Regular Expressions and Finite Automata: Equivalence and Conversion

1. From Description to Machine: Regular Expression to NFA

The first half of our proof shows that for any regular expression, there exists a finite automaton that recognizes the same language. Since we know NFAs and DFAs are equivalent, we'll make our lives easier by constructing a Nondeterministic Finite Automaton with $ϵ$-transitions (NFA-$ϵ$). The flexibility of $ϵ$-transitions is ideal for piecing together smaller machines into a larger one.

The proof strategy is inductive, following the recursive definition of a regular expression:

1. Base Cases: We'll show how to build simple NFAs for the atomic regular expressions: a single symbol $a$, the empty string $ϵ$, and the empty set $\varnothing$.
2. Inductive Step: We'll show how to combine existing NFAs for regular expressions $R_1$ and $R_2$ to create new NFAs for their union ($R_1\cup R_2$), concatenation ($R_1{R}_2$), and Kleene star ($R_1^{\ast }$).

This construction is often called Thompson's Algorithm.

The following video provides a clear, step-by-step walkthrough of this constructive proof.

To summarize the inductive step, the following diagram shows the rules for combining smaller machines into larger ones. This visual should serve as a helpful reference for Thompson's construction.

Caption: The rules for Thompson's construction. Each rule takes the NFA(s) for smaller expressions (R, S) and shows how to wire them together with new start/final states and $ϵ$-transitions to build the NFA for the compound expression.

Example: Building an NFA for (a∪b)*a

The video you just watched also includes a great example of applying these rules to a more complex expression. Working through this will solidify your understanding.

2. From Machine to Description: Finite Automaton to Regular Expression

Now, we prove the other direction: for any finite automaton, there is a regular expression that describes its language. This direction is often considered less intuitive, but the algorithm is just as systematic.

The core idea is to start with a finite automaton and progressively remove its states. As we remove a state, we update the transitions between the remaining states with regular expressions that account for the paths that went through the removed state. We continue this process until only the start state and a single final state remain. The label on the transition connecting them will be our final regular expression.

To formalize this, we use the concept of a Generalized NFA (GNFA), which is an NFA where the transitions are labeled with regular expressions instead of single symbols.

The State Elimination Algorithm

The algorithm consists of three main stages:

1. Preprocessing: Convert the given automaton (DFA or NFA) into a specific GNFA format. We add a new, unique start state with an $ϵ$-transition to the original start state, and a new, unique final state with $ϵ$-transitions from all original final states. This ensures our machine has exactly one start state with no incoming edges and one final state with no outgoing edges.

*Caption: An illustration of part of the preprocessing step. Original final states (`qf1`, `qf2`, `qf3`) are connected via $\epsilon$-transitions to a single new final state (`qf`).*

2. State Elimination: Repeatedly pick an intermediate state to "rip out." For any pair of states q_in and q_out with transitions passing through the state being removed, q_rip, we create a new direct transition from q_in to q_out. The label of this new transition is a regular expression that captures all possible paths from q_in to q_out via q_rip.

3. Finalization: When only the start state and final state are left, the regular expression on the single transition connecting them is the result.

The following reading explains this state-ripping procedure in detail.

The formula you just read is key. If we are removing q_rip, the new regular expression for the transition from q_in to q_out is:

$R_{\text{in}\to \text{out}}^{\prime }=R_{\text{in}\to \to \text{out}}\cup R_{\text{in}\to \text{rip}}(R_{\text{rip}\to \text{rip}}{)}^{\ast }{R}_{\text{rip}\to \text{out}}$

Here, $R_{\text{in}\to \to \text{out}}$ is the label on the existing direct path from q_in to q_out (if any), and the second part captures all paths that go through q_rip one or more times.

The Algorithm in Action

To see how this abstract procedure works, let's turn back to Professor Stajano's lecture, where he introduces this method from a slightly different, more high-level perspective before diving into an example.

As a final illustration, the document you read earlier also has a fantastic, clear, step-by-step graphical example.

Conclusion

In this lesson, we have formally established the equivalence of regular expressions and finite automata by providing concrete algorithms for converting between them. This completes the proof of Kleene's Theorem.

Here are the main takeaways:

• Two Sides of the Same Coin: Regular languages can be defined either by a computational device (DFA/NFA) or by a descriptive text pattern (regular expression). These two formalisms are equivalent in power.
• Thompson's Construction (RE $\to$ NFA): An elegant inductive algorithm that builds an NFA by mirroring the recursive structure of a regular expression.
• State Elimination (FA $\to$ RE): A method that systematically removes states from an automaton, encoding the lost paths into increasingly complex regular expressions on the remaining transitions.
• Constructive Proof: Both conversions are algorithms, providing a constructive proof of equivalence. This has practical significance, as these algorithms (or variants of them) are used in compilers, text editors, and search tools.

We now have a complete picture of the world of regular languages. We know how to define them with machines and with expressions, and how to translate between these representations. But what are the boundaries of this world? What can't these simple models do? In our next lesson, we will introduce a powerful tool, the Pumping Lemma, to prove that certain languages are not regular, thereby exploring the limits of finite memory computation.