Context-Sensitive Languages

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Context-sensitive languages (CSLs) bridge the gap between the well-studied context-free languages and the more complex recursively enumerable languages. In this chapter, we will cover the concepts of context-sensitive languages, exploring their definitions, properties, and relationships to other language classes in the Chomsky hierarchy.

Context-Free Languages: Foundation for Understanding CSLs

Before we enter context-sensitive languages, it's essential to understand the context-free languages (CFLs), as they form the foundation for our discussion.

What is Context-Free Grammar?

A context-free language is generated by a context-free grammar (CFG). In a CFG, production rules have the form: A → X, Where −

• A is a variable (non-terminal)
• X is any string of terminals or variables

Properties of Context-Free Languages

The key characteristic of CFLs is that the replacement of A with X is independent of the surrounding context. This property gives CFLs their name they are "free" of context constraints.

CFLs correspond to pushdown automata (PDAs) in the Chomsky hierarchy, which are more powerful than finite automata but less powerful than linear-bounded automata.

Context-Sensitive Languages: More Powerful CFLs

The context-sensitive languages extend the concept of CFLs by allowing production rules to depend on the context in which variables appear. This seemingly small change leads to a significant increase in expressive power. Let us understand CSG in greater detail.

What is Context-Sensitive Grammars (CSGs)?

A context-sensitive grammar has production rules of the form: αAβ → αXβ, where −

• α, β are strings of terminals and/or variables (can be empty)
• A is a variable
• X is a non-empty string of terminals or variables

Properties of Context-Sensitive Languages

Listed below are some of the important properties of context-sensitive languages −

• Context Preservation − The production process maintains the same context (α and β) on both sides, ensuring that the replacement of A with X only occurs within the defined context.
• Non-Contracting − The grammar's property X cannot be empty, ensuring it doesn't reduce string length during derivation. However, the start variable S can generate an empty string if it's part of the language.
• Increased Expressive Power − CSLs can describe patterns that CFLs cannot, such as matching multiple repeated substrings.

The Chomsky Hierarchy and CSLs

Context-sensitive languages are located at the third level of the Chomsky hierarchy, between context-free and recursively enumerable languages.

The relationship between different classes of languages is as follows −

• Regular Languages ⊂ Context-Free Languages ⊂ Context-Sensitive Languages ⊂ Recursively Enumerable Languages.
• CSLs have the added power to describe patterns that CFLs cannot.

Now let us see CSL through an example.

Example of a Language That is context-free or Not

A classic example of a language that is context-sensitive but not context-free is: L = {abc | n ≥ 0}

The language is composed of strings with equal numbers of a's, b's, and c's, which cannot be generated by context-free grammars due to their inability to count and ensure equal numbers of three different symbols.

To illustrate the power of context-sensitive grammars, let's construct a CSG that generates the language L = {abc | n ≥ 0}.

The production rules are as follows −

• S → ε (to handle the case n = 0)
• S → S'
• S' → ABC
• AB → BAB
• BA → ACA
• CA → CB
• CB → AB
• Bb → Bb
• A → a
• B → b
• C → c

This grammar works through a series of transformations −

• Rule 1 handles the empty string case (n = 0).
• Rules 2-3 initialize the string with one occurrence of each variable (A, B, C).
• Rules 4-7 allow for the rearrangement of variables. These rules effectively "bubble" the A's to the left and the C's to the right, maintaining the correct order and equal numbers of each variable.
• Rule 8 is crucial: it ensures that B's are replaced with lowercase b's only when adjacent to an existing lowercase b. This prevents premature conversion of B's and maintains the structure of the string.
• Rules 9-11 convert the variables to their corresponding terminals once they are in the correct position.

The grammar maintains the equal numbers of A's, B's, and C's while rearranging them into the correct order. The context-sensitive nature of the rules allows for this precise control over the string's structure.

Conclusion

Context-sensitive languages offer a significant improvement over context-free languages by considering surrounding context in production rules. It enables the description of complex patterns beyond the reach of CFLs.

In this chapter, we first highlighted the problem with context-free languages, followed by the definition and properties of context-sensitive languages. In addition, we presented an example to demonstrate how context-sensitive languages are more efficient than context-free languages.