可以将任何有限状态自动机转换为正则表达式吗？

Question

Let's assume that we have a fsa as follows:

fsa = {0:{'a': 1, else: 2},1:{'b': 1, else: 2},2:{else: 2}}

This means: at state 0, if input is 'a', it goes to state 1, otherwise it goes to state 2; at state 1, if input is 'b', it goes to state 1, otherwise it goes to state 2; at state 2, for any input, it goes to state 2.

Assume, state 0 is a starting state, state 1 is an accepting state and state 2 is the failure state. Then this fsa can be translated as regex "ab*". In fact, there are a few algorithms that translate fsa to regex, such as Brzozowski algebraic method.

My question: can any fsa defined in the above form be translated into a regex? Is there any limitation?

Answer 1

Yes, they are mathematically equivalent. ' The equivalence of regular expressions and finite automata is known as Kleene's theorem. '

Answer 2

没有限制-所有有限状态自动机都等效于某些正则表达式，并且所有正则表达式都等效于某些有限状态自动机。

Answer 3

Yes, they are equivalent. From the wikipedia page on regular languages , these are all equivalent definitions of regular languages.

1. it is the language of a regular expression
2. it is the language accepted by a nondeterministic finite automaton (NFA)
3. it is the language accepted by a deterministic finite automaton (DFA)
4. it can be generated by a regular grammar
5. it is the language accepted by an alternating finite automaton
6. it can be generated by a prefix grammar
7. it can be accepted by a read-only Turing machine
8. it can be defined in monadic second-order logic (Büchi-Elgot-Trakhtenbrot theorem)
9. it is recognized by some finite monoid, meaning it is the preimage of a subset of a finite monoid under a homomorphism from the free monoid on its alphabet

Answer 4

Yes. A every finite automaton will have a corresponding regular expression. Kleene's theorem proves this result. The theorem is proved by using the principle of mathematical induction by partitioning the Finite Automata into the union of a number of smaller Finite Automatas. The proof can be found here .