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 a set is 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.

Why is the set of all languages uncountable?

Notice that while each string in a language must have finite length, the language itself may have infinitely many strings as illustrated in the second example just given. Our proof will be based on the fact that a contradiction is obtained if L{0,1} is countable.

Is recursive language type 0?

Recursive languages are: A proper superset of context free languages. Always recognizable by pushdown automata. Also called type 0 languages.

Are all languages recursively enumerable?

All regular, context-free, context-sensitive and recursive languages are recursively enumerable.

Which set is recursively enumerable?

3 Answers. Here’s one definition: A recursively enumerable set is a set where you can write a program that will output each element in the set: E1, E2, E3… it’s okay if this program never stops. People usually talk about this in the context of languages.

Can a language be uncountable?

Language is always uncountable as a general concept or phenomenon.

Is the set of all languages uncountable?

According to the notes: The set of all regular expressions over Σ is infinite and countable. The set of all languages over Σ is infinite and uncountable.

When recursively enumerable language is recursive?

Recursively Enumerable Languages. A language is called Recursively Enumerable if there is a Turing Machine that accepts on any input within the language. Reminder: A language is called Recursive if there is a Turing Machine that accepts on any input within the language and rejects on any other input.

Is recursively enumerable closed under complement?

Recursive enumerable languages are not closed under complementation.It signifies that Y′ may/may not be recursive enumerable. But the answer will be Y′ is not recursive Enumerable. Why? If a language and its complement are both recursively enumerable, then both are recursive.

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.

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.

How to prove that H A L T is not recursively enumerable?

If ⟨ M, x ⟩ ∉ H A L T, then our machine is simply required (by the alternate characterization of recursive enumerability) to not accept. It indeed does not accept. Since ⟨ M, x ⟩ ∉ H A L T, the computation M ( x) never halts, so neither does our machine. Now, we can prove that H A L T ¯ is not recursively enumerable.

Are L1 and L2 recursive languages?

Union: If L1 and If L2 are two recursive languages, their union L1∪L2 will also be recursive because if TM halts for L1 and halts for L2, it will also halt for L1∪L2. Concatenation: If L1 and If L2 are two recursive languages, their concatenation L1.L2 will also be recursive.