4.3 Context-Free Languages (Type-2 Grammars)

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Questions and Answers

What type of languages are context-free languages?

Type-1

Type-0

Type-3

Type-2 (correct)

Which component must be present in every production rule of a context-free grammar?

Multiple non-terminal symbols on the left-hand side

A regular expression on the right-hand side

One non-terminal symbol on the left-hand side (correct)

At least one terminal symbol on the left-hand side

How is the language a^n b^n, where n is a natural number, classified in terms of regularity?

Context-free (correct)

Context-sensitive

Unrestricted

Regular

What is denoted by the symbol ε in the context-free grammar S aSb S ε?

The empty string Signup and view all the answers

How are derivations of words represented in context-free grammars?

As syntax trees Signup and view all the answers

What does it mean if a syntax tree has different leaves with the same content?

There are multiple ways to derive the same word Signup and view all the answers

What is the purpose of the pumping lemma for context-free languages?

To provide a necessary condition for a language to be context-free Signup and view all the answers

How is the pumping lemma for context-free languages different from the one for regular languages?

It requires repetitive application of production rules Signup and view all the answers

What does it mean if a language satisfies the pumping lemma for context-free languages?

It provides no information about the nature of the language Signup and view all the answers

In which scenario can the pumping lemma be used to prove that a language is not context-free?

If the lemma does not hold for the given language Signup and view all the answers

What decision problem can be solved for context-free languages using an algorithm?

Checking if a word is contained in the language Signup and view all the answers

Why is the equivalence problem not decidable for context-free languages?

As there is no algorithm available to decide if two context-free languages generate the same language Signup and view all the answers

What is the purpose of a context-free grammar that has a unique syntax tree for every word of the language?

To ensure unambiguous parsing Signup and view all the answers

Which of the following is NOT a characteristic of an ambiguous grammar?

It is often used in programming languages Signup and view all the answers

What is the role of the Backus–Naur form (BNF) in the context of programming languages?

It is used to define the syntax of programming languages Signup and view all the answers

What is a request for comments (RFC) document used for?

To provide technical specifications and organizational notes for the Internet Signup and view all the answers

What is a uniform resource identifier (URI)?

A unique sequence of characters that identifies a logical or physical resource used by web technologies Signup and view all the answers

What is one of the disadvantages of ambiguous grammars?

They can lead to confusion and potential errors Signup and view all the answers

What is the role of the Kleene star in the context of regular expressions?

It represents the repetition of an expression zero or more times Signup and view all the answers

What is the purpose of the augmented Backus–Naur form (ABNF)?

It is a variant of the Backus–Naur form used in many RFCs of the IETF Signup and view all the answers

What is the relationship between Prolog programs and context-free grammars?

Prolog programs have a context-free grammar structure, but with restrictions Signup and view all the answers

What is the alphabet of a context-free grammar for regular expressions composed of?

Both the terminal symbols a1, a2, ... of the alphabet of the regular expression and the terminal symbols , , , and * Signup and view all the answers

Study Notes

Context-Free Languages (Type-2 Grammars)

Context-free languages (CF languages) are widely used in computer science.
A grammar is context-free if all production rules are of the form: l → r, where l ∈ V and r ∈ V ∪ Σ*.
The language a^n b^n : n ∈ ℕ is not regular, but can be generated via the context-free grammar S → aSb, S → ε.

Context-Free Grammars and Syntax Trees

Context-free grammars describe a simple, hierarchical structure used in various situations.
Every derivation of a word from such a grammar can be represented as a syntax or parse tree.
A syntax tree may contain different leaves with the same content, indicating different ways to derive the same word.

The Pumping Lemma for Context-Free Languages

The pumping lemma for context-free languages is similar to the pumping lemma for regular languages, but a bit more complicated.
If a language L is context-free, then there exists an integer m ≥ 1 such that every word z ∈ L with a length of m or more symbols can be written as z = uvwxy with substrings u, v, w, x, y such that:
- The substrings v, w, and x have a combined length of, at most, m.
- At least one of the words v and x is nonempty.
- For every natural number n, the word uvnwxny is contained in L.

Decision Problems

The word problem is decidable for context-free languages, meaning there is an algorithm to check if a given word is contained in the language.
The emptiness and finiteness problems are also decidable for context-free languages.
However, the equivalence problem is not decidable, meaning there is no algorithm to determine if two given context-free languages generate the same language.

Ambiguity in Context-Free Grammars

A context-free grammar that has a unique syntax (or parse) tree for every word of the language is unambiguous.
A grammar is considered ambiguous if a string can have more than one parse tree.
Ambiguous grammars may not be allowed in certain contexts, and programming languages usually have unambiguous grammars.

Recursive Definitions and Context-Free Grammars

Recursive definitions often provide a context-free grammar that can be represented with a tree.
Examples include the underlying structure of Prolog programs and the definition of regular expressions via a context-free grammar.

The Backus-Naur Form (BNF)

The Backus-Naur form (BNF) is a way to describe context-free languages.
BNF was used to describe the syntax of the programming language Algol 60.
Variants of BNF, such as augmented Backus-Naur form (ABNF), are used in many request for comments (RFCs) of the Internet Engineering Task Force (IETF).

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Test your knowledge on context-free languages (Type-2 grammars) used in computer science. Explore production rules, non-terminal symbols, and the pumping lemma related to context-free grammars.