自动化正则表达式

Question

我有这个家庭作业问题。

Which pair of regular expressions are equivalent?
a) (ab)* and a*b*
b) r(rr)* and (rr)*r
c) r+ and r*r
d) (b) and (c)

我被告知答案是(d)。我认为(b)和(c)也应该是答案。有人可以为我澄清一下吗？

Answer 1

你应该尝试的第一件事是在每种语言中写出一些简单的字符串。如果你发现一个是RE的语言，而不是另一个，你可以检查以确定，如果是，你就完成了。以下是这些的结果：

(a)
 - (ab)*: e, ab, abab, ababab, ...
 - a*b* : e, a, b, aa, ab, bb, ...
 guess: a is in L(a*b*) but not (ab)*.
 check: (ab)* only generates strings with the same number of a's as b's.
 L((ab)*) != L(a*b*)

(b)
 - r(rr)*: r, rrr, rrrrr, rrrrrrr, ...
 - (rr)*r: r, rrr, rrrrr, rrrrrrr, ...
 guess: these look the same.
 proof: the first generates all and only strings of r's of odd length.
        the second generates all and only strings of r's of odd length.
        these languages are the same.
        alternatives:
        - derive DFAs L and R and show DFAs for L \ R and R \ L accept
          the empty language.

(c)
 - r+ : r, rr, rrr, rrrr, ...
 - r*r: r, rr, rrr, rrrr, ...
 guess: these look the same.
 proof: the first generates all and only non-empty strings of r's.
        the second generates all and only non-empty strings of r's.
        these languages are the same.
        alternatives:
        - derive DFAs L and R and show DFAs for L \ R and R \ L accept
          the empty language

基于以上所述，最正确的答案似乎是（d）。