Problem 19 - Discrete Mathematics with Applications

Question

Draw the transition diagram of the FSA, \(M=\left(S, A, I, f, s_{0}\right),\) where \(I=\) \(\\{\mathrm{a}, \mathrm{b}\\},\) and: $$\begin{aligned} &S=\left\\{s_{0}, s_{1}, s_{2}, s_{3}\right\\}, A=\left\\{s_{2}\right\\}\\\ &\begin{array}{|c||cc|} \hline & {\boldsymbol{f}} \\ \hline \boldsymbol{S/I} & \mathbf{a} & \mathbf{b} \\ \hline {s_{0}} & s_{0} & s_{1} \\ s_{1} & s_{1} & s_{2} \\ s_{2} & s_{2} & s_{3} \\ s_{3} & s_{3} & s_{3} \\ \hline \end{array} \end{aligned}$$

Step-by-Step Solution

Answer

The transition diagram for the given FSA can be represented as follows: 1. Draw four circles representing the states \(s_0\), \(s_1\), \(s_2\), and \(s_3\). 2. Mark the initial state \(s_0\) with an incoming arrow. 3. Double-circle the state \(s_2\) to indicate it as an accepting state. 4. Add the following transitions: a. \(s_0 \xrightarrow{a} s_0\) b. \(s_0 \xrightarrow{b} s_1\) c. \(s_1 \xrightarrow{a} s_1\) d. \(s_1 \xrightarrow{b} s_2\) e. \(s_2 \xrightarrow{a} s_2\) f. \(s_2 \xrightarrow{b} s_3\) g. \(s_3 \xrightarrow{a} s_3\) h. \(s_3 \xrightarrow{b} s_3\)

Step 1: Draw the states

First, draw four circles representing the states \(s_0\), \(s_1\), \(s_2\), and \(s_3\). Label each circle with its corresponding state name.

Step 2: Mark the initial state

Now, mark the initial state \(s_0\) with an incoming arrow.

Step 3: Indicate the accepting state(s)

Next, double-circle the state \(s_2\) to indicate that it is an accepting state.

Step 4: Add the transitions

Finally, add the transitions between the states according to the provided transition table. Draw arrows between the states and label them with the corresponding input symbol from the alphabet \(I = \{a, b\}\): 1. From state \(s_0\), draw a loop with label 'a'. 2. From state \(s_0\) to state \(s_1\), draw an arrow with label 'b'. 3. From state \(s_1\), draw a loop with label 'a'. 4. From state \(s_1\) to state \(s_2\), draw an arrow with label 'b'. 5. From state \(s_2\), draw a loop with label 'a'. 6. From state \(s_2\) to state \(s_3\), draw an arrow with label 'b'. 7. From state \(s_3\), draw a loop with label 'a' and another loop with label 'b'. Now, you should have a complete transition diagram for the given FSA.

Key Concepts

Transition Diagram

A transition diagram is a visual representation of a Finite State Automaton (FSA). It depicts the different states of the automaton and how the automaton moves from one state to another based on input symbols. Each state is usually represented as a circle. The movement between these states is shown with arrows, known as transitions. These arrows are labeled with the input symbols that trigger the transitions. This diagram helps in understanding the behavior of the automaton by clearly displaying how it reacts to different inputs.

• States are nodes in the diagram.
• Transitions are directed arrows between nodes.
• Transition labels indicate the input causing the transition.

For the FSA given, the transition diagram includes states from \(s_0 \) to \(s_3\) with specified transitions depending on inputs 'a' and 'b'. Loops on a state indicate a transition that leaves and re-enters the same state.

Accepting State

An accepting state, sometimes known as a final state, is crucial in determining whether a particular string or sequence of inputs is accepted by a finite state automaton. In a transition diagram, accepting states are often marked by double circles, distinguishing them from regular states.When the FSA processes an input string, if it ends in an accepting state, the string is considered accepted by the automaton. If it ends elsewhere, the string is not accepted. Accepting states essentially define the successful conditions for input processing.

• Only double-circled states can accept input strings.
• Accepting states can receive transitions just like other states.

In our example, state \( s_2 \) is marked as the accepting state. If the FSA reaches \( s_2 \) after processing the input, the input is accepted.

Initial State

The initial state in a finite state automaton is the starting point for input processing. This state is usually marked with an incoming arrow that does not originate from any other state. It signifies where the FSA begins processing an input string.An effective way to depict the initial state in a transition diagram is by drawing an arrow pointing toward it from nowhere else, indicating the beginning of the process.

• Only one initial state per FSA.
• Determines where input interpretation starts.

In our given example, \( s_0 \) is the initial state. All input strings start their journey from \( s_0 \) and proceed through transitions based on incoming input.

Transition Table

A transition table is a structured way to define the state transitions in a finite state automaton. Unlike a diagram, a transition table presents this information in a tabular form, listing each state, input, and corresponding transition to another state.The table typically has:

• Rows representing current states.
• Columns for possible input symbols.
• Cells showing resulting states from given state-input combinations.

Using a transition table, one can readily identify which state the FSA will move to based on a current state and an input symbol.For the described FSA, the transition table is:- From \(s_0\), 'a' leads to \(s_0\), 'b' leads to \(s_1\).- From \(s_1\), 'a' leads to \(s_1\), 'b' leads to \(s_2\).- From \(s_2\), 'a' leads to \(s_2\), 'b' leads to \(s_3\).- From \(s_3\), both 'a' and 'b' create loops back to \(s_3\).This table is an efficient way to understand and verify transitions quickly.