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?

<p>A language can be generated by multiple grammars. (A)</p> Signup and view all the answers

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

<p>It represents the set of strings produced by grammar $G$. (C)</p> Signup and view all the answers

What characterizes a type-0 grammar?

<p>It has productions where the left side can have an arbitrary number of variables. (C)</p> Signup and view all the answers

Which statement is true regarding type-1 grammars?

<p>They generate context-sensitive languages. (C)</p> Signup and view all the answers

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

<p>Languages where there exists no type-2 grammar that generates them. (B)</p> Signup and view all the answers

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

<p>A type-1 grammar. (D)</p> Signup and view all the answers

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

<p>It consists of non-terminal symbols that can generate other strings. (A)</p> Signup and view all the answers

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

<p>A type-3 grammar is also a type-2 grammar. (B), A type-1 grammar is also a type-0 grammar. (C)</p> Signup and view all the answers

Which of the following languages belong to the Chomsky hierarchy?

<p>Regular languages (B), Context-sensitive languages (C), Context-free languages (D)</p> Signup and view all the answers

Which property about languages in the Chomsky hierarchy is true?

<p>Some context-free languages are not regular. (A), Every context-free language is context-sensitive. (C)</p> Signup and view all the answers

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

<p>It is generated by a type-i grammar and cannot be produced by any grammar of higher type. (A)</p> Signup and view all the answers

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

<p>Context-sensitive grammar (A), Context-free grammar (B)</p> Signup and view all the answers

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

<p>Context-free languages are recursively enumerable languages. (B), Context-free languages are context-sensitive languages. (C)</p> Signup and view all the answers

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

<p>R is a subset of all context-free languages. (B)</p> 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?

<p>Context-sensitive grammar (C)</p> Signup and view all the answers

What is the definition of productions in grammar?

<p>The rules which generate all the strings over the language. (D)</p> Signup and view all the answers

Which of the following represents a one-step production?

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

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

<p>A one-step production from µ to ν. (C)</p> Signup and view all the answers

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

<p>A finite set of productions that determine how strings are formed. (B)</p> Signup and view all the answers

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

<p>{S, B, C} (C)</p> Signup and view all the answers

What does the symbol R represent in the grammar definition?

<p>The set of production rules. (D)</p> Signup and view all the answers

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

<p>One-step production. (B)</p> 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?

<p>It can lead to the derivation of valid strings. (D)</p> Signup and view all the answers

What does a grammar describe in relation to a language?

<p>How to generate strings over a language (D)</p> Signup and view all the answers

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

<p>G includes terminal and non-terminal sets, a starting rule, and a finite set of rules (D)</p> Signup and view all the answers

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

<p>There is no intersection between terminal and non-terminal sets (C)</p> Signup and view all the answers

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

<p>To provide a single entry point for deriving strings (A)</p> Signup and view all the answers

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

<p>A language is defined by the set of rules that generate strings (C)</p> Signup and view all the answers

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

<p>{she, he, eats, drinks, chocolate, juice} (C)</p> Signup and view all the answers

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

<p>To govern the structure of string generation (C)</p> Signup and view all the answers

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

<p>The example illustrates a specific structure of producing sentences (D)</p> Signup and view all the answers

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

<p>Type-2 grammar (B)</p> Signup and view all the answers

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

<p>L(G) is a context-free language. (C)</p> Signup and view all the answers

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

<p>The right-hand side must consist of a single terminal symbol followed by an optional non-terminal. (C)</p> 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)?

<p>Regular language (C)</p> Signup and view all the answers

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

<p>Complement and intersection (D)</p> Signup and view all the answers

What does the Kleene closure operation on language L1 produce?

<p>The set of strings formed by concatenating L1 with itself any number of times. (D)</p> Signup and view all the answers

Which property is true for languages of type-i?

