问题
I did some googling on this and nothing really definitive popped up.
Let's say I have two languages A and B.
A = { w is a subset of {a,b,c}* such that the second to the last character of w is b }
B = { w is a subset of {b,d}* such that the last character is b }
How would one define this? I think the alphabet would be the union of both, making it {a,b,c,d} but aside from that, I wouldn't know how to make a DFA of this.
If anyone could shed some light on this, that would be great.
回答1:
From a set-theoretic perspective, each language is just a set of strings over some alphabets Σ1 and Σ2. If you intersect two languages, the only strings that can possibly be in the resulting set are those strings that are made of characters in Σ1 ∩ Σ2, since any characters not in that set must belong to strings purely in one set.
As for how to build a DFA for this - I would suggest starting off by answering this question - given any regular language L over alphabet Σ, how would you modify that DFA to have language Σ ∪ { x }, where x ∉ Σ? One way to do this would be to add in a new "dead state" with a transition to itself on all characters in Σ ∪ { x }, then to add a transition from every state in the DFA to the dead state on character { x }. You can then use this to transform the DFAs for the original languages over Σ1 and Σ2 to have alphabets Σ1 ∪ Σ2. Once you've done that, you can use the normal algorithm for intersecting two DFAs to compute their intesection and thus get back a DFA for the intersection of the two languages.
Hope this helps!
来源：https://stackoverflow.com/questions/15213955/what-is-the-intersection-of-two-languages-with-different-alphabets