Problem 13 - Discrete Mathematics with Applications

Question

Construct a transition table for each FSM.

Step-by-Step Solution

Answer

To create a transition table for a given FSM, first identify the system's states and input alphabet and create an empty table with these values as row and column headers. Then, fill in the table with the next state for every possible combination of current state and input symbol. For example, consider the FSM with states {S0, S1, S2}, input alphabet {a, b}, and the following transitions: S0 on input a -> S1, S0 on input b -> S2, S1 on input a -> S0, S1 on input b -> S2, S2 on input a -> S2, and S2 on input b -> S1. The transition table would look like this: a b +----+----+ | S1 | S2 | S0 +----+----+ | S0 | S2 | S1 +----+----+ | S2 | S1 | S2 +----+----+ This table shows the next state for each combination of current state and input symbol.

Step 1: Understand the given FSM

- Identifying the FSM's states: Enumerate all the distinct states the system can be in - Identifying the input alphabet: List the set of possible inputs the system can receive

Step 2: Create an empty transition table

- List all the states as row headers - List all the input symbols as column headers Next, let's create a simple FSM's transition table as an example. #Example FSM Specifications:# States: {S0, S1, S2} Input alphabet: {a, b} Transitions: - S0 on input a -> S1 - S0 on input b -> S2 - S1 on input a -> S0 - S1 on input b -> S2 - S2 on input a -> S2 - S2 on input b -> S1

Step 3: Construct the transition table

a b +----+----+ | S1 | S2 | S0 +----+----+ | S0 | S2 | S1 +----+----+ | S2 | S1 | S2 +----+----+ In this transition table, we can see the next state for each combination of current state and input symbol. For example, if the FSM is in state S0 and the next input symbol is 'a', the table shows that the next state is S1. Follow the steps above to construct a transition table for any Finite State Machine.

Key Concepts

Transition Table

A transition table is a tool used to represent the behavior of a Finite State Machine (FSM) in a structured and easy-to-understand way. It consists of rows and columns that help us map the transitions between states.

• The rows of the table represent the current states of the FSM.
• The columns represent the inputs from the input alphabet.
• Each cell in the table indicates the next state, given the current state and input.

By referring to a transition table, you can easily determine what the resulting state will be for each possible input and current state. This makes it easier to visualize and manage complex systems.

States

States in an FSM represent the different conditions or statuses that the system can be in at any given time.
Each state is typically represented by a unique identifier such as S0, S1, S2, etc.

• States are the building blocks of FSMs.
• They serve as anchors or checkpoints for processing inputs through the system.
• The transition path between states is regulated by input symbols.

For instance, in the example provided earlier, there are three states: S0, S1, and S2. Understanding the distinct roles of each state is crucial for navigating the FSM.

Input Alphabet

The input alphabet refers to the set of symbols that the FSM can recognize and respond to.
In the context of FSM, inputs trigger transitions from one state to another.

• The input alphabet is usually denoted by a set, like {a, b}.
• Inputs are the external signals or data received by the FSM.
• They dictate the execution flow through the states.

Knowing the possible inputs an FSM can encounter helps in effectively constructing and interpreting the transition table. This ensures all scenarios are accounted for in the design.

FSM Transition

FSM transitions describe the movement from one state to another based on a given input.
These transitions are crucial for defining the operation of the FSM.

• Each transition is triggered by a specific input from the input alphabet.
• Transitions are defined clearly in the transition table.
• They illustrate how an FSM responds dynamically to inputs.

For example, if the FSM is in state S0 and receives input 'a', the transition defined in the table directs it to move to state S1. Understanding transitions helps to predict FSM behavior.

Transition Function

The transition function is a formal mechanism that maps a state and an input to the next state.
It is often denoted as δ (delta), where δ(Current State, Input) = Next State.

• The function provides a mathematical description of the FSM's transitions.
• It ensures consistency in behavior each time the same input is encountered in a particular state.
• Using this function, we can precisely calculate the resulting state for any input sequence.

Thus, the transition function is a core concept for implementing FSMs, helping to create a predictable and structured response to various inputs.