What are Automata?

The term "automata" (plural of "automaton") refers to abstract self-operating computing devices. These are mathematical models of machines that perform computations by transitioning through a series of states based on input. The automatic door is a good example.

Key Concepts

States:

An automaton can be in only one state at any given time.

A state represents a specific configuration or condition of the machine.

Inputs:

Automata receive input symbols from a predefined set of symbols, called an alphabet.

Inputs trigger state transitions.

Transitions:

A transition is a rule that dictates how the automaton moves from one state to another based on the current state and the input symbol.

Transitions are the heart of an automaton’s behavior, determining how it responds to different inputs.

Start State:

Every automaton has a designated start state, which is the state the automaton is in at the beginning of any computation.

Accepting States:

Some automata have a set of accepting states. If the automaton ends its computation in an accepting state, the input is considered "accepted" by the automaton.

Accepting states are used to define languages recognized by the automaton.

Why Study Automata Theory?

Understanding Computation:

Automata theory provides a way to model and analyze the fundamental capabilities and limitations of computers.

Compiler Design:

Concepts from automata theory are used in the design of compilers, particularly in lexical analysis and parsing. Formal languages and grammars are used in programming languages and compilers .

Formal Language Theory:

Automata are closely related to formal languages. Each class of automata corresponds to a class of formal languages that they can recognize. Grammars, or formal grammars, define the rules that specify the syntax of a programming language .

Algorithm Design:

The principles of automata theory can be applied to the design of algorithms for various applications, such as pattern matching, text processing, and network protocols.

Simple Example: A Light Switch

 

Consider an automaton that models a simple light switch. This automaton has two states: "ON" and "OFF". The input is a "FLIP" signal, which represents toggling the switch.

1. States

The automaton has two states:

·       ON: The light is on.

·       OFF: The light is off.

Automata can only be in one state at a time  

2. Inputs

The automaton receives one input symbol:

FLIP: This signal represents the action of flipping the light switch.

3. Transitions

The transitions define how the automaton changes from one state to another based on the input.

If the current state is "OFF" and the input is "FLIP", the automaton transitions to the "ON" state.

If the current state is "ON" and the input is "FLIP", the automaton transitions to the "OFF" state.

4. Start State

The start state is the state in which the automaton begins its operation.

OFF: Initially, the light is off.

5. Accepting States

In this example, we can define the "ON" state as the accepting state.

ON: If the automaton ends in this state, it means the light is on.

Example Sequence

Let’s consider a sequence of inputs: FLIP, FLIP, FLIP

Start in the "OFF" state.

Receive "FLIP": Transition to the "ON" state.

Receive "FLIP": Transition to the "OFF" state.

Receive "FLIP": Transition to the "ON" state.

Connection to Automata Theory

This simple example illustrates the fundamental concepts of automata theory:

Finite States: The light switch has a finite number of states (two in this case).

Input-Driven Transitions: The state changes are driven by the input signal "FLIP".

Deterministic Behavior: For each state and input, there is exactly one transition that defines the next state.

 

Another Example: Automatic Door Controller

A classic example of a finite automaton is the controller for an automatic door. Let’s break down how it works:

States: The controller can be in one of two states: "OPEN" or "CLOSED," corresponding to the state of the door.

Inputs: The controller receives input from pressure pads in front ("FRONT") and behind ("REAR") the doorway . The possible input conditions are "FRONT", "REAR", "BOTH" (people on both pads), and "NEITHER" (no one on either pad) .

Transitions: The controller changes states based on the input:

If the door is "CLOSED" and the input is "FRONT" or "BOTH", the controller transitions to the "OPEN" state.

If the door is "OPEN" and the input is "NEITHER", the controller transitions to the "CLOSED" state after a delay.

The "REAR" input ensures the door stays open long enough for someone to pass through completely and prevents the door from hitting someone behind it.

The Automatic Door Controller in Detail

The automatic door controller transitions between states based on specific inputs. Let’s look at the state transitions:

States: "OPEN" or "CLOSED" .

Inputs: "FRONT", "REAR", "BOTH", and "NEITHER".

Transitions:

CLOSED State:

Receiving "NEITHER" or "REAR" keeps the controller in the "CLOSED" state .

Receiving "BOTH" also keeps the controller in the "CLOSED" state to avoid knocking someone over .

Receiving "FRONT" causes a transition to the "OPEN" state .

OPEN State:

Receiving "FRONT", "REAR", or "BOTH" keeps the controller in the "OPEN" state.

Receiving "NEITHER" causes a transition back to the "CLOSED" state.

Example Sequence

Consider the following sequence of inputs: FRONT, REAR, NEITHER, FRONT, BOTH, NEITHER, REAR, and NEITHER. Starting from the "CLOSED" state, the controller would go through these states:

CLOSED (starting) -> OPEN -> OPEN -> CLOSED -> OPEN -> OPEN -> CLOSED -> CLOSED -> CLOSED