Pick all true statements about formal languages:

Question

BlackTom AI · Accepted Answer

A finite language can always be described by a finite regular expression or accepted by a finite automaton, so "All finite languages are regular" is true.

Relevant concepts:
- Regular languages are those accepted by finite automata (DFA/NFA), or equivalently described by regular expressions or regular grammars.
- A finite language is a language containing a finite number of strings.
- Closure properties: regular languages are closed under finite union; each single string is regular.

Step-by-step reasoning and construction:
1. Let L = {w1, w2, ..., wk} be any finite language (k is finite). Each individual string wi is a regular language because a deterministic finite automaton (DFA) can be built that reads exactly wi and ends in an accepting state. Concretely, for wi of length n create n+1 states forming a path labeled by the symbols of wi; only the final state is accepting.
2. Regular languages are closed under finite union. Therefore the union of the k regular single-string languages {w1} ∪ {w2} ∪ ... ∪ {wk} is regular.
3. Equivalently, form the regular expression w1|w2|...|wk (or w1 + w2 + ... + wk). This regular expression denotes L, so L is regular.
4. So every finite language admits a regular expression and a finite automaton; hence all finite languages are regular.

Why the other options are incorrect:
- "All finite languages are non-regular." This contradicts the construction above; finite languages are regular, not non-regular.
- "All infinite languages are non-regular." False: many infinite languages are regular, e.g., a*, (a+b)*, or the set of all strings with an even number of a's. These are infinite and regular.
- "All infinite languages are regular." Also false: some infinite languages are not regular; for example L = {a^n b^n | n ≥ 0} is infinite but non-regular (can be shown by the pumping lemma).

Conclusion: The only correct statement is "All finite languages are regular," which matches the provided answer.

Pick all true statements about formal languages:多项选择题

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