# Are there countably many Turing machines?

## Are there countably many Turing machines?

Since every Turing machine can be encoded as a finite string over a finite alphabet Q ∪ Σ ∪ Γ ∪ {R, L}, the set of all Turing machines is countable. On the other hand, every language over Σ is uniquely described as an infinite 0,1-sequence.

### Why are there countably many Turing machines?

Because each Turing machine can recognize a single language and there are more languages than Turing machines, some languages are not recognized by any Turing machine” (178).

#### Why are Turing machines countably infinite?

The first key observation is that the set of all Turing machines is countable. Each Turing machine has a string encoding, and the set of all strings Σ∗ is countable. Intuition: While there are infinitely many strings, any single string has finite length. Futhermore, there are a finite number of strings of that length.

Are there Countably many decidable languages?

As there are only countably many Turing machines over a finite alphabet, there can only be countably many decidable languages over that same alphabet. of n zeroes. These are all decidable, and there are infinitely many of them.

Are Turing machines infinite?

More specifically, it is a machine (automaton) capable of enumerating some arbitrary subset of valid strings of an alphabet; these strings are part of a recursively enumerable set. A Turing machine has a tape of infinite length on which it can perform read and write operations.

## What is a countable language?

1. Countable and Uncountable Sets. Two sets A and B are the same size if there is one-to-one correspondence (one-to-one, onto mapping) from A to B. A set A is countable if either it is finite or it has the same size as the set of integers {1, 2, 3, … }. Every language is countable.

### Are there infinite decidable languages?

No, there are many infinite languages that are decidable. One trivial example is the language {n € N | a^n} , i.e. the language of words that only contain the letter “a”. This language can be matched by the regular expression a* .

#### Is there a language for every Turing machine?

The TM will then either accept the string, reject the string, or loop on the machine. The language of a TM is defined as the set of all the strings it accepts. Not every language is the language of a Turing machine – that’s one of the landmark results of theoretical computer science.

What is countable and uncountable set in automata?

A set S is countable if there is a bijection f:N→S. An infinite set for which there is no such bijection is called uncountable. Proposition 1.19. Every infinite set S contains a countable subset. Proposition 1.19.

Are integers countably infinite?

For example, the set of integers {0,1,−1,2,−2,3,−3,…} is clearly infinite. However, as suggested by the above arrangement, we can count off all the integers. Counting off every integer will take forever.

## What is automata infinite language?

(An infinite language is a language with infinitely many strings in it. {an | n ≥ 0}, {ambn | m, n ≥ 0}, and {a, b}∗ are all infinite regular languages.) First, note that this can only be true for infinite regular languages. (

### Can a Turing machine accept an infinite language?

Yes, a Turing machine can decide that langauge: it just looks at the first character and accepts or rejects without even needing to look at the rest of the string.

#### How many tuples are there in a Turing machine?

7-tuple
Formally, a Turing machine (TM) is a 7-tuple consisting of states Q, alphabet Σ, tape alphabet Γ, transition δ, and starting/accept/reject states q0, qaccept and qreject.

What is Turing machine in automata?

Definition. A Turing Machine (TM) is a mathematical model which consists of an infinite length tape divided into cells on which input is given. It consists of a head which reads the input tape. A state register stores the state of the Turing machine.

What are the states in a Turing machine?

A given Turing machine has a fixed, finite set of states. One of these states is designated as the start state. This is the state in which the Turing machine begins a computation. Another special state is the halt state.