(a) Formal Languages

A formal language is a set of strings constructed from a finite set of symbols (called the alphabet) according to well-defined rules. These languages are used in theoretical computer science, automata theory, and programming languages.

Key points:

• Defined over an alphabet (Σ), e.g., Σ = {0,1}
• Consists of valid strings formed from Σ
• Used in compilers, parsers, and artificial intelligence
• Can be classified into regular, context-free, context-sensitive, and recursively enumerable languages

Example:

• L = {w | w starts with 'a' and ends with 'b'}, where Σ = {a, b}

(b) Regular Grammar

A regular grammar is a type of formal grammar used to generate regular languages. It consists of production rules that follow a specific format and can be represented by finite automata.

Key points:

• Production rules are right-linear or left-linear:
• Right-linear: A → aB or A → a
• Left-linear: A → Ba or A → a
• Corresponds to finite automata (DFA or NFA)
• Used in lexical analysis, pattern matching

Example:

• Grammar for strings containing only "a" followed by "b":

S → aA  
A → b | ε

(c) Push Down Automata (PDA)

A Push Down Automaton (PDA) is a computational model that extends finite automata by adding a stack. It can recognize context-free languages (CFLs).

Key points:

• Has states, input tape, stack, and transition functions
• Can push, pop, or read from the stack
• Used in parsing programming languages and syntax analysis
• More powerful than finite automata, but less powerful than Turing machines

(d) Linear Bounded Automata (LBA)

A Linear Bounded Automaton (LBA) is a type of Turing machine that operates on a finite tape, which is proportional to the input length. It recognizes context-sensitive languages (CSLs).

Key points:

• Unlike a Turing machine, LBA cannot use an infinite tape
• Can read, write, and move within input limits
• Used in complex language processing and decision problems
• More powerful than Push Down Automata (PDA) but less than Turing machines

Example:

• LBA can recognize L = {aⁿbⁿcⁿ | n > 0}, which is not possible with PDA.

Chomsky Hierarchy of Grammars

The Chomsky hierarchy, introduced by Noam Chomsky in 1956, classifies formal languages into four types based on their grammatical restrictions and the computational power required to recognize them. The four levels are Type-0, Type-1, Type-2, and Type-3.

1. Type-0: Recursively Enumerable Languages (Turing Machine)

• Definition: These languages are generated by unrestricted grammars, meaning there are no constraints on the production rules.
• Grammar Rule Format: α→β,where α and β are any strings of terminals and non-terminals, with α≠ϵ.\alpha \to \beta, \quad \text{where } \alpha \text{ and } \beta \text{ are any strings of terminals and non-terminals, with } \alpha \neq \epsilon.
• Machine Used: Turing Machine (TM)
• Power: Most powerful model; can recognize all computable languages.
• Example:
• L = {aⁿbⁿcⁿdⁿ | n > 0}
• No finite automaton or pushdown automaton can recognize this language.

2. Type-1: Context-Sensitive Languages (Linear Bounded Automaton)

• Definition: These languages are generated by context-sensitive grammars (CSG), where production rules must maintain or increase string length.
• Grammar Rule Format: αAβ→αγβ,where ∣γ∣≥∣A∣.\alpha A \beta \to \alpha \gamma \beta, \quad \text{where } |\gamma| \geq |A|.
• Machine Used: Linear Bounded Automaton (LBA) (a restricted Turing Machine)
• Power: Can recognize more languages than PDA but is weaker than a Turing Machine.
• Example:
• L = {aⁿbⁿcⁿ | n > 0}
• A PDA cannot recognize this, but an LBA can.

3. Type-2: Context-Free Languages (Push Down Automaton)

• Definition: These languages are generated by context-free grammars (CFGs), where every production rule has a single non-terminal on the left-hand side.
• Grammar Rule Format: A→γ,where A is a non-terminal and γ is a string of terminals and/or non-terminals.A \to \gamma, \quad \text{where } A \text{ is a non-terminal and } \gamma \text{ is a string of terminals and/or non-terminals.}
• Machine Used: Push Down Automaton (PDA)
• Power: Used in compilers and programming language syntax.
• Example:
• L = {aⁿbⁿ | n > 0}
• Can be recognized using a PDA with a stack.

