DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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.

Languages over an alphabet

Definition

Let $Σ$ be an alphabet. A language over $Σ$ is a subset $L \subseteq Σ^{*}$ , where $Σ^{*}$ is the set of finite words of Computation alphabets, words, the empty word, and $Σ^{*}$ and $\subseteq$ is the subset relation of Subset $x \subseteq y$ , proper subset $x ⊊ y$ , and the separation notation ${z \in x : φ (z)}$ .

Thus a language is a set of words over a fixed alphabet, not a set of alphabet symbols.

Remarks

The ambient alphabet matters. The same set of written strings can be viewed as a language over different alphabets only when it is contained in each corresponding $Σ^{*}$ .
The two extreme languages over $Σ$ are $\emptyset$ and $Σ^{*}$ .

Depends on

Used by

Acceptance of a word by a DFA and the recognized language Definition
Boolean operations on languages over a fixed alphabet Definition
Every finite language is regular Proposition

Dependency tree · two levels

8 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 E. Savage, Models of Computation: Exploring the Power of Computing (standard reference, not scraped)
Michael Sipser, MIT 18.404J Theory of Computation, Lecture 1 (standard reference, not scraped)