DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

Language concatenation, powers, and Kleene star

Definition

Fix an alphabet $Σ$ , and let $L, K \subseteq Σ^{*}$ be languages over $Σ$ (Languages over an alphabet).

Their concatenation is $L K : = {u v : u \in L and v \in K},$ where $u v$ is word concatenation as in Computation alphabets, words, the empty word, and $Σ^{*}$ .

Define the powers of $L$ by $L^{0} : = {ε}$ and, for each $n \geq 1$ , let $L^{n}$ be the set of words of the form $u_{1} u_{2} \dots u_{n}$ with each $u_{i} \in L$ .

The Kleene star of $L$ is $L^{*} : = ⋃_{n \in N} L^{n} .$

Remarks

The word $ε$ belongs to $L^{*}$ for every language $L$ , even when $L = \emptyset$ , because it already lies in $L^{0}$ .
The positive closure $L^{+} : = ⋃_{n \geq 1} L^{n}$ satisfies $L^{*} = L^{+} \cup {ε}$ .

Depends on

Used by

Dependency tree · two levels

12 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)
Michael Sipser, MIT 18.404J Theory of Computation, Lecture 1 (standard reference, not scraped)