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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 context-free grammar?
Which of the following languages is an example of a context-free language?
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What is the type of languages that can be generated by a context-sensitive 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)
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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
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Examples:
- Language of all strings consisting of only 0s and 1s
- Language of all strings with an even number of 0s
Context-free Languages
- Type 2: Languages that can be generated by a context-free grammar (CFG)
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Closure properties:
- Union: The union of two context-free languages is context-free
- Concatenation: The concatenation of two context-free languages is context-free
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Examples:
- Language of all palindromes
- Language of all strings consisting of balanced parentheses
Context-sensitive Languages
- Type 1: Languages that can be generated by a context-sensitive grammar (CSG)
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Closure properties:
- Union: The union of two context-sensitive languages is context-sensitive
- Concatenation: The concatenation of two context-sensitive languages is context-sensitive
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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)
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Closure properties:
- Union: The union of two recursively enumerable languages is recursively enumerable
- Concatenation: The concatenation of two recursively enumerable languages is recursively enumerable
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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)
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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
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Examples:
- Language of all strings consisting of only 0s and 1s
- Language of all strings with an even number of 0s
Context-free Languages
- Type 2: Generated by a context-free grammar (CFG)
- Closure properties: Union and Concatenation are context-free
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Examples:
- Language of all palindromes
- Language of all strings consisting of balanced parentheses
Context-sensitive Languages
- Type 1: Generated by a context-sensitive grammar (CSG)
- Closure properties: Union and Concatenation are context-sensitive
-
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)
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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.
