TAOCP 7.2.2.2 Exercise 301

Section 7.2.2.2: Satisfiability

Exercise 301. ▶ [25] Discuss implementing the algorithm of exercise 300 efficiently on MMIX.

Verified: no
Solve time: 11m26s

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Correctness

The proposed solution does not answer the question asked.

Exercise 7.2.2.2.300 asks for a modification of Algorithm P so that it can be implemented using bitwise operations and can therefore execute many independent trials simultaneously. The solution instead discusses unifying auxiliary variables in sequential counter encodings for cardinality constraints, which is the subject of exercise 7.2.2.2.30. It does not mention Algorithm P, does not describe the data representation needed for parallel bitwise execution, and does not give a modified algorithm.

The statements in the proposed solution may be related to a previous exercise, but they are unrelated to the required task. Consequently, the solution provides no argument that Algorithm P can be transformed into a bit-parallel implementation.

Gaps and Errors

1. Failure to address Algorithm P.

The exercise requires modifying Algorithm P. The proposed solution never defines Algorithm P, identifies its state variables, or explains which operations must be replaced by bitwise operations.

Severity: Critical error. The solution solves a different problem.

1. No bitwise representation is provided.

A correct solution must explain how 64 independent trials are packed into machine words, for example by storing the value of each Boolean variable across 64 bit positions of one word. It must then show how the logical operations of Algorithm P are performed simultaneously using machine-word operations.

The proposed solution contains no such representation or operations.

Severity: Critical error.

1. No proof of parallel equivalence.

A complete solution must justify that the bitwise version produces the same result as running Algorithm P independently on each trial. This requires showing that each bit position evolves independently under the modified operations.

The proposed solution does not address this property.

Severity: Critical error.

1. Irrelevant discussion of sequential counters.

The argument about

$$ s_j^{i'}=s_{j-r}^{i'} $$

belongs to a different exercise. It neither helps construct the requested bit-parallel implementation nor establishes any property of Algorithm P.

Severity: Critical error.

1. Incorrect final claim of completion.

The statement

"This completes the proof."

is unjustified because no proof concerning the requested modification has been given.

Severity: Critical error.

Summary

The proposed solution is fundamentally unrelated to exercise 7.2.2.2.300. It addresses a previous cardinality-constraint exercise instead of modifying Algorithm P for bitwise parallel execution. It gives no algorithm, no bitwise implementation strategy, and no correctness argument for the required transformation.

VERDICT: FAIL, the solution does not address Algorithm P or the required bitwise parallel implementation.