PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-12

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If a PDA has at most one accepting computation on each word, the standard PDA-to-CFG construction is unambiguous

Statement

Let $M$ be a PDA such that every word has at most one accepting computation in $M$ . Then the fragment-variable grammar $G_{M}$ from the standard PDA-to-CFG construction is unambiguous.

Facts & Assumptions

Given: A PDA $M$ with at most one accepting computation on each word.

[L1]

By Ambiguous grammars, unambiguous grammars, and inherently ambiguous languages, a grammar is unambiguous when each generated word has exactly one leftmost derivation.

[L2]

By Every PDA has an equivalent context-free grammar, the standard grammar $G_{M}$ is built from fragment variables and generates exactly the language of $M$ .

Proof

technique · direct

1.1

In the grammar $G_{M}$ , every production choice records one concrete local decision in an accepting computation of $M$ : either one normalized PDA move, or one split state used to divide a two-symbol fragment into two smaller fragments. Thus a leftmost derivation of a word in $G_{M}$ determines an accepting computation of $M$ on that word.

L2given

2.1

Conversely, the accepting computation of a word determines all of those local choices uniquely, because the first moment when each top stack symbol disappears fixes the fragment split and therefore fixes the next production in the fragment-variable grammar. So one accepting computation determines exactly one leftmost derivation in $G_{M}$ .

L2step 1.1

3.1

If some generated word had two different leftmost derivations in $G_{M}$ , steps 1.1 and 2.1 would yield two different accepting computations of $M$ on that word, contradicting the hypothesis. Therefore [L1] shows that $G_{M}$ is unambiguous.

L1step 1.1step 2.1∎

Depends on

Used by

Deterministic context-free languages are unambiguous 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)