TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 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.

A language is decidable if and only if it and its complement are recognizable

Statement

For every language $L \subseteq Σ^{*}$ , the following are equivalent:

$L$ is decidable.
Both $L$ and its complement $Σ^{*} ∖ L$ are recognizable.

Facts & Assumptions

Given: A language $L \subseteq Σ^{*}$ .

[L1]

By Decidable and recognizable languages, a decider halts on every input and gives the correct yes-or-no answer, while a recognizer only needs to accept the members.

[L2]

By Boolean operations on languages over a fixed alphabet, the complement of $L$ is the language $L^{c} = Σ^{*} ∖ L$ over the same alphabet.

Proof

technique · direct

1.1

If $L$ is decidable, then a decider for $L$ is already a recognizer for $L$ , and swapping its accepting and rejecting halting states gives a recognizer for $L^{c}$ . Thus item 1 implies item 2 by [L1] and [L2].

L1L2given

1.2

Assume $M$ recognizes $L$ and $N$ recognizes $L^{c}$ . On input $w$ , simulate one step of $M$ , then one step of $N$ , then two steps of each, and so on. If $M$ accepts at some stage, accept; if $N$ accepts at some stage, reject. Because every word lies in exactly one of $L$ and $L^{c}$ , exactly one machine must eventually accept, so this interleaving always halts with the correct answer. Therefore $L$ is decidable by [L1].

L1L2construct

2.1

Steps 1.1 and 1.2 prove both directions.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

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 17: More undecidable languages; reductions (standard reference, not scraped)