The question is to design a DFA that accepts a language in which the count of substring "$110$" is no less than the count of substring "$011$". For example, $110$, "0101", "0111101110" is good but "011011" is bad.
The restriction of the language equivalently means whenever a "011" is encountered, only this pattern "$0111^*0$" is allowed, so that a "$011$" is matched with a "$110$". There is no other restriction.
I tried to give a regular expression as $(1^*0111^*0)^* + 1^*0^*$, but I tried very hard but failed to convert this to a DFA. Is there a systematic way to convert such a regular expression to a DFA?
