Context Free Grammars vs. Context Sensitive Grammars

Attribute	Context Free Grammars	Context Sensitive Grammars
Definition	Consist of a set of production rules that describe all possible strings in a language	Consist of a set of production rules where the left-hand side can be replaced by the right-hand side in certain contexts
Rule Format	Rules are of the form A -> α, where A is a non-terminal symbol and α is a string of terminals and non-terminals	Rules are of the form αAβ -> αγβ, where A is a non-terminal symbol and α, β, γ are strings of terminals and non-terminals
Derivation	Derivations are done by replacing non-terminals with the right-hand side of production rules	Derivations are done by replacing non-terminals with the right-hand side of production rules in specific contexts
Language Recognition	Can recognize regular languages and some context-free languages	Can recognize all context-free languages and some context-sensitive languages

Introduction

Context Free Grammars (CFGs) and Context Sensitive Grammars (CSGs) are two types of formal grammars used in the field of theoretical computer science to describe the syntax of programming languages, natural languages, and other formal languages. While both types of grammars are used to generate strings of symbols, they differ in terms of the rules and restrictions that govern their production.

Definition

A Context Free Grammar is a formal grammar in which each production rule consists of a non-terminal symbol on the left-hand side and a sequence of terminals and/or non-terminals on the right-hand side. The rules in a CFG are context-free, meaning that the left-hand side non-terminal can be replaced by the right-hand side sequence of symbols regardless of the context in which it appears. This property makes CFGs easy to analyze and parse using algorithms such as the CYK algorithm.

On the other hand, a Context Sensitive Grammar is a formal grammar in which the production rules are more restrictive than in a CFG. In a CSG, the left-hand side of a production rule can be replaced by the right-hand side sequence of symbols only in specific contexts defined by the grammar. This context sensitivity allows CSGs to describe more complex languages that cannot be captured by CFGs, such as natural languages with agreement constraints.

Expressiveness

One of the key differences between CFGs and CSGs is their expressiveness in terms of the languages they can generate. CFGs are capable of generating a wide range of languages, including regular languages, context-free languages, and some context-sensitive languages. However, there are certain languages that cannot be described by a CFG, such as languages with nested structures or languages with cross-serial dependencies.

CSGs, on the other hand, are more powerful than CFGs in terms of expressiveness. They can generate a larger class of languages, including all context-free languages and many context-sensitive languages. This increased expressiveness comes at the cost of complexity, as CSGs are harder to analyze and parse than CFGs due to their context-sensitive rules.

Formal Definition

In a CFG, the production rules are of the form A → α, where A is a non-terminal symbol and α is a string of terminals and/or non-terminals. The start symbol of the grammar is used to derive the language by applying production rules until a string of terminals is generated. The language generated by a CFG is the set of all strings that can be derived from the start symbol.

In a CSG, the production rules are of the form αAβ → αγβ, where A is a non-terminal symbol, α and β are strings of terminals and/or non-terminals, and γ is a non-empty string. The context-sensitive nature of the rules allows for the replacement of A by γ only in the context defined by α and β. This additional constraint makes CSGs more powerful than CFGs in terms of language generation.

Algorithmic Complexity

One of the practical implications of the differences between CFGs and CSGs is their algorithmic complexity. Parsing a string generated by a CFG can be done efficiently using algorithms such as the CYK algorithm, which has a time complexity of O(n^3) for a string of length n. This efficiency is due to the context-free nature of CFGs, which allows for bottom-up parsing.

On the other hand, parsing a string generated by a CSG is more challenging due to the context-sensitive nature of the grammar rules. While there are algorithms for parsing CSGs, such as the Earley parser, they are generally more complex and have higher time complexity than algorithms for CFGs. This increased complexity is a trade-off for the increased expressiveness of CSGs.

Applications

CFGs and CSGs have various applications in computer science and linguistics. CFGs are commonly used to describe the syntax of programming languages, where the rules of the grammar correspond to the syntax of the language. They are also used in natural language processing for tasks such as parsing and machine translation.

CSGs, on the other hand, are used in more specialized applications where the context-sensitive nature of the grammar is necessary. For example, CSGs are used in formal language theory to study the properties of context-sensitive languages and in computational linguistics to model linguistic phenomena that cannot be captured by CFGs. Overall, both types of grammars play a crucial role in the study of formal languages and their applications.