Transition Graphs

Transition Graphs

Transition graphs (TGs) are a type of abstract machine used in automata theory. They extend the concept of finite automata (FAs) by allowing transitions based on reading substrings of input rather than just single letters. This flexibility enables TGs to accept a broader range of languages.

Definition of Transition Graphs

A transition graph is defined by three components:

1. A finite set of states, including at least one start state and possibly several final states.
2. An alphabet I of possible input letters.
3. A finite set of transitions, represented as edges, showing how to move from one state to another based on reading specified substrings of input letters.

Reading Input in Transition Graphs

In a TG, an input string can be processed in various ways, depending on how the operator chooses to read the letters. This means that the same input string can lead to different paths through the TG. For example, consider the string "baa". If we read the first letter "b" and then the next two letters "aa" together, we can reach a final state. Alternatively, if we read each letter individually, we might end up in a non-final state.

Examples of Transition Graphs

Let’s look at some examples of TGs:

Example 1: Accepting the String "baa"

Consider a TG with the following states:

• State 1: Start state
• State 2: Intermediate state
• State 3: Final state

The transitions might be:

• From State 1 to State 2 on input "b".
• From State 2 to State 3 on input "aa".

This TG accepts the string "baa" because it can follow the path: State 1 → State 2 → State 3.

Example 2: Accepting Multiple Paths

Now, consider a TG that accepts the string "baab". It can accept this string in two ways:

• Path 1: State 1 → State 2 (on "ba") → State 3 (on "ab").
• Path 2: State 1 → State 2 (on "baa") → State 4 (on "b").

This demonstrates that some strings can have multiple paths leading to acceptance, which is a key feature of TGs.

Crashing States

A crashing state occurs when the input string reaches a state with no outgoing edges. In this case, the execution terminates, and the input must be rejected. In contrast to FAs, where every state has outgoing edges for each possible input letter, TGs may have states that do not allow further transitions.

Generalized Transition Graphs

Generalized transition graphs (GTGs) are an extension of TGs. In GTGs, the edges can be labeled with regular expressions instead of just strings. This allows for even more complex input processing. For instance, an edge might be labeled with a regular expression that accepts any string of letters from a defined alphabet.

Example of a Generalized Transition Graph

Consider a GTG where:

• State 1 is the start state.
• State 2 is the final state.

Edges might be labeled with regular expressions like:

• From State 1 to State 2 on input (a + b)*, which means any combination of "a" and "b".

This allows the GTG to accept a wide range of input strings.

Nondeterminism in Transition Graphs

Nondeterminism in TGs refers to the presence of multiple paths for processing an input string. This means that the outcome of processing can depend on the choices made during execution. For example, given the input "abb", the TG might allow the operator to choose to read the first two letters together or one at a time, leading to different paths through the machine.

Summary

• A transition graph consists of states, an alphabet, and transitions based on input substrings.
• Input processing can lead to multiple paths and outcomes.
• Crashing states occur when no outgoing edges are available.
• Generalized transition graphs allow for edges labeled with regular expressions.
• Nondeterminism allows for multiple processing paths for the same input.

Check your understanding

1. What are the three components of a transition graph?
2. Explain the difference between a crashing state and a final state.
3. How does a generalized transition graph differ from a standard transition graph?
4. What does nondeterminism mean in the context of transition graphs?