Regular language

Mathematical statement

The DFA certificate must cover every finite word over the declared alphabet. Mod-count and parity languages use a state invariant valid for arbitrary word length; a finite language can use a prefix-trie DFA construction. Finitely many tested words do not prove regularity or language equivalence.

Representative formula (exact TeX)

R(L)

Assumptions

• The alphabet is finite and the DFA transition function is complete and total.
• This slice adds no regex, NFA, epsilon transition, minimization or pumping-lemma machinery.

Symbol scope

• R — regular-language certificate
• L — language as a set of words

Intuition, definition, and misconceptions

Existence of a complete recognizing DFA is a whole-language certificate.

Parity and mod-count constructions use an invariant for every prefix.

Finite languages admit explicit DFA constructions; passing finitely many tests is not enough.

A small test suite does not establish regularity or equality of two languages.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Deterministic finite automaton — A DFA supplies the recognition certificate.
• Set — A language is a set of words.
• Sequence — A word is a finite ordered symbol sequence.
• Decision problem — Membership is accepted or rejected.

Verification boundary

Only DFAs with explicit invariants or finite-language constructions are handled; no other computational model is inferred.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add NFAs, regular expressions and closure constructions.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.