Write a regular expression defining strings that begin with an a and end with a b and can contain any number (including zero) of c's or d's in the middle. Every c that is in the string must be followed by at least one d.

Question

Write a regular expression defining strings that begin with an a and end with a b and can contain any number (including zero) of c's or d's in the middle. Every c that is in the string must be followed by at least one d.

Construct a deterministic finite state automaton with no more than four states that accepts the language defined by the regular expression in problem 1.

Answer 1

Try

my $pat="^ad*(cd+)*b\$"

Use this pattern to match strings according to the rules specified.

Note that $pat will match a qualifying string, but does not define a string.

Hope someone else can answer the second part.

Answer 2

Explanation of the pattern follows:

^a specifies the string begins with "a"
b\$ specifies that the string ends with "b"
d* allows 0 or more embedded d's.
(cd+)* requires a "c" to be followed by at least one "d". This can happen 0 or more times.

Answer 3

To construct the regular expression, we can follow these steps:

1. Start by matching the beginning of the string with the character "a": `a`

2. Next, we can use the pattern `(c*d*)*` to match any number (including zero) of c's or d's in the middle, where every c must be followed by at least one d.

3. Finally, we end the pattern by matching the character "b": `b`

Putting it all together, the regular expression is: `a(c*d*)*b`

Now, let's construct a deterministic finite state automaton (DFA) with no more than four states that accepts the language defined by this regular expression.

Here is the step-by-step process:

1. Start with an initial state, q0.

2. Create a new state, q1, to represent the stage where we have matched the initial "a" character.

3. From q0, add a transition to q1 on the input "a".

4. Create a new state, q2, to represent the stage where we have matched the middle part `(c*d*)*`.

5. From q1, add transitions to q1 on the input "c" or "d". This represents the loop that allows for any number of c's or d's.

6. Add a transition from q1 to q2 on the input "b". This represents the stage where we have matched the final "b" character.

7. Add a transition from q2 to q1 on the input "c". This represents the requirement that every "c" must be followed by at least one "d".

8. Mark q2 as the accept state.

The resulting DFA will have no more than four states and will accept the language defined by the regular expression `a(c*d*)*b`.