91Ó°ÊÓ

In a finite-state automaton, states are fundamental elements that represent different configurations or steps of the system. Think of states as positions in a board game. Each row in a next-state table corresponds to one specific state. For the given exercise, the states that were identified are \(s_{1}\), \(s_{2}\), and \(s_{3}\). These serve as the critical junctures that determine how the automaton processes input and transitions between different phases.

Each state can be imagined as a distinct situation where the automaton waits for an input to decide which state to move to next. It's essential to note that these states are not isolated; they interact with each other through transitions based on input symbols, helping the automaton to function dynamically.

Input symbols in automata act like commands that tell the automaton where to go or what to do next. These are typically labeled using letters or numbers. In our context, the input symbols are identified based on the table's columns. According to the exercise, the automaton has two input symbols, which we can call \(a\) and \(b\).

This is akin to choosing between different paths in a decision tree; at each step (or state), the automaton examines the input symbol and decides the next state by looking at the designated column in the next-state table. This directed path is fundamental in determining the automaton's movement from one state to another.

Transition diagrams, also known as state diagrams, visually illustrate how an automaton transitions from one state to another based on inputs. Each state is represented as a circle, with arrows showing possible transitions. The initial step is to draw the states (e.g., \(s_{1}\), \(s_{2}\), and \(s_{3}\)) as circles. Then, using information from the next-state table, you connect these states with directed arcs labeled with input symbols \((a, b)\).

For instance, in the given exercise, the diagram captures transitions like moving from \(s_{1}\) to \(s_{2}\) upon receiving input \(b\), or looping back to \(s_{1}\) with input \(a\). This graphical representation facilitates an easier understanding of how the automaton behaves with different inputs, acting as a roadmap for its operational flow.

The initial state is where the automaton begins its journey. It is the starting point and is usually represented in diagrams by an arrow with no originating source. In the exercise, the initial state was identified as \(s_{1}\) because it is the first listed in the next-state table.

Accepting states, on the other hand, are the configurations where the automaton successfully completes its operation under given conditions. These states are depicted as double circles in transition diagrams, indicating successful termination. The exercise inferred \(s_{0}\) as an accepting state based on its appearance in transitions but absence as a main header in the table.

Understanding these states is crucial as they define both the start and successful completion points of the automaton's process, shaping how inputs can lead to desired outcomes.