All the following are equivalent EXCEPT which one?

Question

BlackTom AI · Accepted Answer

Regular languages are exactly those describable by regular expressions and recognizable by finite automata (both deterministic and nondeterministic), whereas pushdown automata recognize a strictly larger class (context-free languages).

1) Relevant concepts:
- Regular Expression: algebraic description of regular languages.
- DFA (Deterministic Finite Automaton): state machine with no extra memory recognizing regular languages.
- NFA (Nondeterministic Finite Automaton): like DFA but with nondeterminism; recognizes the same class as DFAs.
- PDA (Pushdown Automaton): finite control plus a stack; recognizes context-free languages (CFLs), strictly more powerful than finite automata.

2) Step-by-step reasoning:
- Kleene’s theorem: regular expressions ↔ finite automata. Any regular expression can be converted to an NFA (Thompson construction), and any NFA can be converted to an equivalent DFA (subset construction).
- Therefore Regular Expression, NFA, and DFA all define exactly the same class: regular languages.
- A PDA can simulate a DFA by simply never using its stack, so PDAs include all regular languages. But PDAs can also recognize non-regular context-free languages such as { a^n b^n | n ≥ 0 }.

3) Intermediate demonstration (no heavy calculation needed):
- Example language L = { a^n b^n }. A PDA can push one stack symbol for each a and pop one for each b, accepting when stack empty.
- No DFA/NFA/regular expression can recognize L because a finite automaton has finitely many states and cannot enforce an unbounded equality between number of a’s and b’s. The pumping lemma for regular languages formalizes this impossibility.

4) Why the correct answer is correct:
- The question asks which one is NOT equivalent to the others. Because Regular Expression, DFA, and NFA are equivalent characterizations of regular languages, but Pushdown Automata accept a strictly larger class (context-free languages), Pushdown Automata is not equivalent to the other three.

5) Why other options are incorrect:
- Regular Expression: equivalent to DFA/NFA by Kleene’s theorem.
- DFA: equivalent to NFA via subset construction and to regular expressions via conversions.
- NFA: equivalent to DFA and regular expressions.

6) Conclusion:
Thus the correct answer is Pushdown Automata, which matches the provided answer.

All the following are equivalent EXCEPT which one?单项选择题

类似问题

Finite automata are equivalent in computational power to a Turing Machine.

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Finite automata are equivalent in computational power to a Turing Machine.

TMs can access any data on the tape without losing any other data but PDAs lose newer data if they access older data.

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