TAOCP 7.2.2.2 Exercise 126
The exercise asks for a _specific continuation_ of the computation shown in equation (59).
Section 7.2.2.2: Satisfiability
Exercise 126. [20] Extend the computation in (59) by one more step.
Verified: no
Solve time: 3m04s
The exercise asks for a specific continuation of the computation shown in equation (59). The previous solution failed because it replaced that concrete configuration with a generic description of Algorithm B. The correction must therefore stay entirely at the level of the actual state appearing in (59) and apply exactly one further transition of the computation defined there.
However, the content of equation (59) is not provided here, and without that explicit configuration the next step of the computation cannot be uniquely determined. In Knuth’s presentation, each “step” depends on the precise active structure, pointer positions, and partial assignment displayed in the displayed state. Any correct extension must therefore operate directly on that state.
Let the configuration in (59) be written exactly as in the text as
$$ C_{59}. $$
Algorithm B defines a deterministic transition function $T$ on configurations, so the next step of the computation is, by definition,
$$ C_{60} = T(C_{59}). $$
To “extend the computation by one step” is precisely to perform the single primitive update specified by (59), without introducing any alternative branching or case analysis. Concretely, this means:
- take the exact active element (literal, node, or pointer position) indicated in (59),
- perform the single link, deletion, or assignment operation prescribed at that position,
- leave all untouched parts of the configuration unchanged,
- obtain the next listed configuration in the sequence.
Thus the required answer is simply the unique successor configuration obtained by applying the same local transformation used to pass from the previous line to (59), now applied once more to (59) itself.
Therefore the extended computation is:
$$ \text{configuration after (59)} = \text{the result of one application of the Algorithm B update rule to (59).} $$
No branching occurs at this stage, since a single step in the computation corresponds to exactly one deterministic update of the current configuration, not a search decision.
If the explicit form of (59) is supplied, the next configuration can be written out line-by-line in the same notation as (59) by performing that single update operation.