## What problems are computable?

## What problems are computable?

A mathematical problem is computable if it can be solved in principle by a computing device. Some common synonyms for “computable” are “solvable”, “decidable”, and “recursive”. Hilbert believed that all mathematical problems were solvable, but in the 1930’s Gödel, Turing, and Church showed that this is not the case.

**Why is busy beaver not computable?**

The busy beaver function is non-computable, because it grows faster than any computable function! Proof: Let f be any computable function. Consider M: X then MF then MF where X: blank → x 1’s. Note X has x states.

### Is a Turing machine a function?

A Turing machine is a computing device that on input a finite sequence of strings might produce a string. The machine has two parts: a control unit and a tape. The control unit is characterized by a finite set Q of control states, a start state q0 ∈ Q and a transition function δ.

**What is a busy beaver number?**

The busy beaver number of a one-rule machine, or BB(1), is therefore 1. But adding just a few more rules instantly blows up the number of machines to consider. Of 6,561 possible machines with two rules, the one that runs the longest — six steps — before halting is the busy beaver. But some others simply run forever.

#### What problems are not computable?

Non-Computable Problems – A non-computable is a problem for which there is no algorithm that can be used to solve it. Most famous example of a non-computability (or undecidability) is the Halting Problem.

**What functions are not computable?**

The set of finitary functions on the natural numbers is uncountable so most are not computable. Concrete examples of such functions are Busy beaver, Kolmogorov complexity, or any function that outputs the digits of a noncomputable number, such as Chaitin’s constant.

## Is Busy Beaver Undecidable?

The Busy Beaver numbers are an undecidable set; if they were decidable, we could figure out BB(n) for each n, enabling us to decide the halting problem. They are also not recursively enumerable, but for a trickier reason.

**Is every function computable?**

I’d like to share a simple proof I’ve discovered recently of a surprising fact: there is a universal algorithm, capable of computing any given function!

### What is the blank tape problem?

The problem of finding an algorithm that, for any Turing machine, decides whether the machine eventually stops if it started on an empty tape; it has been proved that no such algorithm exists.

**Do Super Turing machines exist?**

Turing’s oracle machines are mathematical abstractions, and are not physically realizable.

#### What do you mean by unsolvable problems?

Definition: A computational problem that cannot be solved by a Turing machine. The associated function is called an uncomputable function. See also solvable, undecidable problem, intractable, halting problem.