2 Answers. L is regular not closed for RE since this is **Regularity property**. L is finite not closed for RE since this is finiteness property.

## Why is it called recursively enumerable?

Recursive Enumerable (RE) or Type -0 Language

It means **TM can loop forever for the strings which are not a part of the language**. RE languages are also called as Turing recognizable languages.

## Are recursively enumerable languages infinite?

Proof: The set of strings is an infinite countable set. The set of languages is not countable because it is the powerset of the set of strings. Recursively enumerable languages are countable because TMs are countable. Therefore, recursively enumerable languages ⊂ **all languages**.

### What is a language L called if L is a recursively enumerable language?

In mathematics, logic and computer science, **a formal language** is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.e., if there …

### 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 enumerable languages decidable?

**Yes**. In particular, recursive (decidable) languages are a subset of the recursively enumerable languages, so anything that’s not recursively enumerable isn’t recursive (decidable).

### Does undecidable mean not recursively enumerable?

A language L is undecidable if L is not decidable. Thus, there is no Turing machine M that halts on every input and L(M) = L. **L is recursively enumerable** but not decidable. That is, any Turing machine M such that L(M) = L, M does not halt on some inputs.

### Are recursively enumerable sets closed under intersection?

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

### What language is not recursively enumerable?

**The Diagonalization Language L _{d}** is not Recursively Enumerable. We define L

_{d}, the diagonalization language, as follows: Let w

_{1}, w

_{2}, w

_{3}, . . . be an enumeration of all binary strings. Let M

_{1}, M

_{2}, M

_{3}, . . . be an enumeration of all Turing machines.

### 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.

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

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.

### 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.

### Which language is accepted by Turing machine?

Explanation: The language accepted by Turing machines are called **recursively ennumerable (RE)**, and the subset of RE languages that are accepted by a turing machine that always halts are called recursive.

### Is every enumerable language finite?

If all words of the given language are listed within finite time, the given **language is finite**.

### What is a decidable language?

(definition) Definition: **A language for which membership can be decided by an algorithm that halts on all inputs in a finite number of steps — equivalently**, can be recognized by a Turing machine that halts for all inputs. Also known as recursive language, totally decidable language.

### What is TOC recursive language?

A recursive language is **a formal language for which there exists a Turing machine** that, when presented with any finite input string, halts and accepts if the string is in the language, and halts and rejects otherwise.

### 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.

### 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 accepted by Turing machine?

The **turing machine accepts all the language even** though they are recursively enumerable. Recursive means repeating the same set of rules for any number of times and enumerable means a list of elements.

### How do you show not recursively enumerable?

**Consider the complement**

- Theorem. If a language L and its complement are both RE, they are both recursive.
- Proof. …
- So, if you can prove that L is not recursive but its complement is RE, then L is not RE.
- Theorem. …
- Proof. …
- So, if you can reduce the halting problem for M to your problem, your problem is not RE.

### What is diagonalization language?

The language Ld, the diagonalization language, is **the set of strings Wi such that Wi is not in L(Mi)**. That is, Ld consists of all strings w such that the TM M whose code is w does not accept when given w as input. The reason Ld is called a “diagonalization” language can be seen if we consider the followingfigure.

### What is non 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.

### Is Re closed under intersection?

This can make things seem very nice and symmetric: **r.e. sets are closed under both union and intersection**. However, this is not the case once we start talking about infinite unions and intersections. Obviously the r.e. sets are not closed under arbitrary infinite unions/intersections.