Kleene's Theorem

Kleene's Theorem

Kleene's Theorem is a fundamental result in the theory of finite automata. It connects three different methods for defining a language: regular expressions, finite automata, and transition graphs. The theorem states that if a language can be defined by one of these methods, it can also be defined by the other two. This means that these three methods are equivalent.

Theorem Statement

The theorem can be formally stated as follows:

Any language that can be defined by a regular expression, or a finite automaton, or a transition graph can be defined by all three methods.

This theorem is crucial because it allows us to convert between these representations of languages. The proof of this theorem consists of three parts:

1. Every language that can be defined by a finite automaton can also be defined by a transition graph.
2. Every language that can be defined by a transition graph can also be defined by a regular expression.
3. Every language that can be defined by a regular expression can also be defined by a finite automaton.

Proof of Part 1: Finite Automata to Transition Graphs

This part is straightforward. A finite automaton is itself a type of transition graph. Therefore, if a language is defined by a finite automaton, it is also defined by a transition graph. There is nothing more to prove here.

Proof of Part 2: Transition Graphs to Regular Expressions

This part is more complex and is proven using a constructive algorithm. The goal is to start with a transition graph and produce a regular expression that defines the same language. The algorithm must work for any transition graph and must finish in a finite number of steps.

First, we simplify the transition graph by ensuring it has only one start state and one final state. If the transition graph has multiple start states, we introduce a new start state that connects to all original start states with edges labeled with the empty string (denoted as ε). This allows all inputs to begin at the new unique start state.

Next, if the transition graph has multiple final states, we introduce a new unique final state. We connect all original final states to this new final state with ε-edges. This process does not change the language accepted by the transition graph.

State Elimination and Regular Expression Construction

Now we will build the regular expression step by step. Consider a state in the transition graph that has multiple loops back to itself. If there are loops labeled with regular expressions r1, r2, and r3, we can replace them with a single loop labeled with the regular expression r1 + r2 + r3.

Similarly, if two states are connected by multiple edges in the same direction, we can replace these edges with a single edge labeled with the combined regular expressions.

We can also define a bypass operation. If we have three states in a row connected by edges labeled with regular expressions (or simple strings), we can eliminate the middle state. The new edge will be labeled with the concatenation of the labels of the edges leading into and out of the middle state.

Continue this process until only the unique start and final states remain. The edge connecting these two states will be a regular expression that defines the same language as the original transition graph.

Example of Transition Graph to Regular Expression

Consider a transition graph that accepts all words that start and end with double letters (e.g., aabb, bbaa). The initial transition graph might look like this:

First, we introduce a unique final state and modify the edges according to the algorithm. For example, if the graph allows transitions on aa or bb from the start state to the first state, we can replace this with the regular expression aa + bb. If there are loops on state 1 for single a's or b's, we replace those with a + b.

Next, we eliminate states one by one, combining edges and updating the regular expression accordingly until we reach a single edge from the start state to the final state.

Proof of Part 3: Regular Expressions to Finite Automata

The final part of the proof shows that for each regular expression, we can construct a finite automaton that accepts the same language. This is done using a recursive algorithm based on the structure of regular expressions.

Building Finite Automata from Regular Expressions

Regular expressions can be built using the following rules:

1. For a single character from the alphabet, there exists a finite automaton that accepts only that character.
2. If there are two finite automata, FA1 and FA2, that accept languages defined by regular expressions r1 and r2, respectively, we can construct a new finite automaton FA3 that accepts the language defined by (r1 + r2).
3. If there is a finite automaton FA1 that accepts r1 and a finite automaton FA2 that accepts r2, we can construct a new finite automaton that accepts the concatenation r1r2.
4. If there is a finite automaton that accepts r1, we can create a finite automaton that accepts r1* (the Kleene star operation).

Example of Regular Expression to Finite Automaton

For example, consider the regular expression (aa + bb)(a + b)*. We can build a finite automaton that accepts all strings that start with aa or bb followed by any combination of a's and b's.

The finite automaton would have states representing the transitions for reading a's and b's, leading to accepting states for the combinations defined in the regular expression.

Conclusion

In summary, Kleene's Theorem establishes the equivalence of regular expressions, finite automata, and transition graphs. By understanding how to convert between these representations, you can effectively work with languages in theoretical computer science.

• Kleene's Theorem demonstrates the equivalence of three methods for defining languages.
• The proof consists of three parts: finite automata to transition graphs, transition graphs to regular expressions, and regular expressions to finite automata.
• Constructive algorithms are used to establish these conversions.

Check your understanding

1. What is Kleene's Theorem?
2. How can you convert a finite automaton to a transition graph?
3. What steps are involved in converting a transition graph to a regular expression?
4. Describe the process of creating a finite automaton from a regular expression.