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

Delta₁⁰ sets are exactly the decidable sets Corollary
A recognizer for 0 that loops on every nonmember Counterexample
Computable many-one reductions between languages Definition
Computably enumerable hardness and completeness under computable many-one reductions Definition
Turing reductions via oracle deciders Definition
FALSE: every countable language is decidable False statement
FALSE: recognizable means total halting False statement
A hypothetical decider for A_TM yields a diagonal self-application machine Lemma
Well-formed deterministic one-tape Turing machine descriptions form a decidable language Proposition
A binary language is recognizable if and only if it is computably enumerable Theorem
A language is decidable if and only if it and its complement are recognizable Theorem
Computable many-one reductions transfer decidability and recognizability backward Theorem
Decidable languages are closed under the Boolean operations Theorem
Halting on every input is not recognizable Theorem
Language equivalence of Turing machines is undecidable Theorem
Recognizable languages are closed under union and intersection Theorem
Sigma₁⁰ sets are exactly the computably enumerable sets Theorem
The halting problem is recognizable and undecidable Theorem
The Turing-machine acceptance problem is recognizable Theorem
The Turing-machine acceptance problem is undecidable Theorem
There exists a language that is not recognizable 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)