Deterministic Finite-state Automaton Questions

Question

I have this DFA described as (Q, q1, A, N, F) where

Q = {1,2,3,4},
q1 = 1,
A = {a,b,c},
F = {2,4},
N = {
(1,a) -> 2, (1,b) -> 3, (1,c) -> 4,
(2,a) -> 2, (2,b) -> 4,
(3,a) -> 2, (3,c) -> 4,
(4,b) -> 4, (4,c) -> 4 }

So I have drawn the transition diagram, and that looks fine,

I then need to work out wether or not the following strings are acceptable by this DFA:

1. aabbcc
2. acacac
3. cabbac
4. babbab

and come up with the following

1. Correct
2. Incorrect (can not move from a -> c ?)
3. Incorrect (can not move from c -a ? )
4. Incorrect (cannot move from b -> a)

I am not 100% sure those are correct, but think they are on the right track.

I then need to describe the language this accepts, in english, which I do not see being a problem, but where I need help is describing this language using mathematical notation. Could you please help me to understand this.

Thanks so much for your help

Answer 1

Your answers about the strings acceptability are right, this can easily be seen by trying to follow them out in the diagram.
Now, about the language:

In the 2-nd vertex we can finish with the words corresponding following regular expression:
b?a+ - we can optionally obtain b my moving to 3-rd vertex first, and then pass over a, or we can move to 2-nd vertex by a at once, and there we can add as much as as we want.

Now about finishing the word in the 4-th vertex:

First, how can we reach vertex 4 ?
1. We can first time reach vertex 4 by moving there by c at once, or by moving to 3-rd vertex first, obtaining b, and then to 4-th by c. Hereby we get strings like b?c
2. We can reach vertex 2 with b?a+ (as described in previous case), and then pass over b. Hereby we get strings like b?a+b.
Totally, we can reach to 4-th vertex with any word matching the b?(a+b|c) regexp.

Now, adding arbitrary count of b and c symbols in the end on vertex 4, we get the answer for this case:
b?(a+b|c)(bc)*

Finally, we can result the whole set of words acceptable by this DFA words as the following regex:

b?( a+ | (a+b|c)(bc)*? )