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.
Boolean operations on languages over a fixed alphabet
Definition
Fix an alphabet , and let be languages over (Languages over an alphabet).
Their union, intersection, and difference are the languages
Their complement is taken inside the same ambient :
Remarks
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Complement depends on the chosen ambient alphabet. A language may have different complements when it is regarded inside different .
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These are ordinary set-theoretic operations applied to subsets of .
Depends on
Used by
- Membership, emptiness, finiteness, equivalence, and containment for regular languages Definition
- The GNFA state-elimination update Definition
- The language denoted by a regular expression Definition
- A language is decidable if and only if it and its complement are recognizable Theorem
- Complementing the accepting states complements the recognized language Theorem
- Decidable languages are closed under the Boolean operations Theorem
- NFAs can be constructed for union, concatenation, and Kleene star Theorem
- Recognizable languages are closed under union and intersection 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
3 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)
- Michael Sipser, MIT 18.404J Theory of Computation, Lecture 1 (standard reference, not scraped)