What is a generalized transition graph (GTG)?

A generalized transition graph (GTG) is a graphical representation used when converting regular expressions into finite automata. A GTG provides a structured visual representation of the transitions between states in a finite automaton derived from a given regular expression. It’s particularly useful for understanding the conversion of regular expressions into finite automata.

Each state in a GTG corresponds to a subexpression or a set of subexpressions within the original regular expression. Transitions between states are labeled with regular expressions. These labels describe the possible paths or transitions between different parts of the input string based on the regular expression’s rules. GTGs have accepting states that indicate when the input string matches the corresponding transition’s subexpression.

Properties

A GTG is a collection of the following:

1. A finite set of states, of which at least one is a start state and some (maybe none) are final states

2. An alphabet Σ of input letters

3. Directed edges connecting some pairs of states, each labeled with a regular expression

Examples

For the examples below, $Σ=$ {$a, b$}.

1. A GTG for words without two consecutive $b$ letters is shown in the figure below. The word $b$ is not accepted.

In this GTG, a loop exists at state $1$ to accept words with only $a$ in them. The edge from state $1$ to state $2$ accepts words encountering a $b$ followed by an $a$ ($ba$) or just a sequence of $a$. The edge from state $2$ to state $3$ either accepts $b$ or serves as a free ride to the end state (by using the $\lambda$); this allows for words with a single $b$ to satisfy the criteria without encountering the $bb$ sequence.

2. A GTG for words that begin and end with different letters is shown in the figure below.

In this GTG, the edge from state $1$ to state $2$ is labeled $b(a+b)*a$, indicating that it accepts words that start with $b$, followed by zero or more occurrences of $a$ or $b$, and end with $a$.

The edge from state $1$ to state $3$ is labeled $a(a+b)*b$, indicating that the GTG also accepts words that start with $a$, followed by zero or more occurrences of $a$ or $b$, and end with $b$.

This is another example of how the GTG can be represented.

Conclusion

The purpose of a GTG is to provide a structured way to visualize the steps involved in processing a regular expression and recognize the strings that match the given regular expression pattern. It helps illustrate how the regular expression is decomposed into smaller subexpressions and how it’s processed as part of the finite automaton.