- Automata Theory - Applications
- Automata Terminology
- Basics of String in Automata
- Set Theory for Automata
- Finite Sets and Infinite Sets
- Algebraic Operations on Sets
- Relations Sets in Automata Theory
- Graph and Tree in Automata Theory
- Transition Table in Automata
- What is Queue Automata?
- Compound Finite Automata
- Complementation Process in DFA
- Closure Properties in Automata
- Concatenation Process in DFA
- Language and Grammars
- Language and Grammar
- Grammars in Theory of Computation
- Language Generated by a Grammar
- Chomsky Classification of Grammars
- Context-Sensitive Languages
- Finite Automata
- What is Finite Automata?
- Finite Automata Types
- Applications of Finite Automata
- Limitations of Finite Automata
- Two-way Deterministic Finite Automata
- Deterministic Finite Automaton (DFA)
- Non-deterministic Finite Automaton (NFA)
- NDFA to DFA Conversion
- Equivalence of NFA and DFA
- Dead State in Finite Automata
- Minimization of DFA
- Automata Moore Machine
- Automata Mealy Machine
- Moore vs Mealy Machines
- Moore to Mealy Machine
- Mealy to Moore Machine
- Myhill–Nerode Theorem
- Mealy Machine for 1’s Complement
- Finite Automata Exercises
- Complement of DFA
- Regular Expressions
- Regular Expression in Automata
- Regular Expression Identities
- Applications of Regular Expression
- Regular Expressions vs Regular Grammar
- Kleene Closure in Automata
- Arden’s Theorem in Automata
- Convert Regular Expression to Finite Automata
- Conversion of Regular Expression to DFA
- Equivalence of Two Finite Automata
- Equivalence of Two Regular Expressions
- Convert Regular Expression to Regular Grammar
- Convert Regular Grammar to Finite Automata
- Pumping Lemma in Theory of Computation
- Pumping Lemma for Regular Grammar
- Pumping Lemma for Regular Expression
- Pumping Lemma for Regular Languages
- Applications of Pumping Lemma
- Closure Properties of Regular Set
- Closure Properties of Regular Language
- Decision Problems for Regular Languages
- Decision Problems for Automata and Grammars
- Conversion of Epsilon-NFA to DFA
- Regular Sets in Theory of Computation
- Context-Free Grammars
- Context-Free Grammars (CFG)
- Derivation Tree
- Parse Tree
- Ambiguity in Context-Free Grammar
- CFG vs Regular Grammar
- Applications of Context-Free Grammar
- Left Recursion and Left Factoring
- Closure Properties of Context Free Languages
- Simplifying Context Free Grammars
- Removal of Useless Symbols in CFG
- Removal Unit Production in CFG
- Removal of Null Productions in CFG
- Linear Grammar
- Chomsky Normal Form (CNF)
- Greibach Normal Form (GNF)
- Pumping Lemma for Context-Free Grammars
- Decision Problems of CFG
- Pushdown Automata
- Pushdown Automata (PDA)
- Pushdown Automata Acceptance
- Deterministic Pushdown Automata
- Non-deterministic Pushdown Automata
- Construction of PDA from CFG
- CFG Equivalent to PDA Conversion
- Pushdown Automata Graphical Notation
- Pushdown Automata and Parsing
- Two-stack Pushdown Automata
- Turing Machines
- Basics of Turing Machine (TM)
- Representation of Turing Machine
- Examples of Turing Machine
- Turing Machine Accepted Languages
- Variations of Turing Machine
- Multi-tape Turing Machine
- Multi-head Turing Machine
- Multitrack Turing Machine
- Non-Deterministic Turing Machine
- Semi-Infinite Tape Turing Machine
- K-dimensional Turing Machine
- Enumerator Turing Machine
- Universal Turing Machine
- Restricted Turing Machine
- Convert Regular Expression to Turing Machine
- Two-stack PDA and Turing Machine
- Turing Machine as Integer Function
- Post–Turing Machine
- Turing Machine for Addition
- Turing Machine for Copying Data
- Turing Machine as Comparator
- Turing Machine for Multiplication
- Turing Machine for Subtraction
- Modifications to Standard Turing Machine
- Linear-Bounded Automata (LBA)
- Church's Thesis for Turing Machine
- Recursively Enumerable Language
- Computability & Undecidability
- Turing Language Decidability
- Undecidable Languages
- Turing Machine and Grammar
- Kuroda Normal Form
- Converting Grammar to Kuroda Normal Form
- Decidability
- Undecidability
- Reducibility
- Halting Problem
- Turing Machine Halting Problem
- Rice's Theorem in Theory of Computation
- Post’s Correspondence Problem (PCP)
- Types of Functions
- Recursive Functions
- Injective Functions
- Surjective Function
- Bijective Function
- Partial Recursive Function
- Total Recursive Function
- Primitive Recursive Function
- μ Recursive Function
- Ackermann’s Function
- Russell’s Paradox
- Gödel Numbering
- Recursive Enumerations
- Kleene's Theorem
- Kleene's Recursion Theorem
- Advanced Concepts
- Matrix Grammars
- Probabilistic Finite Automata
- Cellular Automata
- Reduction of CFG
- Reduction Theorem
- Regular expression to ∈-NFA
- Quotient Operation
- Parikh’s Theorem
- Ladner’s Theorem
Regular Expressions, Regular Grammar and Regular Languages
Read this chapter to get a clear understanding of two important concepts in formal languages and automata theory the concept of Regular Expressions and Regular Grammars. Both are crucial in defining and manipulating strings within formal languages, but they do it through different approaches. Let us understand the concepts one by one for a better understanding.