<p>They are closed under mirror and concatenation. (D)</p> Signup and view all the answers

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

<p>It is formulated as a regular language via defined terminal and non-terminal symbols. (B)</p> Signup and view all the answers

Flashcards

String

A sequence of symbols. In the context of formal languages, these symbols can be letters, numbers, or special characters.

Grammar

A set of rules that define how to generate all possible strings (sequences of symbols) that belong to a language.

Language

A collection of all possible strings that can be created using a specific grammar. Basically, it's the set of all 'legal' sentences within a language.

Non-terminals

A set of non-terminal elements that are used in the rules to create strings. Think of them as placeholders or categories.

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Terminals

A set of terminal elements, also called symbols or characters, that are the basic building blocks for creating strings in a language. They are the actual symbols that appear in the final string.

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Start Symbol (S)

The starting symbol in a grammar. It's where the generation of a string begins.

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Rules (R)

A collection of rules that define how non-terminals can be replaced with terminals or other non-terminals. They form the core of a grammar.

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Chomsky Hierarchy

A formal framework for classifying and understanding the capabilities of different types of grammars. It's a hierarchy that ranks them based on their power to describe languages.

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Productions (or Rules, R)

A set of rules that dictate how non-terminals can be replaced by terminals or other non-terminals, forming the heart of a grammar.

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String (in Language Theory)

A sequence of symbols from a finite alphabet.

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Rules (in Formal Grammars)

A finite set of pairs from VVN V x V*, denoted by →. Each pair represents a rule that transforms one string into another.

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Formal Grammar Definition

A formal grammar is defined as a 4-tuple: G = (VT, VN, S, R)

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Terminal Symbols (VT)

A finite set of symbols used in the language, excluding non-terminal symbols. Think of them as the actual words or letters in the language.

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Non-Terminal Symbols (VN)

A finite set of symbols used in the language, representing abstract concepts or categories. They serve as placeholders that can be replaced by other symbols (terminal or non-terminal).

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Productions (in Formal Grammars)

A set of rules for string generation that allows transforming one string into another.

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One-step & Multi-step Productions

Production rules can be applied once or multiple times to generate strings. One-step production applies a rule once to transform the current string. Multi-step production applies a rule multiple times sequentially to transform the same string continuously.

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Context-sensitive grammar

A type of grammar where the length of the left side of a rule (the non-terminal) is always less than or equal to the length of the right side (the combination of terminals and non-terminals).

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Context-free grammar

A type of grammar where the left side of a rule is always a single non-terminal, and the right side can contain a combination of terminals and non-terminals.

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Regular Language

A type of language that can be recognized by a finite-state automaton. It's simple, regular and can be easily defined by a set of rules.

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Context-Free Language

A language that can be accepted by a pushdown automaton. It's more complex than a regular language and can handle nested structures.

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Context-Sensitive Language

A language that can be accepted by a linear bounded automaton. It's even more complex and can handle more intricate structures, such as nested recursion.

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Recursively Enumerable Language

A language that can be accepted by a Turing machine. It's the most powerful type of language and can handle the most complex structures.

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Production

A rule that defines how non-terminals can be replaced with terminals or other non-terminals. They are the building blocks of a grammar.

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Productions (R)

A set of rules that dictate how non-terminals can be replaced by terminals or other non-terminals, forming the heart of a grammar.

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Type 3 grammar (regular grammar)

A type of grammar where the rules are restricted to a specific form. Any language that can be described by this kind of grammar is considered a regular language.

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Type 2 grammar (context-free grammar)

A type of grammar that allows for more complex patterns than type 3 grammars. Any language that can be described by this kind of grammar is called a context-free language.

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Closure Property

The ability of a language to be manipulated using certain operations without changing its type. Common operations include union, concatenation, Kleene closure, and mirror.

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Type-i language

A specific language type that is closed under the following operations: union, concatenation, Kleene closure, and mirror.

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Non-closure property

Languages that cannot be manipulated by the operations of intersection and complement without changing their type.

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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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