Linear Grammar in Automata Theory

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We have explained different types of grammars in automata including regular grammar and contextfree grammar. Related to regular grammar, there is another class of grammars called the linear grammars.

Linear grammar plays a crucial role in the theory of computation. In this chapter, we will explore the fundamentals of linear grammar, its types, and provide practical examples for a better understanding.

Grammar in Automata Theory

Grammar in automata theory refers to a set of rules that define the structure of strings in a language. These rules are used to generate all possible strings in a particular language, thereby defining the language itself.

It has several types, including −

• Type 0 (Unrestricted Grammar) − These have no restrictions on production rules.
• Type 1 (Context-Sensitive Grammar) − Here, the length of the output string must be greater than or equal to the length of the input string.
• Type 2 (Context-Free Grammar) − Production rules have a single non-terminal symbol on the left side.
• Type 3 (Regular Grammar) − These have strict production rules with either a single terminal or a terminal followed by a non-terminal.

Linear grammar falls under Type 3 (Regular Grammar), which has the idea of linear production rules.

What is Linear Grammar?

A linear grammar is a type of formal grammar where each production rule has at most one non-terminal symbol on the right-hand side of the production.

Linear grammar is significant because it describes a subset of regular languages and plays an important role in the design and analysis of finite automata.

Left-Linear and Right-Linear Grammar

Linear grammar can be further divided into two categories −

• Left-Linear Grammar − In this type, the non-terminal appears at the left end of the production rule. For example, a rule like is left-linear because the non-terminal appears on the left.
• Right-Linear Grammar − Here, the non-terminal appears at the right end of the production rule. For example, is right-linear as the non-terminal is on the right.

Both left-linear and right-linear grammars generate regular languages, which can be recognized by finite automata.

Examples of Linear Grammar

Let us see the idea of linear grammar through examples. We will see both the left and right linear grammars one by one for a better understanding.

Example of Left-Linear Grammar

Consider the following left-linear grammar production rules −

$$\mathrm{S \: \rightarrow \: Ab}$$

$$\mathrm{A \: \rightarrow \: Bb}$$

$$\mathrm{B \: \rightarrow \: a}$$

• The start symbol S produces Ab.
• The non-terminal A produces Bb.
• Finally, B produces a.

Example of Right-Linear Grammar

Consider the following right-linear grammar production rules −

$$\mathrm{S \: \rightarrow \: aA}$$

$$\mathrm{A \: \rightarrow \: bB}$$

$$\mathrm{B \: \rightarrow \: c}$$

• The start symbol S produces aA.
• The non-terminal A produces bB.
• Finally, B produces c.

Let us take another practical example. Consider a practical scenario where you need to design a finite automaton to recognize a language consisting of strings ending with a particular character, say "b". You could create a right-linear grammar for this language with the following rules −

$$\mathrm{S \: \rightarrow \: aS}$$

$$\mathrm{S \: \rightarrow \: b}$$

This grammar will generate strings like "ab", "aab", "aaab", etc., all ending with the terminal 'b'.

The Importance of Linear Grammar

Linear grammar is useful in several cases −

• Simplicity in Design − Linear grammar's simplicity makes it ideal for designing simple compilers and parsers, where efficiency is key.
• Finite Automata Representation − Linear grammars can easily be converted to finite automata, which are used in pattern matching, lexical analysis, and designing algorithms.
• Theoretical Foundations − Understanding linear grammar helps in understanding more complex topics like context-free grammars and pushdown automata, which are essential for advanced computation theory.

Conclusion

Linear grammar, particularly left-linear and right-linear grammars, is a fundamental concept in the theory of computation. In this chapter, we explored the concept of linear grammar including left and right linear grammars with examples. We also analysed where we need such grammars in automata theory.