Constructing an NFA that Accepts Strings Ending with a Previously Seen Character

In the realm of theoretical computer science, Non-deterministic Finite Automata (NFAs) play a crucial role in defining and recognizing specific patterns within strings. This article delves into the construction of an NFA that accepts strings over the alphabet {a, b, c, d} and ends with a letter that is already a part of the string.

Introduction

The objective is to design an NFA that can verify whether a given string meets the condition of ending with a letter that has been previously seen in the string. This satisfies certain linguistic and algorithmic requirements, making it an important topic in formal language theory and automata theory.

NFA Construction

To achieve the desired functionality, we will define states and transitions in an NFA setup. Below are the detailed steps and descriptions of the construction process.

Defining States

The NFA we construct will have the following states:

#916;_0: Start state #916;a: State indicating that 'a' has been seen #916;b: State indicating that 'b' has been seen #916;c: State indicating that 'c' has been seen #916;d: State indicating that 'd' has been seen #916;_accept: Accept state

Transitions

The transitions between these states will be based on the input symbols ('a', 'b', 'c', 'd'). Here is the detailed transition information:

From #916;_0: On 'a', go to #916;a On 'b', go to #916;b On 'c', go to #916;c On 'd', go to #916;d From #916;a: On 'a', stay in #916;a On 'b', go to #916;b On 'c', go to #916;c On 'd', go to #916;d From #916;b: On 'a', go to #916;a On 'b', stay in #916;b On 'c', go to #916;c On 'd', go to #916;d From #916;c: On 'a', go to #916;a On 'b', go to #916;b On 'c', stay in #916;c On 'd', go to #916;d From #916;d: On 'a', go to #916;a On 'b', go to #916;b On 'c', go to #916;c On 'd', stay in #916;d From #916;a, #916;b, #916;c, and #916;d: On the same letter (e.g., from #916;a on a transition to #916;_accept) On any other letter that has been seen before (e.g., from #916;a on 'b', 'c', or 'd' should go to #916;_accept)

Accept States

The accept state #916;_accept will accept any string that ends with a letter that has been seen previously in the string.

Diagram Representation

This NFA can be represented in a state diagram where:

Each state is a circle labeled with the state name. Arrows indicate transitions between states based on input symbols. The start state #916;_0 has an arrow pointing to it from nowhere. The accept state #916;_accept is double-circled.

Example Strings

The following strings are examples of those that would be accepted by the NFA:

abba cdabc daa

On the other hand, the following strings would not be accepted:

abcd abdc cc

This NFA effectively captures the requirement that the last character of the string must be one of the characters that has appeared earlier in the string.

Conclusion

In summary, the described NFA provides a tool for identifying strings that end with a character already present in the sequence. Such an NFA is a useful model for understanding and implementing various text processing and pattern recognition tasks.