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Questions and Answers
What is the type of languages that can be recognized by a finite automaton?
Which of the following is a closure property of regular languages?
What is the term for languages that can be generated by a contextfree grammar?
Which of the following languages is an example of a contextfree language?
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What is the type of languages that can be generated by a contextsensitive grammar?
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Which of the following languages is an example of a recursively enumerable language?
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What is the term for languages that cannot be recognized by a Turing machine?
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What is the relationship between the levels of the Chomsky hierarchy?
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Study Notes
Chomsky Hierarchy
Regular Languages
 Type 3: Languages that can be recognized by a finite automaton (FA)

Closure properties:
 Union: The union of two regular languages is regular
 Intersection: The intersection of two regular languages is regular
 Complementation: The complement of a regular language is regular
 Concatenation: The concatenation of two regular languages is regular

Examples:
 Language of all strings consisting of only 0s and 1s
 Language of all strings with an even number of 0s
Contextfree Languages
 Type 2: Languages that can be generated by a contextfree grammar (CFG)

Closure properties:
 Union: The union of two contextfree languages is contextfree
 Concatenation: The concatenation of two contextfree languages is contextfree

Examples:
 Language of all palindromes
 Language of all strings consisting of balanced parentheses
Contextsensitive Languages
 Type 1: Languages that can be generated by a contextsensitive grammar (CSG)

Closure properties:
 Union: The union of two contextsensitive languages is contextsensitive
 Concatenation: The concatenation of two contextsensitive languages is contextsensitive

Examples:
 Language of all strings consisting of a sequence of 0s and 1s with equal numbers of 0s and 1s
 Language of all strings consisting of a sequence of 0s and 1s with the number of 0s being a multiple of the number of 1s
Recursively Enumerable Languages
 Type 0: Languages that can be recognized by a Turing machine (TM)

Closure properties:
 Union: The union of two recursively enumerable languages is recursively enumerable
 Concatenation: The concatenation of two recursively enumerable languages is recursively enumerable

Examples:
 Language of all strings that can be accepted by a TM
 Language of all strings that can be generated by a TM
Unrestricted Languages
 Type 0: Languages that cannot be recognized by a Turing machine (TM)

Examples:
 Language of all strings that are not recursively enumerable
 Language of all strings that are not Turing recognizable
Note: The Chomsky hierarchy is a containment hierarchy, meaning that each level is a subset of the level above it.
Chomsky Hierarchy
Regular Languages
 Type 3: Recognized by a finite automaton (FA)
 Closure properties: Union, Intersection, Complementation, and Concatenation are all regular

Examples:
 Language of all strings consisting of only 0s and 1s
 Language of all strings with an even number of 0s
Contextfree Languages
 Type 2: Generated by a contextfree grammar (CFG)
 Closure properties: Union and Concatenation are contextfree

Examples:
 Language of all palindromes
 Language of all strings consisting of balanced parentheses
Contextsensitive Languages
 Type 1: Generated by a contextsensitive grammar (CSG)
 Closure properties: Union and Concatenation are contextsensitive

Examples:
 Language of all strings consisting of a sequence of 0s and 1s with equal numbers of 0s and 1s
 Language of all strings consisting of a sequence of 0s and 1s with the number of 0s being a multiple of the number of 1s
Recursively Enumerable Languages
 Type 0: Recognized by a Turing machine (TM)
 Closure properties: Union and Concatenation are recursively enumerable

Examples:
 Language of all strings that can be accepted by a TM
 Language of all strings that can be generated by a TM
Unrestricted Languages
 Type 0: Not recognized by a Turing machine (TM)

Examples:
 Language of all strings that are not recursively enumerable
 Language of all strings that are not Turing recognizable
Containment Hierarchy
 Each level is a subset of the level above it
 The hierarchy is a containment relationship between the language types
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Description
Explore the properties and examples of regular languages, including union, intersection, complementation, and concatenation, as part of the Chomsky Hierarchy.