How do you show a language is unrecognizable?

To prove that a given language is non-Turing-recognizable: Either do both of these: • Prove that its complement is Turing-recognizable. Prove that its complement is undecidable.

Can a language be recognizable and unrecognizable?

If some other language S and its complement ¯S are both recognizable, then S and ¯S are decidable. If ¯S is unrecognizable, then then S is undecidable but still recognizable.

Is an undecidable language unrecognizable?

Every unrecognizable language is undecidable, but there are undecidable languages which are recognizable. In fact, the decidable languages are exactly those which are recognizable and co-recognizable (that is, have recognizable complement).

What is a Corecognizable language?

A language is Recognizable iff there is a Turing Machine which will halt and accept only the strings in that language and for strings not in the language, the TM either rejects, or does not halt at all. Note: there is no requirement that the Turing Machine should halt for strings not in the language.

Is ATM a language?

D cannot exist ⇒ H cannot exist Therefore, ATM is not a decidable language. D on accepts if and only if Mi on rejects. So, D on will accept if and only if D on rejects! Therefore, ATM is not a decidable language.

Is ATM complement recognizable?

ATM is recognizable but its complement is not recognizable. (f) Decidable sets are closed under complement. True. We just need to flip the accept states and reject states.

What is unrecognizable language?

Quote Modify. ~ATM is the canonical example of a Turing-unrecognizable language. This means there does not exist a Turing Machine which will accept the set of all machine-string pairs such that M does NOT halt when run on w. The proof of this is very short: Lemma: ATM is Turing-recognizable.

Do all Turing machines recognize a language?

The language recognized by a Turing machine is, by definition, the set of strings it accepts. When an input is given to the machine, it is either accepted or not. Any particular input to that machine is either always accepted (in the language) or always not accepted (not in the language).

Is a TM recognizable?

There is no way to decide whether a TM will accept or eventually terminate. and HALT are recognizable. We can always run a TM on a string w and accept if that TM accepts or halts.

Is ATM Turing recognizable?

Recognizable Language A Turing machine M recognizes language L if L = L(M). We say L is Turing-recognizable (or simply recognizable) if there is a TM M such that L = L(M).

What is HALT TM?

HALTTM is Turing-recognizable since it can be recognized by TM U. HALTTM is not Turing-decidable. Proof: We will reduce ATM to HALTTM. Assume TM R decides HALTTM. We construct TM S that decides ATM as follows: On input < M,w > where M is a TM and w is a string, S first run TM R on < M,w >, if R rejects, rejects.

Is mapping reflexive Reducibility?

The language A is mapping reducible to language B, denoted A ≤m B, if there is a computable function f : Σ∗ → Σ∗ that has the property: (∀w ∈ Σ∗) w ∈ A iff f(w) ∈ B. We say that A reduces to B via the function f. In particular, the relation ≤m has the following properties: 1. ≤m is reflexive and transitive (exercise).