DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

PDA configurations and one-step moves

Definition

Fix a PDA $M = (Q, Σ, Γ, δ, q_{0}, Z_{0}, F)$ .

A configuration of $M$ is a triple $(q, w, α) \in Q \times Σ^{*} \times Γ^{*}$ where $q$ is the current state, $w$ is the unread input, and $α$ is the current stack content. We read the leftmost symbol of $α$ as the top of the stack.

There is a one-step move $(q, a w, Z γ) ⊢_{M} (p, w, β γ)$ when $(p, β) \in δ (q, a, Z)$ , and there is a one-step move $(q, w, Z γ) ⊢_{M} (p, w, β γ)$ when $(p, β) \in δ (q, ε, Z)$ .

Write $⊢_{M}^{*}$ for the reflexive-transitive closure of $⊢_{M}$ .

Depends on

Nondeterministic pushdown automata

Used by

Acceptance by empty stack for a PDA Definition
Acceptance by final state for a PDA Definition
The PDA fragment variable [p,Z,q] Definition
A DPDA has at most one computation on each input Lemma

Dependency tree · two levels

4 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)