- 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
Context-Free Grammar vs Regular Grammar
In this chapter, we will cover an important concept in grammar: the context-free grammar and regular grammars. We will highlight the differences between regular grammars and context-free grammars.
We have already learnt what regular and context-free grammars are and how to build derivation trees (also known as parse trees). Now, let's explore how these two types of grammars differ.
Regular Grammar
A regular grammar is formally represented as a four-part system, written as: (V, T, P, S).
Where,
- V − Represents the set of non-terminals. These are like placeholders or variables in our grammar, typically denoted by uppercase letters.
- T − Represents the set of terminals. These are the actual input symbols of our language, often denoted by lowercase letters.
- P − Represents the set of production rules. These rules dictate how we can replace non-terminals with other symbols (terminals or non-terminals) to generate strings in our language.
- S − Represents the start symbol, a special non-terminal from which all derivations begin.
Context-Free Grammar
Just like regular grammar, a context-free grammar is also represented as a four-part system: (V, T, P, S). Here,
- V − Represents the non-terminals (uppercase letters).
- T − Represents the terminals (lowercase letters).
- P − Represents the production rules, although the rules for writing these productions differ between context-free and regular grammars (more on that later).
- S − Represents the start symbol, the starting point of our derivations.
The definitions seem almost identical! The key difference lies in the production rules, which we'll discuss in detail further on.
Parsing
Regular Grammar and Parsing − Here's a crucial difference, the regular grammars are not suitable for parsing. Remember how parsing involves breaking down an expression into its individual tokens (like we did with parse trees) Regular grammars lack the power to handle this process effectively.
Context-Free Grammar and Parsing − On the other hand, context-free grammars are perfectly suited for parsing. This capability makes them incredibly useful for analyzing the structure of languages.
This fundamental difference in parsing ability has significant implications for the applications of these grammars.
Automata Construction
Regular Grammars − We use regular grammars to construct Deterministic Finite Automata (DFAs). DFAs are computational models used to recognize patterns within strings.
Context-Free Grammars − We use context-free grammars to construct Pushdown Automata (PDAs). PDAs are more powerful than DFAs and can handle more complex languages. We'll explore PDAs in detail in future articles.
Representing Programming Languages
Regular Grammars − While useful for simpler tasks, regular grammars cannot fully represent the syntax of any programming language. Their limited expressive power restricts their use in this domain.
Context-Free Grammars − Context-free grammars, with their ability to be parsed and their more flexible production rules, can fully represent the syntax of any programming language. This feature makes them essential for compiler construction, where understanding the structure of programming languages is paramount.
Chomsky Hierarchy and Subsets
Chomsky Hierarchy − The Chomsky Hierarchy classifies grammars based on their generative power.
- Regular grammars are classified as Type 3 grammars.
- Context-free grammars are classified as Type 2 grammars.
Subsets − Importantly, every regular grammar can be considered a context-free grammar. In other words, regular grammars form a subset of context-free grammars.
However, the reverse is not true. Not every context-free grammar can be simplified into a regular grammar. This difference in expressiveness highlights the broader capabilities of context-free grammars.
Production Rules
Regular Grammar Production Rules − The production rules in regular grammars are highly restricted. They follow one of these forms −
- A → a:A non-terminal (A) can be replaced by a single terminal (a).
- A → aB: non-terminal (A) can be replaced by a terminal (a) followed by another non-terminal (B).
- A → Ba: non-terminal (A) can be replaced by a non-terminal (B) followed by a terminal (a).
Context-Free Grammar Production Rules − Context-free grammars offer much more flexibility in their production rules. The right-hand side of a production rule can be −
- A single terminal: A → a
- A single non-terminal: A → B
- A combination of terminals and non-terminals: A → aBcD
This flexibility in combining terminals and non-terminals gives context-free grammars their power to represent complex language structures.
Conclusion
In this chapter, we highlighted the differences between context-free grammar or CFG with regular grammars. Context-free grammars are more powerful than regular grammars due to their flexible production rules and parsing ability. They can represent programming language syntax and be used in compiler construction.