DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

How 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.

Decidable and recognizable languages

Definition

Let $L \subseteq Σ^{*}$ be a language.

$L$ is recognizable when some deterministic Turing machine recognizes $L$ in the sense of Languages recognized and decided by a Turing machine.
$L$ is decidable when some deterministic Turing machine decides $L$ in the sense of Languages recognized and decided by a Turing machine.

After fixing any effective binary encoding of the finite alphabet $Σ$ (and hence of $Σ^{*}$ ) and the evident encoding of ${0, 1}$ , equivalently $L$ is decidable exactly when its characteristic function $χ_{L} : Σ^{*} \to {0, 1}$ is computable with respect to those encodings. Recognizability only asks for effective halting on the positive instances and forbids acceptance on the negative instances.

Depends on

Used by

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)