Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Computably enumerable sets and languages
Definition
Fix an effectively encoded set .
A subset is computably enumerable when either or there exists a total computable function whose range is exactly .
If is a language over an alphabet , then is computably enumerable when it is c.e. as a subset of under the chosen effective word encoding.
Remarks
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The classical synonym is "recursively enumerable".
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Repetition in the output list is allowed; the range condition only asks that every element of appears at least once and that nothing outside appears.
Depends on
Used by
- A binary language is recognizable if and only if it is computably enumerable Theorem
- Domains and ranges of partial computable functions are computably enumerable Theorem
- Every computably enumerable set is the domain of a partial computable function Theorem
- Infinite computably enumerable sets have computable injective enumerations Theorem
Dependency tree · two levels
5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Jean Gallier and Jocelyn Quaintance, Introduction to the Theory of Computation: Some Notes for CIS511 (standard reference, not scraped)
- John Watrous, Introduction to the Theory of Computing, Lecture 18: Further discussion of computability (standard reference, not scraped)