ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-12

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.

Running the emptiness and finiteness tests on small grammars

Example

Use the three grammars

$G_{\emptyset} : S \to A B, A \to a, B \to b B,$ $G_{\infty} : S \to A, A \to a A b ∣ a b,$ $G_{f i n} : S \to a A ∣ b, A \to c ∣ d .$

Facts & Assumptions

Given: The three displayed grammars.

[L1]

By CFG emptiness and finiteness are decidable, the emptiness and finiteness questions for CFG's are decidable by generating-variable and dependency-graph tests.

Verification

technique · direct

1.1L1given

In $G_{\emptyset}$ , the variable $B$ never derives a terminal word, so the start symbol $S$ is not generating. The emptiness test from [L1] therefore reports $L (G_{\emptyset}) = \emptyset$ .

1.2L1given

In $G_{f i n}$ , the dependency graph is acyclic and the grammar generates exactly the three words $a c$ , $a d$ , and $b$ . So the finiteness test from [L1] reports "finite" on this grammar.

2.1L1step 1.1

In $G_{\infty}$ , the variable $A$ lies on a productive cycle through the rule $A \to a A b$ , so the dependency-graph test from [L1] reports infinitude. Indeed $L (G_{\infty}) = {a^{n} b^{n} : n \geq 1}$ is infinite.

3.1step 1.1step 1.2step 2.1∎

These three tiny inputs realize the empty, infinite, and finite outcomes of the theorem.

Depends on

CFG emptiness and finiteness are decidable

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

H. Conrad Cunningham, CSci 311, Models of Computation, Chapter 8 (standard reference, not scraped)