Problem 17 - 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: $$\begin{array}{rl} S=\left\\{s_{0}, s_{1}, s_{2}\right\\}, \quad A=\left\\{s_{2}\right\\} \\ f\left(s_{0}, \mathrm{a}\right)=s_{0} & f\left(s_{0}, \mathrm{b}\right)=s_{1} \quad f\left(s_{1}, \mathrm{a}\right)=s_{0} \quad f\left(s_{1}, \mathrm{b}\right)=s_{2} \\ f\left(s_{2}, \mathrm{a}\right)=s_{0} & f\left(s_{2}, \mathrm{b}\right)=s_{2} \end{array}$$

Step-by-Step Solution

Answer

The transition diagram for the given FSA can be drawn as follows: 1. Draw three circles labeled as \(s_0\), \(s_1\), and \(s_2\). 2. Add an arrow pointing to \(s_0\) to indicate initial state. 3. Add a double circle around \(s_2\) to indicate the accepting state. 4. Add transitions as per the given function: - Edge 'a' from \(s_0\) to \(s_0\). - Edge 'b' from \(s_0\) to \(s_1\). - Edge 'a' from \(s_1\) to \(s_0\). - Edge 'b' from \(s_1\) to \(s_2\). - Edge 'a' from \(s_2\) to \(s_0\). - Edge 'b' from \(s_2\) to \(s_2\). This completes the transition diagram for the FSA.

Step 1: Draw the states

We have 3 states: \(s_0, s_1, s_2\). Draw three circles and label them as \(s_0\), \(s_1\), and \(s_2\).

Step 2: Add the initial state

The initial state is \(s_0\). Place an arrow pointing to \(s_0\) to indicate it is the initial state.

Step 3: Add the accepting states

We have one accepting state: \(A = \{s_2\}\). Add a double circle around state \(s_2\) to indicate it is an accepting state.

Step 4: Add the transition edges:

According to the given transition function \(f\), there are six transitions. 1. f(s_0, a) = s_0 - Draw an edge labeled 'a' from \(s_0\) to \(s_0\) (using a loop). 2. f(s_0, b) = s_1 - Draw an edge labeled 'b' from \(s_0\) to \(s_1\). 3. f(s_1, a) = s_0 - Draw an edge labeled 'a' from \(s_1\) to \(s_0\). 4. f(s_1, b) = s_2 - Draw an edge labeled 'b' from \(s_1\) to \(s_2\). 5. f(s_2, a) = s_0 - Draw an edge labeled 'a' from \(s_2\) to \(s_0\). 6. f(s_2, b) = s_2 - Draw an edge labeled 'b' from \(s_2\) to \(s_2\) (using a loop). With all the states, initial state, accepting state, and transitions represented, the transition diagram of the FSA is complete.

Key Concepts

Finite State Automaton

A finite state automaton (FSA), sometimes called a finite state machine, is a theoretical model of computation used to design both computer programs and sequential logic circuits. It is composed of a finite number of states and is used to recognize patterns or sequences of inputs from a given set.

An FSA operates by transitioning from one state to another in accordance with a function that takes as inputs the current state and an element of the input set. These transitions are determined by a set of rules described by the transition function. The process begins at a designated starting state and consumes an input; based on the transition rules, it either stays in the same state or moves to another state.

This model is particularly useful for modeling systems that can be in one of a limited number of configurations at a given time, such as traffic lights, vending machines, or text parsers. The primary beauty of an FSA lies in its simplicity and the ease with which it can be represented visually, which helps in understanding and designing complex algorithmic behavior through simpler building blocks.

Accepting States

In the context of a finite state automaton, accepting states, also known as final states, play a crucial role in defining the language that the automaton recognizes. An accepting state is one in which the automaton signifies that the input sequence it has processed so far is accepted, meaning that the sequence is part of the language the FSA is equipped to recognize.

In diagrams, these states are often represented with a double circle. The set of all accepting states is denoted by 'A', and a language is said to be 'accepted' by the automaton if, after processing an input string, the automaton comes to rest in one of these accepting states.

For example, in a simple FSA designed to check if a binary number is even, the accepting state would be the one that the automaton enters after reading an input string that ends in the binary digit '0'. It's important to note that FSAs can have multiple accepting states, depending on the complexity and requirements of the language or pattern they are designed to recognize.

Transition Function

The transition function is the mathematical description of the rules that govern how an FSA moves from one state to another in response to inputs. It defines the logic and conditions under which state transitions occur.

Typically denoted by the letter 'f' or the Greek letter delta 'δ', the transition function takes two parameters: the current state and the current input symbol. The result of this function is the next state to which the automaton will move. In essence, the transition function provides a mapping from pairs of a state and input symbol to a state.

The complete list of possible transitions, including the input symbols and the resulting states, make up the transition table of the FSA. Creating a clear and comprehensive transition function is key for an FSA to operate correctly, as it ensures that the automaton responds consistently to each given input and makes it possible to predict its behavior for any sequence of inputs within the defined language.