What is a language that is recursively enumerable but not recursive?

The Universal Language L Lu is recursively enumerable but not recursive. Lu is the set of binary strings that consist of encoded pairs (M, w) such that M is an encoding of a Turing machine and w is an encoding of a binary input string accepted by that Turing machine.

How do you prove something is recursively enumerable?

A language is recursively enumerable if there exists a Turing machine that accepts every string of the language, and does not accept strings that are not in the language. (Strings that are not in the language may be rejected or may cause the Turing machine to go into an infinite loop.)

Are Turing machines recursively enumerable?

The set of halting turing machines is recursively enumerable but not recursive. Indeed, one can run the Turing Machine and accept if the machine halts, hence it is recursively enumerable.

What languages are not recursively enumerable?

An example of a language which is not recursively enumerable is the language L of all descriptions of Turing machines which don’t halt on the empty input.

What is the difference between recursive and recursively enumerable language?

The main difference is that in recursively enumerable language the machine halts for input strings which are in language L. but for input strings which are not in L, it may halt or may not halt. When we come to recursive language it always halt whether it is accepted by the machine or not.

Which of the following is Recognised by recursively enumerable language?

Explanation: A language L is recursively enumerable if there is a turing machine that accepts L, and recursive if there is a TM that recognizes L. (Sometimes these languages are alse called Turing-acceptable and Turing-decidable respectively).

How do you prove a set is enumerable?

A set A is said to be enumerable if there exists a surjection N → A. e(n)=(−1)n mod 2 ((n + 1) div 2) for all n ∈ N.

How do you know if a language is recursive?

Equivalently, a formal language is recursive if there exists a total Turing machine (a Turing machine that halts for every given input) that, when given a finite sequence of symbols as input, accepts it if it belongs to the language and rejects it otherwise. Recursive languages are also called decidable.

When can we say that a language L is Turing recognizable or recursively enumerable?

A language L is recursively enumerable/Turing recognizable if there is a Turing Machine M such that L(M) = L. A language L is decidable if there is a Turing machine M such that L(M) = L and M halts on every input. Thus, if L is decidable then L is recursively enumerable. Definition 1.

What do you mean by recursive language prove that complement of recursive language is recursive?

Complements of Recursive and Recursively Enumerable Languages. A recursive language is one that is accepted by a TM that halts on all inputs. The complement of a recursive language is recursive. If a language L and its complement are RE, then L is recursive.

Is a set enumerable?

The set S is computably enumerable. That is, S is the domain (co-range) of a partial computable function.

How do you prove a language is not recursive?

Theorem. If a language L and its complement are both RE, they are both recursive. Proof. Decide whether w ∈ L by enumerating L and its complement in parallel and accept/reject as soon as w appears in one of the enumerations. ◻ So, if you can prove that L is not recursive but its complement is RE, then L is not RE.

What is recursive enumerable (re) language?

Recursive Enumerable (RE) or Type -0 Language RE languages or type-0 languages are generated by type-0 grammars. An RE language can be accepted or recognized by Turing machine which means it will enter into final state for the strings of language and may or may not enter into rejecting state for the strings which are not part of the language.

What is Turing machine acceptance of re language?

An RE language can be accepted or recognized by Turing machine which means it will enter into final state for the strings of language and may or may not enter into rejecting state for the strings which are not part of the language. It means TM can loop forever for the strings which are not a part of the language.

Can We prove that L is not recursive but its complement is?

So, if you can prove that L is not recursive but its complement is RE, then L is not RE. Theorem. Let M be the class of Turing machines equipped with an oracle for the ordinary Turing machine halting problem.