DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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 PDA fragment variable $[p, Z, q]$

Definition

Let $M = (Q, Σ, Γ, δ, q_{0}, Z_{0}, F)$ be a PDA that accepts by empty stack.

For states $p, q \in Q$ and a stack symbol $Z \in Γ$ , the fragment variable $[p, Z, q]$ denotes the language of words $x \in Σ^{*}$ such that for every stack suffix $α \in Γ^{*}$ , $(p, x, Z α) ⊢_{M}^{*} (q, ε, α) .$

Thus $[p, Z, q]$ describes exactly the words that remove the top symbol $Z$ , change state from $p$ to $q$ , and leave the rest of the stack untouched.

Depends on

Used by

PDA fragment variables compose Lemma
Every PDA has an equivalent context-free grammar Theorem

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

Jean Gallier and Jocelyn Quaintance, Introduction to the Theory of Computation: Some Notes for CIS511 (standard reference, not scraped)

The PDA fragment variable [p,Z,q][p,Z,q]