Objectives:

- Analyze the equivalence of regular expressions and FA.

- To see that regular expressions and Finite Automata are equivalent we
must show that every regular expression must have an equivalent FA and that
every FA must have an equivalent regular expression.

We will start by taking a regular expression and see how we can construct an FA to recognize the same language. The process is explained in detail on pp 78-80 of your text.Open JFLAP and select the Regular Expression button. Enter the expression a*+ab. We will see how we can construct an NFA to recognize this language:

a) Select Convert to NFA. You are given a machine that has a single initial and final state with the transition labeled with the regular expression. Clearly this is not a legal NFA since it has a regular expression on a transition. This is a**generalized transition graph**(GTG).

b) Select Do Step. This will "De-or" the regular expression by removing the + .Explain why the resulting "De-ORed" machine accepts either a* OR ab.

How does de-oring work in general?

c) Select Do Step again. This will "De-concatenate".Explain how the de-concatenate works.

d) Select Do Step a final time to "De-star".Explain how the de-starring works - We
now must show that every FA must have an equivalent regular expression.
The process is explained in detail on pp80-85 of your text and involves
building a GTG with only two states (an intial and final state) and a
regular expression of the transition between them.

Select the Finite Automata button and open the file ex4.1.

Under the Convert menu select convert FA to RE.

The first step in the conversion is to make sure that every state has exactly one transition to every other state. This may involve combining transitions with an OR or adding transitions with the empty set as a label (meaning the transition does not lead to acceptance of any string). Add the empty transitions using the transition tool.

Next we need to remove non-initial and final states. Select the state collapser tool and click on state q1. A table of transitions will be shown. This table contains how the transition labels of the machine need to change (by regular expressions) if state q1 is removed. For example the transition from q0 to itself will be either the old transition of b OR the labels of q0 to q1 concatenated with q1 to itself as many times as we like, concatenated with the transition from q1 back to q0 (b + ab*a).

Explain the new transition from q0 to q2. Hint: The new transition is the old label OR the label that follows the path from q0 to q1 to q2.

Click finalize on the table window. If we had more states to remove we would continue this process until we have only the initial and final states left.

Now the regular expression expressed by the FA is displayed at the top of the window.Explain why the FA represents that regular expression.

Since we can convert a FA to an RE and we can convert an RE to an FA, it must be the case that these two models of computation represent the same family of languages -- the Regular Languages!