Abstract
The algorithm in this chapter uses and extends many of the techniques that we have seen so far. Also, it introduces a few more, such as “coding,” “witness lists,” and the “window of opportunity.”
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Notes
- 1.
Our switch statement is similar to the switch statement in the programming language Java.
- 2.
My students voted unanimously for the switch statement here.
- 3.
See the remarks about wrinkles in the Afternotes section.
- 4.
See Chap. 15 for more about delaying tactics.
- 5.
This graph is acyclic, mercifully.
- 6.
I tried to pick the names of these stages mnemonically: r is for r(ealization), t for (inser)t(ion), and u for u(nrealization). Also, I wanted their alphabetical order to match their numerical order, to help keep (14.4) in mind.
- 7.
We don’t benefit from this extra strength, because Fact 5 is used only in the proof of Fact 6, which concerns only nodes in \(\textit{TP}\).
- 8.
This is impossible, by Fact 7. However, we cannot use Fact 7 here, because our proof of Fact 7 uses Fact 6.
- 9.
Thus, the variable \(n_\sigma (k)\) has a final value, whereas \(m_\sigma \) might not (although it does have a finite liminf).
- 10.
The sole purpose of lines 2 and 3 of the unrealization subroutine is to ensure that \(n \not \in B_1\) here.
- 11.
Equation (14.19) is similar to line 4 of Algorithm 14.2, which was used in the proof of Fact 5.
- 12.
Nevertheless, infinitely many odd numbers do find their way into \(B_1\) (see Exercise 5).
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Supowit, K.J. (2023). Witness Lists (Density Theorem). In: Algorithms for Constructing Computably Enumerable Sets. Computer Science Foundations and Applied Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-26904-2_14
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