ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

The language ${a^{n} b^{n} : n \geq 0}$ is deterministic context-free and unambiguous

Example

The language ${a^{n} b^{n} : n \geq 0}$ is recognized by the DPDA that pushes one stack symbol for each $a$ , then pops one for each $b$ , rejects any attempt to read an $a$ after a $b$ , and accepts exactly when the endmarker arrives with only the bottom marker left.

It is also generated by the grammar $S \to a S b ∣ ε,$ which is unambiguous.

Facts & Assumptions

Given: The usual stack-matching DPDA and the grammar $S \to a S b ∣ ε$ .

[L1]

By Deterministic pushdown automata and deterministic context-free languages, a language recognized by such a marked-input DPDA is deterministic context-free.

[L2]

By Deterministic context-free languages are unambiguous, every deterministic context-free language has an unambiguous grammar.

Verification

technique · direct

1.1

The displayed machine is deterministic because before the first $b$ it has only the unique push move on $a$ , after the first $b$ it has only the unique pop move on $b$ , and the endmarker is accepted only when the stack has returned to the bottom marker. Therefore it recognizes exactly ${a^{n} b^{n} : n \geq 0}$ .

L1given

1.2

The grammar $S \to a S b ∣ ε$ generates the same language and has exactly one derivation for each word $a^{n} b^{n}$ , because the number of leading $a$ 's uniquely determines how many times the recursive rule must be used before the final $ε$ step.

given

2.1

So this language is deterministic context-free, and [L2] is realized concretely by an unambiguous grammar.

L2step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)
Dexter C. Kozen, Automata and Computability (standard reference, not scraped)

The language {anbn:n≥0}{anbn:n≥0} is deterministic context-free and unambiguous