A recursive language can’t go into an infinite loop, it has to clearly reject the string, but a recursively enumerable language can go into an infinite loop. So, every recursive language is also recursively enumerable. Thus, the statement ‘Every subset of a recursively enumerable set is recursive’ is **false**.

## What is the difference between recursive and recursively enumerable languages?

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.

## Why is it called recursively enumerable?

All recursive languages are also recursively enumerable **because you can just enumerate every string, and then output it if it’s in your set**. Recursive languages are also called decidable because you can decide for sure if an element is in it or not.

### How do you know if a language is recursive?

A language is recursive if **there exists a Turing machine that accepts every string of the language and rejects every string (over the same alphabet) that is not in the language**. Note that, if a language L is recursive, then its complement -L must also be recursive.

### Is Empty set recursive?

The empty set ∅ and **S are recursive**. 2. If R, S ⊂ S are recursive then so are R ∪ S and R ∩ S. … If S is recursive then so is its complement ¯ S = S S.

### Are recursively enumerable language L can be recursive if?

Explanation: A language L is recursively enumerable if and **only if it can be enumerated by some turing machine**. A recursive enumerable language may or may not be recursive.

### Is the family of recursively enumerable languages closed under intersection?

Recursively enumerable languages are also **closed** under intersection, concatenation, and Kleene star.

### Are recursive languages closed under complement?

Recursively **enumerable languages are closed under complement**. … For recursively enumerable languages, M_{1v2} fails to prove closure under union. M1 may not halt, and so M2 may not be run by M_{1v2} on w.

### 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.

### What is recursive set in TOC?

In computability theory, a set of natural numbers is called computable, recursive, or decidable if **there is an algorithm which takes a number as input, terminates after a finite amount of time** (possibly depending on the given number) and correctly decides whether the number belongs to the set or not.

### How do you prove a 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.

### What is a non recursive language?

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.

### Are recursive languages closed under union?

a) Union Recursive and **Recursively Enumerable languages are closes under union**. Let’s built a Turing Machine M which is going to simulate M1 and M2 on the input it gets. M will accept if either accept.

### What are the properties of recursive and recursively enumerable languages?

**Recursive and Recursive Enumerable Languages in TOC**

- 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.

### Which of the following is complement of a recursive language is recursive?

The complement of a recursive language is recursive. If **a language L and its complement are RE**, then L is recursive. A language can be RE but its complement need not be RE.

### Are regular languages recursive?

A recursive language is one that can be decided by a Turing machine, which has a (potentially) infinite tape for memory. Regular languages are recursive because **you can make a TM equivalent to a FA by not using the tape**.

### Which of the following is true the complement of recursive language is recursive?

(C) The complement of a recursive language is either **recursive or recursively enumerable**.

### Which automata is more powerful?

The most general and powerful automata is **the Turing machine**.

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

**The Universal Language L**

L_{u} is recursively enumerable but not recursive. L_{u} 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.

### What is meant by recursive enumerable languages?

A recursively enumerable language is **a formal language for which there exists a Turing machine (or other computable function) which will enumerate all valid strings of the language**. … Contrast this to recursive languages, which require that the Turing machine halts in all cases.

### What are recursive functions give three examples?

Simple examples of a recursive function include **the factorial**, where an integer is multiplied by itself while being incrementally lowered. Many other self-referencing functions in a loop could be called recursive functions, for example, where n = n + 1 given an operating range.

### Does TM accept empty language?

If a TM**(Turing Machine) accepts NO input string**(even the blank), then its language is empty. If a TM ONLY accepts the blank string(meaning that there is nothing on the tape except for the default blank characters), then its language has only one item and it is the blank string.

### Is an empty language decidable?

Languages (1) and (2) are, respectively, {0, 1}* and the empty language, both of which **are decidable** (so there are TMs that always halt that accept those languages).