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Chomsky Hierarchy Language Operations and Properties. The Chomsky Hierarchy. You don’t have to know this. The Chomsky Hierarchy. Partially Computable Languages. {M, H(<M>)}. Computable Languages. {a n b n c n , n ≥ 0}. Context Free Languages. {a n b n , n ≥ 0}. Regular Languages.

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the chomsky hierarchy
The Chomsky Hierarchy

You don’t

have to

know this

the chomsky hierarchy1
The Chomsky Hierarchy

Partially Computable Languages

{M, H(<M>)}

Computable Languages

{an bn cn , n ≥ 0}

Context Free Languages

{an bn , n ≥ 0}

Regular Languages

{am bn , m,n ≥ 0}

regular languages
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.
context free languages
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.
computable languages
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.
partially computable language
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…)
closure under operations
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.
computable and partially computable languages
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.
computable languages are also partially computable
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.

properties of computable and partially computable languages
Properties of Computable and Partially Computable Languages
  • If a language L and its complement are both partially computable then the language is computable

Both 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.