Introduction To Grammar in Theory of Computation

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What is Grammar in Computation?

In the literary sense of the term, grammars denote syntactical rules for conversation in natural languages. Linguistics have attempted to define grammars since the inception of natural languages like English, Sanskrit, Mandarin, etc.

The theory of formal languages finds its applicability extensively in the fields of Computer Science. Noam Chomsky gave a mathematical model of grammar in 1956 which is effective for writing computer languages.

Representation of Grammar

A grammar G can be formally written as a 4-tuple (N, T, S, P) where −

• N or V_N is a set of variables or non-terminal symbols.
• T or ∑ is a set of Terminal symbols.
• S is a special variable called the Start symbol, S ∈ N
• P is Production rules for Terminals and Non-terminals. A production rule has the form α → β, where α and β are strings on V_N ∪ ∑ and least one symbol of α belongs to V_N.

Example 1

Grammar G1 −

({S, A, B}, {a, b}, S, {S → AB, A → a, B → b})

Here,

• S, A, and B are Non-terminal symbols;
• a and b are Terminal symbols
• S is the Start symbol, S ∈ N
• Productions, P : S → AB, A → a, B → b

Example 2

Grammar G2 −

(({S, A}, {a, b}, S,{S → aAb, aA → aaAb, A → ε } )

Here,

• S and A are Non-terminal symbols.
• a and b are Terminal symbols.
• ε is an empty string.
• S is the Start symbol, S ∈ N
• Production P : S → aAb, aA → aaAb, A → ε

Basic Elements of Grammar

Grammar is composed of two basic elements

Terminal Symbols - Terminal symbols are the components of the sentences that are generated using grammar and are denoted using small case letters like a, b, c etc.

Non-Terminal Symbols - Non-Terminal Symbols take part in the generation of the sentence but are not the component of the sentence. These types of symbols are also called Auxiliary Symbols and Variables. They are represented using a capital letter like A, B, C, etc.

Example 1

Consider a grammar

$$\mathrm{G \:=\: (V ,\: T ,\: P ,\: S)}$$

Where,

• V = { S , A , B } Non-Terminal symbols
• T = { a , b } Terminal symbols
• P = { S → ABa , A → BB , B → ab , AA → b } Production rules
• S = { S } Start symbol

Example 2

Consider a grammar

$$\mathrm{G \:=\: (V, \:T, \:P, \:S)}$$

Where,

• V = {S, A, B} non terminal symbols
• T = { 0,1} terminal symbols
• P = { S → A1B A → 0A| ε B → 0B| 1B| ε } Production rules
• S = {S} start symbol.

Types of grammar

The different types of grammar −

The diagram representing the types of grammar in the theory of computation (TOC) is as follows −

Derivations from a Grammar

Strings may be derived from other strings using the productions in a grammar. If a grammar G has a production α → β, we can say that x α y derives x β y in G. This derivation is written as −

x α y ⇒G x β y

Example

Let us consider the grammar −

G2 = ({S, A}, {a, b}, S, {S → aAb, aA → aaAb, A → ε } )

Some of the strings that can be derived are −

S ⇒ aAb using production S → aAb

⇒ aaAbb using production aA → aAb

⇒ aaaAbbb using production aA → aaAb

⇒ aaabbb using production A → ε