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CHOMSKY HIERARCHYLANGUAGE OPERATIONS AND PROPERTIES

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THE CHOMSKY HIERARCHY

Type

Language Grammar Automaton

0 Partially Computable

Unrestricted DTM - NTM

1 Context Sensitive

Context Sensitive

Linearly Bounded Automaton

2 Context Free Context Free NPDA

3 Regular right regular, left regular

DFA, NFA

0 Type 1 Type 2 Type 3 Type

You don’thave to know this

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THE CHOMSKY HIERARCHY

Context Free Languages

Computable Languages

Partially Computable Languages

{an bn , n ≥ 0}

Regular Languages

{an bn cn , n ≥ 0}

{am bn , m,n ≥ 0}

{M, H(<M>)}

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REGULAR LANGUAGES

To prove that a language is regular: Find a DFA (NFA, NFAε) that recognizes it. Find a regular expression that represents is. Find a (right or left) regular grammar that

generates it.

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CONTEXT FREE LANGUAGES

To prove that a language is context free: Find a NPDA that recognizes it. Find a context free grammar that generates

it.

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COMPUTABLE LANGUAGES

To prove that a language is computable you must find a Turing Machine that decides membership in the language:

If the string is in the language then the machine should accept.

If the string is not in the language then the machine should reject.

The machine shouldn’t loop for any input.

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PARTIALLY COMPUTABLE LANGUAGE

To prove that a language is partially computable:

Find a Turing Machine that recognizes it: if the string is in the language the machine

should accept. if the string is not in the language the machine

should loop. Give an unrestricted grammar that

represents it (you won’t be asked to do that…)

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CLOSURE UNDER OPERATIONS

• Regular Languages are closed under: Union, Concatenation, Star, Intersection and Complement

• Context Free Languages are closed under: Union, Concatenation, Star. They are not closed under: Intersection, Complement.

• Computable Languages are closed under: Union, Concatenation, Star, Intersection, Complement

• Partially Computable Languages are closed under: Union, Concatenation, Star, Intersection. They are not closed under Complement. video.edhole.com

COMPUTABLE AND PARTIALLY COMPUTABLE LANGUAGES

• Computable languages most of the times are called Decidable Languages because a Turing Machine can decide membership in those languages (it can either accept or reject a string). They are also called Recursive.

• Partially Computable Languages most of the times are called Recognizable because a Turing Machine can recognize a string in the language (accept it) but it might not be able to decide if a string is not in the language (it might loop). They are also called Recursively Enumerable.

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COMPUTABLE LANGUAGES ARE ALSO PARTIALLY COMPUTABLE

Proof:If L is Computable then there is a Turing Machine M

that decides membership in L:– M accepts on x if x is in L– M rejects on x if x is not in L

Change M to M’ in the following way:– For all the combinations (q,a) for which the machine

rejects add the transition δ(q,a) = (q, a, S)

Now the new machine M’:– Accepts on x if x is in L– Loops on x is x is not in L

So M’ recognizes L.video.edhole.com

PROPERTIES OF COMPUTABLE AND PARTIALLY COMPUTABLE LANGUAGES

If a language L and its complement are both partially computable then the language is computableBoth L and its complement have TMs M and M’ that

recognize them. To decide if x is in L: Run both machines M and M’ in parallel. Eventually one will halt. If M halts accept. If M’ halts reject.

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