Problem 30 - Discrete Mathematics with Applications

Question

Create a NDFSA that accepts the regular language over \(\\{a, b\\}\) of strings that: End with \(b b\)

Step-by-Step Solution

Answer

The NDFSA that accepts the regular language over \(\{\mathrm{a}, \mathrm{b}\}\) of strings ending with 'bb' has the following states: Q = {q0, q1, q2}; the alphabet is Σ = {a, b}; and the initial state is q0. Define the following transition rules: 1) δ(q0, a) = {q0}; 2) δ(q0, b) = {q0, q1} (Non-deterministic transition); and 3) δ(q1, b) = {q2}. The final state is q2.

Step 1: Define the alphabet and initial state

Create an NDFSA with the following states: Q = {q0, q1, q2}. The alphabet of this NDFSA will be Σ = {a, b}. Set the initial state as 'q0'.

Step 2: Determine transitions between states

Define the following transition rules: 1. δ(q0, a) = {q0} 2. δ(q0, b) = {q0, q1} (Non-deterministic transition) 3. δ(q1, b) = {q2} Here's an explanation of the transitions: - If the NDFSA reads an 'a' while in state q0, it remains in q0 as 'a' will not change the string's ending with 'bb'. - If it reads a 'b' while in state q0, we take a non-deterministic approach by moving to both state q0 and q1. This is because we can't tell if reading this 'b' will ultimately lead to a string that ends with 'bb' or not. - If it reads another 'b' while in state q1, it moves to state q2. At this point, the last two characters read are 'bb', and the NDFSA is in a position to accept the string if it ends here. It is also important to note that the transitions for states q1 and q2 on input 'a' are not defined, as they are not relevant to us in accepting strings ending with 'bb'.

Step 3: Identify and mark final states

The only final state for this NDFSA is q2 because, at this point, the last two characters read are 'bb'. Mark this state as the final state. The resulting NDFSA will accept the regular language over {a, b} of strings that end with 'bb'.

Key Concepts

Understanding Regular Language

A regular language is a type of language that can be expressed using a set of rules known as regular expressions. In the context of Non-Deterministic Finite Automata (NDFSA), a regular language over the alphabet \(\{a, b\}\) could be a language consisting of strings that meet specific criteria. For instance, the language we are discussing consists of strings that end with 'bb'.
Regular languages can be recognized by finite automata. These automata are constructed to accept or reject a string based on whether it fits the defined patterns or rules of the language.
The key characteristics include the ability to repeat certain sequences and require specific endings or patterns, like ending with specific characters as in this exercise. Regular languages are integral in computer science for pattern matching and text processing tasks.

Decoding Transition Rules

Transition rules define how an NDFSA moves from one state to another when reading an input symbol. These are fundamental in determining how strings are processed by the machine.
In the exercise, several key transition rules are established:

• From state \(q0\), on reading 'a', it stays at \(q0\).
• From state \(q0\), on reading 'b', it non-deterministically moves to both \(q0\) and \(q1\).
• From state \(q1\), on reading 'b', it moves to \(q2\).

These rules are crafted to capture the end 'bb' pattern. The rule where \(q0\) moves to \(q0\) and \(q1\) is non-deterministic, meaning the machine creates both possibilities to see which path leads to acceptance.
Understanding these rules is crucial as they dictate whether the string will be accepted based on the last characters read.

Identifying the Final State

In an NDFSA, a final state is one where the automaton halts in an accepting condition. It signifies that the string being processed matches the required pattern.
For the task at hand, the final state is \(q2\). This is because reaching \(q2\) means the last two characters read were 'bb'. If the string ends at \(q2\), the NDFSA accepts it.
Identifying and marking the final state is crucial because only strings that can end in these states are considered part of the language. The automaton is designed to begin at an initial state, process through various transitions, and ideally arrive at the final state, thus confirming its acceptance of the string.