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Questions and Answers
What does it mean if a string $
u$ is produced by a grammar $G$?
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 $S$?
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?
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?
Which of the following statements about grammars and languages is true?
What is the correct interpretation of the notation $L(G)$ in relation to grammars?
What is the correct interpretation of the notation $L(G)$ in relation to grammars?
What characterizes a type-0 grammar?
What characterizes a type-0 grammar?
Which statement is true regarding type-1 grammars?
Which statement is true regarding type-1 grammars?
Which of the following best describes a context-sensitive language?
Which of the following best describes a context-sensitive language?
In the context of grammar classifications, which grammar generates the language L(G1)?
In the context of grammar classifications, which grammar generates the language L(G1)?
What is the role of the variable VN in the grammar representations?
What is the role of the variable VN in the grammar representations?
Which statement correctly reflects the relationship between grammar types in the Chomsky hierarchy?
Which statement correctly reflects the relationship between grammar types in the Chomsky hierarchy?
Which of the following languages belong to the Chomsky hierarchy?
Which of the following languages belong to the Chomsky hierarchy?
Which property about languages in the Chomsky hierarchy is true?
Which property about languages in the Chomsky hierarchy is true?
What defines a type-i language in the Chomsky hierarchy?
What defines a type-i language in the Chomsky hierarchy?
Given the sequence $a^n b^n$ where $n
e 0$, which type of grammar can recognize it?
Given the sequence $a^n b^n$ where $n e 0$, which type of grammar can recognize it?
Which language type is explicitly included in the proper inclusions defined in the Chomsky hierarchy?
Which language type is explicitly included in the proper inclusions defined in the Chomsky hierarchy?
In the Chomsky hierarchy, what does the notation R ⊂ CF imply?
In the Chomsky hierarchy, what does the notation R ⊂ CF imply?
Consider the language defined by $a^n b^n c^n$. Which grammar type is required to generate this language?
Consider the language defined by $a^n b^n c^n$. Which grammar type is required to generate this language?
What is the definition of productions in grammar?
What is the definition of productions in grammar?
Which of the following represents a one-step production?
Which of the following represents a one-step production?
What does the notation µ ⇒ ν signify in the context of grammar?
What does the notation µ ⇒ ν signify in the context of grammar?
In the context of formal grammars, what does the term 'rules' refer to?
In the context of formal grammars, what does the term 'rules' refer to?
Which set represents the non-terminals in the given grammar example?
Which set represents the non-terminals in the given grammar example?
What does the symbol R represent in the grammar definition?
What does the symbol R represent in the grammar definition?
If the production rule is defined as S → aSBC, what type of production is this?
If the production rule is defined as S → aSBC, what type of production is this?
Which of the following statements about the non-terminal B is TRUE in the context of the given grammar?
Which of the following statements about the non-terminal B is TRUE in the context of the given grammar?
What does a grammar describe in relation to a language?
What does a grammar describe in relation to a language?
Which of the following correctly defines the components of a grammar quadruple G?
Which of the following correctly defines the components of a grammar quadruple G?
In the context of formal language theory, which statement about terminal and non-terminal elements is correct?
In the context of formal language theory, which statement about terminal and non-terminal elements is correct?
What is the purpose of the starting rule S in a grammar?
What is the purpose of the starting rule S in a grammar?
Which of the following best describes a language in the context of grammar?
Which of the following best describes a language in the context of grammar?
Which example of a terminal set is provided in the content?
Which example of a terminal set is provided in the content?
What role does the finite set of rules R serve in grammar?
What role does the finite set of rules R serve in grammar?
Which of the following statements is true regarding the provided example grammar?
Which of the following statements is true regarding the provided example grammar?
What is the designation of grammar G2 in the Chomsky hierarchy?
What is the designation of grammar G2 in the Chomsky hierarchy?
Which of the following statements about language L(G) is correct?
Which of the following statements about language L(G) is correct?
What is the primary characteristic of a type-3 grammar as represented in G3?
What is the primary characteristic of a type-3 grammar as represented in G3?
Which type of language is L(G) when there exists a regular grammar G3 such that L(G3) = L(G)?
Which type of language is L(G) when there exists a regular grammar G3 such that L(G3) = L(G)?
According to the closure properties, which operations are type-1 languages not closed under?
According to the closure properties, which operations are type-1 languages not closed under?
What does the Kleene closure operation on language L1 produce?
What does the Kleene closure operation on language L1 produce?
Which property is true for languages of type-i?
Which property is true for languages of type-i?
What defines L(G3) in terms of its grammar structure?
What defines L(G3) in terms of its grammar structure?
Flashcards
String
String
A sequence of symbols. In the context of formal languages, these symbols can be letters, numbers, or special characters.
Grammar
Grammar
A set of rules that define how to generate all possible strings (sequences of symbols) that belong to a language.
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
Non-terminals
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Terminals
Terminals
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Start Symbol (S)
Start Symbol (S)
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Rules (R)
Rules (R)
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Chomsky Hierarchy
Chomsky Hierarchy
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Productions (or Rules, R)
Productions (or Rules, R)
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String (in Language Theory)
String (in Language Theory)
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Rules (in Formal Grammars)
Rules (in Formal Grammars)
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Formal Grammar Definition
Formal Grammar Definition
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Terminal Symbols (VT)
Terminal Symbols (VT)
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Non-Terminal Symbols (VN)
Non-Terminal Symbols (VN)
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Productions (in Formal Grammars)
Productions (in Formal Grammars)
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One-step & Multi-step Productions
One-step & Multi-step Productions
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Context-sensitive grammar
Context-sensitive grammar
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Context-free grammar
Context-free grammar
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Regular Language
Regular Language
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Context-Free Language
Context-Free Language
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Context-Sensitive Language
Context-Sensitive Language
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Recursively Enumerable Language
Recursively Enumerable Language
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Production
Production
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Productions (R)
Productions (R)
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Type 3 grammar (regular grammar)
Type 3 grammar (regular grammar)
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Type 2 grammar (context-free grammar)
Type 2 grammar (context-free grammar)
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Closure Property
Closure Property
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Type-i language
Type-i language
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Non-closure property
Non-closure property
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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.
- Type-0 (Recursively enumerable languages)
- 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 (∩)
- Complement (¬)
Applications of Formal Language Theory
- Programming tools
- TLM-project
- Python package (Chomsky-Hierarchy)
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