4. Type-3: Regular Languages (Finite Automaton)

• Definition: These languages are generated by regular grammars, which follow strict left-linear or right-linear rules.
• Grammar Rule Format:
• Right-linear: A→aBA \to aB or A→aA \to a
• Left-linear: A→BaA \to Ba or A→aA \to a
• Machine Used: Finite Automaton (DFA/NFA)
• Power: Least powerful, but fastest and most efficient.
• Example:
• L = {w | w contains an even number of 0s}
• Can be recognized by a Deterministic Finite Automaton (DFA).

The Chomsky hierarchy helps classify languages based on their complexity and the automata needed to recognize them. Higher types (Type-0 and Type-1) are more powerful but less efficient, while Type-3 (Regular Languages) are efficient but less powerful.

Theory of Automata:- Branch of Computer Science Study of Simple Machine that process. input and: fellow rules to produce output or Make Decisions. help us understand how Computers & programs Work 

at a basic level by breaking down task into procedure, Theory of Automata helps us understand how machines process and recognize patterns, forming the basis for computer algorithms

Applications of Automata Theory

1. Compiler Design: Lexical analysis, syntax checking, and parsing.
2. Artificial Intelligence: Natural language processing (NLP).
3. Robotics & Control Systems: Finite state control systems.
4. Text Search & Pattern Matching: Used in search engines and text editors.
5. Cryptography & Security: Automata-based cryptographic protocols.

A Finite Automaton (FA) is a theoretical machine used in formal language theory and automata theory to model computations. It is a type of automaton with a finite number of states, which transitions between those states based on input symbols. It is typically used to recognize patterns or process input strings based on predefined rules.

States (Q): A finite set of states. One of these states is the start state and possibly one or more accepting states.

Alphabet (Σ): A finite set of symbols that the automaton can process as input (e.g., {a, b}).

Transition Function (δ): A function that describes how the automaton moves from one state to another based on input symbols. For a DFA, this is a function from the current state and an input symbol to a new state.

Start State (q₀): The state where the automaton begins its computation.

Accepting/Final States (F): A set of states where if the automaton halts, the input string is considered accepted.

Transition Graph (in Automata Theory)

A transition graph (or transition diagram) is a graphical representation of a finite automaton (FA) where the states are represented as nodes (vertices), and the transitions between the states are represented as directed edges (arcs). This is a visual way of describing how an automaton processes an input string based on its state transitions.

M=(Q,Σ,δ,q0​,F)

Q: The finite set of states.

Σ (Sigma): The alphabet (a finite set of symbols that the automaton reads).

δ (delta): The transition function that describes the state transitions. This is a function that takes a state and an input symbol and returns the next state.

q₀: The start state (an element of Q, where the automaton begins processing input).

F: The set of accepting states (a subset of Q, where the automaton ends after processing the input and accepts the string if it is in F).

Components of a Transition Graph:

1. States (Nodes): Each state in the automaton is represented as a node (or vertex). The set of states is typically denoted as Q.
2. Alphabet (Edges): The edges of the graph represent the transitions based on the input symbols from the alphabet Σ.
3. Start State: The state where the automaton begins its computation is typically represented by an incoming arrow pointing to it (without any other node pointing to it).
4. Accepting States: The accepting (or final) states are often shown with a double circle to distinguish them from non-accepting states.
5. Transitions (Edges): The edges are labeled with the input symbols that trigger the transitions. Each edge represents a change from one state to another based on an input symbol.

A transition graph is a visual representation of a finite automaton (FA) where states are nodes, and transitions are edges labeled with input symbols.

It helps to understand how an automaton processes input strings and transitions from one state to another based on those inputs.

It is a powerful way to visualize the operation of both deterministic and non-deterministic finite automata.

Formal Definition of Finite Automaton (FA)

A Finite Automaton (FA) is formally defined as a 5-tuple:

M=(Q,Σ,δ,q0,F)

Q: A finite set of states.

Σ: A finite set of input symbols (alphabet).

δ: A transition function that dictates how the automaton moves between states based on the input symbol.

q₀: The starting state from which the automaton begins its computation.

