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.
Acceptance of a word by a DFA and the recognized language
Definition
Let be a DFA, and let be its extended transition function. This function exists and is unique by The extended transition function exists and is unique.
A word is accepted by when
The language recognized by is This is a language over in the sense of Languages over an alphabet.
Remarks
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Acceptance depends on the chosen start state and accepting set, not just on the transition graph.
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Two different DFAs may recognize the same language.
Depends on
Used by
- Regular languages Definition
- Computing δ^*(q,uv) as δ^*(δ^*(q,u),v) Example
- Removing unreachable states preserves the recognized language Lemma
- The published prefix automata extend canonically to DFAs for factor-avoidance languages Proposition
- Complementing the accepting states complements the recognized language Theorem
- The product construction gives a DFA for language difference Theorem
- The product construction gives DFA's for union and intersection 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
- Jean Gallier and Jocelyn Quaintance, Introduction to the Theory of Computation: Some Notes for CIS511 (standard reference, not scraped)
- John Watrous, Introduction to the Theory of Computing, Lecture 2 (standard reference, not scraped)