Alphabeta Math
LemmaStatement: Literature-sourcedProof: 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.

A DPDA has at most one computation on each input

Statement

Let M be a DPDA. For every marked input w⊣, any two finite computations of M starting from the initial configuration on w⊣ are comparable by extension. Consequently, there is at most one maximal computation on that marked input.

Facts & Assumptions

Given: A DPDA M and an input word w⊣.

[L1]

By Deterministic pushdown automata and deterministic context-free languages, once the current state and top stack symbol are fixed, there is at most one enabled move, and an enabled ε-move excludes all simultaneous input-reading moves.

[L2]

By PDA configurations and one-step moves, a computation is a sequence of one-step moves between configurations.

Proof

technique · direct
1.1

From any configuration of M, the pair consisting of the current state and top stack symbol determines at most one legal next step by [L1].

L1given
2.1

Therefore two computations on the same input and with the same initial configuration must agree on their first move whenever either has a first move. Repeating the same argument after each common prefix shows inductively that they agree step by step for as long as both are defined.

L1L2step 1.1induction
3.1

Thus any two finite computations from the initial configuration are comparable by extension. Their union, when the computation continues indefinitely, or their common terminal extension, when it halts, is the unique maximal computation on w⊣.

L2step 2.1∎

Depends on

Used by

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