TAOCP 7.2.2.2 Exercise 253

Exercise 253. ▶ [18] What are the next two clauses learned if decision '5' follows next after (114)?

Verified: no
Solve time: 6m50s

Solution

The data supplied are not sufficient to derive the two learned clauses. The exercise refers to the continuation of the worked clause-learning example in Section 7.2.2.2 after equation (114), but the clauses and the trail information used after that point are not included in the material provided here.

The proposed derivation cannot be repaired by merely changing the final clauses, because the missing information is exactly the information needed to perform the calculation. A clause-learning step requires the current clause database, the current assignment, the reason clause attached to each propagated literal, and the resolution rule used in the example. None of these is determined by the statement of Exercise 253 alone.

In particular, after the decision literal $5$, the next learned clause is obtained by the following fixed procedure from the example:

$$ \text{decision }5 \longrightarrow \text{unit propagation} \longrightarrow \text{conflict clause} \longrightarrow \text{resolution sequence} \longrightarrow \text{learned clause}. $$

To justify the first learned clause, every propagated literal must be derived from a clause that is present in the database at that stage. If a clause

$$ C\vee l $$

is the reason for a propagated literal $l$, then all literals in $C$ must already be assigned false when $l$ is placed on the trail. The proposed solution does not establish these premises, so its propagation sequence cannot be accepted.

The same issue occurs in the conflict analysis. A resolution step is valid only when the two parent clauses contain complementary occurrences of the pivot literal. For example, a claimed step

$$ C\vdash D $$

must be accompanied by two clauses $C_1$ and $C_2$ such that $D$ is exactly their resolvent. The proposed sequence contains resolution steps whose parent clauses are not specified, so the resulting clauses are not derived.

The second learned clause requires the complete continuation after the first backjump. Its derivation depends on the new learned clause, the assignments forced by it, and the next conflict encountered by the solver. Those clauses and assignments are likewise absent from the supplied text.

Therefore the previous solution is not a valid solution, but the missing information prevents a rigorous replacement derivation. The complete clause-learning example containing equation (114), the clause database at that point, and the subsequent propagation rules are required before the two learned clauses can be computed.

This completes the proof of insufficiency of the supplied data. ∎