The Basics of Grammar
To get the regular expression and regular grammars, we must focus the concept of grammars at first. In simple terms, a grammar acts as a set of rules governing the structure and formation of sentences within a language. In normal human languages, Grammar dictates how words combine to form meaningful sentences, ensuring clear communication.
Similarly, in the computer science, grammars provide a framework for constructing and interpreting languages, specifically programming languages. The mathematical model of grammar is actually in writing computer languages to ensure structured and unambiguous programming, enabling computers to understand and execute instructions effectively.
Chomsky Hierarchy and Formal Grammars
If we see the Chomskys hierarchy for formal grammars, Chomsky proposed a hierarchical classification of grammars known as the Chomsky Hierarchy, categorizing them based on their generative power.
Noam Chomsky introduced a mathematical model of grammar that can be used for writing computer languages. He identified four types of grammars −
- Type 0 Grammar (Unrestricted Grammar),
- Type 1 Grammar (Context-Sensitive Grammar),
- Type 2 Grammar (Context-Free Grammar), and
- Type 3 Grammar (Regular Grammar).
We are discussing on type 3 Grammars or the regular grammars, form the basis for regular expressions. These grammars have a specific structure in their production rules. This limits the types of languages they can define.
The languages generated by regular grammars are called Regular Languages. These languages can be represented using finite state machines (FSA), which are used to design Type 3 grammars.
Regular Grammar and Regular Languages
The regular grammar is a type of formal grammar used to describe a regular language.
Regular languages are the simplest in Chomsky's hierarchy of formal languages which makes them easy to understand and implement in computer programs.
Regular Expressions are particularly useful in tasks like −
- Lexical analysis in compilers − Identifying the basic building blocks (keywords, identifiers, operators) of a program.
- Pattern matching in text editors − Finding specific patterns of characters within a text document.
- Validating input formats − Ensuring that user input conforms to a predefined format.
Types of Regular Grammar
Regular grammars can be further classified into two main types −
Right Linear Grammar
In right linear grammars, the non-terminal symbol in a production rule always appears at the rightmost position. For example −
$$\mathrm{A \:→\: xB \:\:or\:\: A \:→ \:x}$$
where 'A' and 'B' are non-terminal symbols and 'x' is a terminal symbol.
Left Linear Grammar
In left linear grammars, the non-terminal symbol resides in the leftmost position. For example −
$$\mathrm{A \:→\: Bx \:\:or\:\: A\: →\: x}$$
Where 'A' and 'B' are non-terminal symbols and 'x' is a terminal symbol.
Difference between Regular Expressions and Regular Grammars
The following table compares and contrasts the important features of Regular Expressions and Regular Grammars −
| Feature | Regular Expressions | Regular Grammars |
|---|---|---|
| Purpose | Represents patterns within strings | Defines rules for generating regular languages |
| Notation | Algebraic, using symbols and operators | Production rules with variables and terminals |
| Representation | Concise, often shorter for complex patterns | More verbose, especially for complex patterns |
| Power | Equivalent to regular grammars (Type 3) | Equivalent to regular expressions (Type 3) |
Regular expressions and regular grammars both have the same expressive power, representing the same set of languages. Regular expressions provide a more concise and readable way to represent patterns, while regular grammars offer a formal framework for understanding the structure of regular languages.
Conclusion
In this chapter, we explained the concept of regular language and regular grammars. In short, both regular expressions and regular grammars are powerful tools in automata theory for defining and manipulating patterns of text.
In addition, we understood the regular grammars that provide a formal and structured framework, particularly useful in compiler design and formal language theory.