TAOCP 7.1.4 Exercise 108
The solution does not correctly establish the required inequality.
Section 7.1.4: Binary Decision Diagrams
Exercise 108. [HM24] [HM24] Let b(n, s) be the number of n-variable Boolean functions with B(f) ≤s. Prove that (s −3)! b(n, s) ≤(n(s −1)2)s−2 when s ≥3, and explore the ramifications of this inequality when s = ⌊2n/(n + 1/ln 2)⌋. Hint: See the proof of Theorem 7.1.2S.
Verified: no
Solve time: 7m35s
Correctness
The solution does not correctly establish the required inequality. The upper bound on the number of labeled BDD structures is in the right direction, but the multiplicity argument introducing the factor $(s-3)!$ is not justified and is not valid in general. The conclusion of the inequality depends critically on that factor, so the proof is incomplete.
Gaps and Errors
The step claiming that each function with $B(f)=s$ is counted at least $(s-3)!$ times is unsupported and incorrect.
A topological ordering of a DAG underlying a reduced ordered BDD is not arbitrary. The number of admissible linear extensions depends on the partial order induced by the edges, and it is not bounded below by $(s-3)!$ in general. Many permutations of internal nodes do not yield valid labelings consistent with the required ordering constraints, so the argument overcounts the multiplicity in an uncontrolled way.
This is a critical error because the factor $(s-3)!$ is essential to the stated inequality; without a correct justification of this multiplicity, the inequality does not follow.
The earlier counting step that each internal node has at most $n(s-1)^2$ choices is acceptable as an overestimate, but it only bounds labeled structures for a fixed ordering. It does not connect correctly to the number of distinct functions.
The subsequent Stirling-based asymptotic analysis is irrelevant because it relies on an unproven inequality.
Summary
The argument fails at the combinatorial counting step that relates BDD labelings to functions via a factorial multiplicity factor. This gap is fundamental and invalidates the derivation of the stated bound.
VERDICT: FAIL, the claimed $(s-3)!$ multiplicity of valid node orderings is unjustified and incorrect.