Grammar Theory Quiz

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Questions and Answers

What does it mean if a string $ u$ is produced by a grammar $G$?

It consists solely of non-terminal symbols.

It cannot be derived from any other grammar in a similar form.

It is a random sequence of symbols from the terminal set.

It can always be derived from the start symbol $S$. (correct)

In the context of grammars, what is the significance of the symbol $S$?

$S$ indicates the end of a production sequence.

$S$ is a placeholder for non-terminal symbols only.

$S$ represents the set of all strings that can be generated.

$S$ is the starting symbol from which productions begin. (correct)

How is a multi-step production denoted in grammar?

By writing $ u eq ho$ to show the non-equivalence.

By using a double arrow to indicate a sequence of transformations. (correct)

With a notation similar to $ u$ produces $ u *$.

Using a single arrow, as in $ u ightarrow ho$.

Which of the following statements about grammars and languages is true?

A language can be generated by multiple grammars. Signup and view all the answers

What is the correct interpretation of the notation $L(G)$ in relation to grammars?

It represents the set of strings produced by grammar $G$. Signup and view all the answers

What characterizes a type-0 grammar?

It has productions where the left side can have an arbitrary number of variables. Signup and view all the answers

Which statement is true regarding type-1 grammars?

They generate context-sensitive languages. Signup and view all the answers

Which of the following best describes a context-sensitive language?

Languages where there exists no type-2 grammar that generates them. Signup and view all the answers

In the context of grammar classifications, which grammar generates the language L(G1)?

A type-1 grammar. Signup and view all the answers

What is the role of the variable VN in the grammar representations?

It consists of non-terminal symbols that can generate other strings. Signup and view all the answers

Which statement correctly reflects the relationship between grammar types in the Chomsky hierarchy?

A type-3 grammar is also a type-2 grammar. Signup and view all the answers

Which of the following languages belong to the Chomsky hierarchy?

Regular languages Signup and view all the answers

Which property about languages in the Chomsky hierarchy is true?

Some context-free languages are not regular. Signup and view all the answers

What defines a type-i language in the Chomsky hierarchy?

It is generated by a type-i grammar and cannot be produced by any grammar of higher type. Signup and view all the answers

Given the sequence $a^n b^n$ where $n e 0$, which type of grammar can recognize it?

Context-sensitive grammar Signup and view all the answers

Which language type is explicitly included in the proper inclusions defined in the Chomsky hierarchy?

Context-free languages are recursively enumerable languages. Signup and view all the answers

In the Chomsky hierarchy, what does the notation R ⊂ CF imply?

R is a subset of all context-free languages. Signup and view all the answers

Consider the language defined by $a^n b^n c^n$. Which grammar type is required to generate this language?

Context-sensitive grammar Signup and view all the answers

What is the definition of productions in grammar?

The rules which generate all the strings over the language. Signup and view all the answers

Which of the following represents a one-step production?

If µ = σαγ and ν = ηβθ with (α → β) in R. Signup and view all the answers

What does the notation µ ⇒ ν signify in the context of grammar?

A one-step production from µ to ν. Signup and view all the answers

In the context of formal grammars, what does the term 'rules' refer to?

A finite set of productions that determine how strings are formed. Signup and view all the answers

Which set represents the non-terminals in the given grammar example?

{S, B, C} Signup and view all the answers

What does the symbol R represent in the grammar definition?

The set of production rules. Signup and view all the answers

If the production rule is defined as S → aSBC, what type of production is this?

One-step production. Signup and view all the answers

Which of the following statements about the non-terminal B is TRUE in the context of the given grammar?

It can lead to the derivation of valid strings. Signup and view all the answers

What does a grammar describe in relation to a language?

How to generate strings over a language Signup and view all the answers

Which of the following correctly defines the components of a grammar quadruple G?

G includes terminal and non-terminal sets, a starting rule, and a finite set of rules Signup and view all the answers

In the context of formal language theory, which statement about terminal and non-terminal elements is correct?

There is no intersection between terminal and non-terminal sets Signup and view all the answers

What is the purpose of the starting rule S in a grammar?

To provide a single entry point for deriving strings Signup and view all the answers

Which of the following best describes a language in the context of grammar?

A language is defined by the set of rules that generate strings Signup and view all the answers

Which example of a terminal set is provided in the content?

