof of a different TAOCP exercise, but it completely fails to address Exercise 7.2.2.2.266. A correct solution must modify the CDCL step C6 decision rule to incorporate random variable selection with probability $p$.
VERDICT: FAIL, the solution addresses a different exercise and does not analyze the required CDCL step C6 modification.