Introduction To Grammar in Theory of Computation
Last Updated : 23 Apr, 2025
In Theory of Computation, grammar refers to a formal system that defines how strings in a language are constructed. It plays a crucial role in determining the syntactic correctness of languages and forms the foundation for parsing and interpreting programming languages, natural languages, and other formal systems.
This article provides an in-depth exploration of:
- The types of grammars.
- Their components.
- The process of string derivation using grammar rules.
Grammar in Computation
Grammar is a formal system that defines a set of rules for generating valid strings within a language. It serves as a blueprint for constructing syntactically correct sentences or meaningful sequences in a formal language.
Grammar is basically composed of two basic elements:
- Terminal Symbols: Terminal symbols are those that are the components of the sentences generated using grammar and are represented using small case letters like a, b, c, etc.
- Non-Terminal Symbols: Non-terminal symbols are those symbols that take part in the generation of the sentence but are not the component of the sentence. Non-Terminal Symbols are also called Auxiliary Symbols and Variables. These symbols are represented using a capital letters like A, B, C, etc.
Representation of Grammar
Any Grammar can be represented by 4 tuples - <N, T, P, S>
- N - Finite Non-Empty Set of Non-Terminal Symbols.
- T - Finite Set of Terminal Symbols.
- P - Finite Non-Empty Set of Production Rules.
- S - Start Symbol (Symbol from where we start producing our sentences or strings).
Production Rules
A production or production rule in computer science is a rewrite rule specifying a symbol substitution that can be recursively performed to generate new symbol sequences. It is of the form α-> β where α is a Non-Terminal Symbol which can be replaced by β which is a string of Terminal Symbols or Non-Terminal Symbols.
Example 1
Consider Grammar G1 = <N, T, P, S>
T = {a,b} #Set of terminal symbols
1 2 3 4 5
P = { A -> Aa, A -> Ab, A -> a ,A -> b, A -> 𝜺} #Set of all production rules
S = {A} #Start Symbol
As the start symbol S is equivalent to A then we can produce Aa, Ab, a, b, 𝜺 strings. These strings can further produce strings where A can be replaced by the strings mentioned in the production rules. Hence this grammar can be used to produce strings of the form (a+b)*.
Derivation of Strings
A->a #using production rule 3
OR
A->Aa #using production rule 1
Aa->ba #using production rule 4
OR
A->Aa #using production rule 1
Aa->AAa #using production rule 1
AAa->bAa #using production rule 4
bAa->ba #using production rule 5
Example 2
Consider Grammar G2 = <N, T, P, S>
N = {A} #Set of non-terminals Symbols
T = {a} #Set of terminal symbols
1 2 3 4
P = {A -> Aa, A -> AAa, A -> a, A -> 𝜀} #Set of all production rules
S = {A} #Start Symbol
As the start symbol is S then we can produce Aa, AAa, a, which can further produce strings where A can be replaced by the Strings mentioned in the production rules and hence this grammar can be used to produce strings of form (a)*.
Derivation of Strings
A->a #using production rule 3
OR
A->Aa #using production rule 1
Aa->aa #using production rule 3
OR
A->Aa #using production rule 1
Aa->AAa #using production rule 1
AAa->Aa #using production rule 4
Aa->aa #using production rule 3
Equivalent Grammars
Two grammars are said to be equivalent if they generate the same language. For instance, if Grammar 1 and Grammar 2 both generate strings of the form (𝑎+𝑏)∗, they are considered equivalent.
Types of Grammars
There are several types of Grammar. We classify them on the basis mentioned below.
- Type of Production Rules: The form and complexity of the production rules define the grammar type, such as context-free, context-sensitive, regular, or unrestricted grammars.
- Number of Derivation Trees: The number of ways a string can be derived from the grammar. Ambiguous grammars have multiple derivation trees for the same string.
- Number of Strings: The size and nature of the language generated by the grammar.
Similar Reads
Introduction to Theory of Computation Automata theory, also known as the Theory of Computation, is a field within computer science and mathematics that focuses on studying abstract machines to understand the capabilities and limitations of computation by analyzing mathematical models of how machines can perform calculations.Why we study
7 min read
Last Minute Notes - Theory of Computation The Theory of Computation (TOC) is a critical subject in the GATE Computer Science syllabus. It involves concepts like Finite Automata, Regular Expressions, Context-Free Grammars, and Turing Machines, which form the foundation of understanding computational problems and algorithms.This article provi
13 min read
Theory of Computation (TOC) for GATE Theory of Computation (TOC) is a key subject in the GATE CSE exam. Here's a complete tutorial on the Theory of Computation for the GATE CSE exam. Whether you're revising or starting fresh, this tutorial will help you prepare effectively.If you have less time to study topic-wise in detail, you may re
4 min read
Equivalence in Theory of Computation In the theory of computation, equivalence refers to the idea that two computational models, representations, or systems can recognize or process the same language or solve the same class of problems. It is a fundamental concept that ensures consistency across different methods of computation.Why Is
4 min read
Relationship between grammar and language in Theory of Computation In the Theory of Computation, grammar and language are fundamental concepts used to define and describe computational problems. A grammar is a set of production rules that generate a language, while a language is a collection of strings that conform to these rules. Understanding their relationship i
4 min read