{she, he, eats, drinks, chocolate, juice} Signup and view all the answers

What role does the finite set of rules R serve in grammar?

To govern the structure of string generation Signup and view all the answers

Which of the following statements is true regarding the provided example grammar?

The example illustrates a specific structure of producing sentences Signup and view all the answers

What is the designation of grammar G2 in the Chomsky hierarchy?

Type-2 grammar Signup and view all the answers

Which of the following statements about language L(G) is correct?

L(G) is a context-free language. Signup and view all the answers

What is the primary characteristic of a type-3 grammar as represented in G3?

The right-hand side must consist of a single terminal symbol followed by an optional non-terminal. Signup and view all the answers

Which type of language is L(G) when there exists a regular grammar G3 such that L(G3) = L(G)?

Regular language Signup and view all the answers

According to the closure properties, which operations are type-1 languages not closed under?

Complement and intersection Signup and view all the answers

What does the Kleene closure operation on language L1 produce?

The set of strings formed by concatenating L1 with itself any number of times. Signup and view all the answers

Which property is true for languages of type-i?

They are closed under mirror and concatenation. Signup and view all the answers

What defines L(G3) in terms of its grammar structure?

It is formulated as a regular language via defined terminal and non-terminal symbols. Signup and view all the answers

Study Notes

Formal Language Theory - Chapter 4: Grammars and Chomsky Hierarchy

Formal language theory studies how to define languages using grammars.
A grammar is a set of rules that describes how to generate strings over a language.
The universal formula for a sentence in English is: Sentence = Subject + Verb + Object.
A language is defined by the grammar that describes how to generate all the strings within the language.
A grammar is a quadruple G = (VT, VN, S, R), where:
- VT is the set of terminal elements.
- VN is the set of non-terminals (VT ∩ VN = $ and VT U VN = V).
- S ∈ VN is the starting rule.
- R is a finite set of rules.

Rules

A rule is a pair from V*VN V* × V*, denoted by →.
Examples of rules from the presentation are:
- S → aSBC
- S → abC
- CB → BC
- bB → bb
- bC → bc
- CC → cc
Productions are the rules that generate (produce) the strings of a language, denoted by →.
- Productions can be done in one step or multiple steps.
A one-step production is defined as: - μ ⇒ ν iff μ = σαγ and v = ηβθ and (α → β) ∈ R.
- An example of a one-step production is: S → aSBC ⇒ aabCBC
A multi-step production happens in multiple steps.
- μ ⇒ ν iff ∃ pi ∈ VVNV, ∃ pi+1 ∈ V* where μ = p0, ν= Pn and pi ⇒ pi+1.

Languages & Grammars

A language L is the set of strings produced by the grammar G
L(G) = {w/w ∈ V* and S ⇒ w} - w is terminal string, s is start symbol, ⇒ means can produce

Chomsky Hierarchy

Grammars are categorized by their rules' restrictions.
Noam Chomsky categorized grammars into 4 levels:
- Type-0 (Recursively enumerable languages)
  - Any grammar that can be described
- Type-1 (Context-sensitive languages)
  - Restricted that the the number of symbols can't decrease, but can increase
- Type-2 (Context-free languages)
  - Only rules that match a single non-terminal symbol at one place.
- Type-3 (Regular languages)
  - The most restricted, producing strings with a structure/form.
The type of a grammar increases as the restrictions on the rules decrease.

Grammar Proper Inclusions

A type-i grammar is also a type-(i-1) grammar (i.e., every type-3 grammar is also a type-2, type-1, and type-0 grammar)

Language Proper Inclusions

Every regular language is context-free.
Every context-free language is context-sensitive.
Every context-sensitive language is recursively enumerable.

Language Type

A language is type-i if it can be generated by a type-i grammar.

Closure Properties

The set of type-i languages is closed:
- Under union (∪)
- Under concatenation (⋅)
- Under Kleene closure (∗)
The set of type-i languages is not closed under:
- Complement (¬)
  - The complement (¬) of a regular language is not necessarily regular.
- Intersection (∩)

Applications of Formal Language Theory

Programming tools
- TLM-project
- Python package (Chomsky-Hierarchy)

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Test your understanding of formal grammars and their classifications with this quiz. Explore concepts such as types of grammars, production rules, and the significance of specific symbols within grammatical frameworks. Perfect for students of computational linguistics or theoretical computer science.