Alphabeta MathHow 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.
Alternating polynomial time and the class AP
Definition
An alternating Turing machine is a nondeterministic Turing machine whose nonhalting states are partitioned into existential and universal states.
Acceptance is defined recursively on the computation tree:
- an accepting halting configuration accepts;
- a rejecting halting configuration rejects;
- an existential configuration accepts when at least one legal successor accepts;
- a universal configuration accepts when every legal successor accepts.
The class AP is the set of languages decided by alternating Turing machines whose every computation branch halts within polynomial time.
Remarks
- Deterministic polynomial time is the special case with only one successor at every step.
- Alternation differs from ordinary nondeterminism only at universal states, where all continuations must accept.
Depends on
Used by
- AP equals PSPACE Theorem
Dependency tree · two levels
6 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
- Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)