F: A set of accepting (final) states where the automaton accepts the string if it ends in one of these states.

finite automaton (FA) is a mathematical model used to recognize certain types of languages. It is defined by a 5-tuple consisting of states, an alphabet, a transition function, a start state, and a set of accepting states.

Non-Deterministic Finite Automaton (NFA)

A Non-Deterministic Finite Automaton (NFA) is a type of finite automaton where, for a given state and input symbol, the automaton may have multiple possible transitions, or even no transition at all. This means that at any point in time, the automaton might "choose" from multiple states to transition into, rather than being forced to follow exactly one path as in a Deterministic Finite Automaton (DFA).

Formal Definition of NFA

An NFA is formally defined as a 5-tuple:

M=(Q,Σ,δ,q0,F)

Q: A finite set of states.

Σ (Sigma): A finite set of input symbols, known as the alphabet.

δ (delta): The transition function.

• This is the function that defines how the automaton moves from one state to another based on the input symbol. In an NFA, the transition function maps a state and an input symbol to a set of possible states (since there may be more than one possible next state).
• Mathematically, δ is defined as:
• δ:Q×Σ→2pwrQ

q₀: The start state.

F: The set of accepting states (or final states).

Key Characteristics of an NFA:

• Multiple Transitions: For a given state and input symbol, the NFA may have multiple possible next states.
• ε (epsilon) Transitions: An NFA can have ε-transitions (also called epsilon or empty string transitions), which allow the automaton to move from one state to another without consuming any input symbol (i.e., the input string remains unchanged).
• Non-determinism: At any point, the NFA may transition into more than one state or no state at all, making its behavior non-deterministic.

Difference Between DFA and NFA:

• DFA (Deterministic Finite Automaton): For every state and input symbol, there is exactly one possible transition to another state. There is no ambiguity or choice involved in the transitions.
• NFA (Non-Deterministic Finite Automaton): For a given state and input symbol, there can be multiple possible next states or no transition at all. The NFA can "choose" a path (even though it’s non-deterministic).

NFA to DFA Conversion:

It’s important to note that every NFA can be converted into an equivalent DFA. The process of converting an NFA into a DFA is known as the subset construction or powerset construction algorithm. The resulting DFA will accept the same language as the NFA, but it will not have multiple possible transitions for the same input symbol.

A Deterministic Finite Automaton (DFA) is a type of finite automaton that has a specific structure in which, for each state and input symbol, there is exactly one transition to a new state. This means that the behavior of the automaton is deterministic: at any given state and input symbol, the next state is uniquely determined.

Formal Definition of DFA

A DFA is formally defined as a 5-tuple:

M=(Q,Σ,δ,q0,F)

Q: A finite set of states.

• These are the different conditions or configurations the automaton can be in at any given moment.

Σ (Sigma): A finite set of input symbols, also known as the alphabet.

• The symbols that the automaton reads as input.

δ (delta): The transition function.

• This defines how the automaton moves between states based on the current state and the input symbol.
• In a DFA, the transition function δ is a total function, meaning that for each state and each symbol in the alphabet, there is exactly one state to transition to. Mathematically:
• δ:Q×Σ→Q
• This means that for every state q in Q and each input symbol a in Σ, the transition function δ will return exactly one state.

q₀: The start state.

• This is the state where the automaton starts processing the input string. It is an element of Q.

F: The set of accepting states (or final states).

• This is a subset of Q. If the automaton ends in one of these accepting states after processing the entire input string, the string is accepted.

Key Characteristics of a DFA:

• Determinism: For each state and input symbol, there is exactly one possible next state.
• No ε-transitions: Unlike NFAs, DFAs cannot transition without consuming an input symbol (no epsilon transitions).
• Memory of past states: A DFA has memory of its current state, and its behavior is completely determined by the current state and input symbol.

Properties of a DFA:

1. Deterministic: Every input symbol leads to exactly one state from the current state.
2. Finite State: The DFA has a finite number of states.
3. Efficiency: Since the transitions are deterministic, it is computationally efficient to simulate the DFA.
4. No Backtracking: The DFA does not revisit previous states once it has moved forward.