Chomsky Hierarchy Language Operations and Properties The Chomsky

Chomsky Hierarchy Language Operations and Properties

The Chomsky Hierarchy Type Language Grammar Automaton 0 Partially Computable Unrestricted DTM - NTM 1 Context Sensitive Linearly Bounded Automaton 2 Context Free NPDA 3 Regular right regular, left regular DFA, NFA You don’t have to know this

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

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 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 • 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 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 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 not in L So M’ recognizes L.

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.