What are the limits (if any) of computers?

To answer this question, we need to clarify the notions of **computer**
and **computation**

A good theory should be as simple as possible, but not simpler.

—Albert Einstein

A **computation** is any process that can be described by a set of
unambiguous instructions.

Alan Turing invented the idea of a **Turing Machine **in
1935-36 to describe computations.

- a Turing Machine is a purely theoretical device
- control unit can be in one of a finite number of
**states** - each tape cell holds one of a finite number of
**symbols** - read/write head overwrites current symbol on each step
- read/write head moves one square to the
**left**or**right**on each step

State |
Symbol |
New State |
New Symbol |
Move |

1 | 0 |
1 | 1 |
R |

1 | 1 |
1 | 0 |
R |

1 | b |
2 | b |
R |

Start State: 1

Halt State: 2

This Turing machine can be viewed as a **function** that takes an input
sequence and returns the corresponding inverted sequence (all **1**'s
replaced by **0**'s and vice versa).

**1100** `-->` **0011** **01**
`-->` **10**

Turing machine description:

(1, **0**, 1,
**1**, R)

(1, **1**, 1, **0**,
R)

(1, **b**, 2, **b**, R)

Turing Machine Simulator from *The Analytical
Engine*

Is this really enough to compute *everything*?

Consider:

- Memory is not a problem (tape is infinite)
- Efficiency is not a problem (purely theoretical question)
- Data representation is not a problem (we can use binary, or whatever symbols we like)
- All attempts to characterize computation have turned out to be equivalent
- Partial recursive functions (Gödel, Kleene)
- Lambda calculus (Church)
- Post production systems (Post)
- Turing machines (Turing)
- Random-access machines

Church-Turing
ThesisAnything that can be computed can be computed by
a Turing machine |

Choice of programming language doesn't really matter — all are "Turing equivalent"

When we talk about Turing machines, we're really talking about computer programs in general.

CorollaryIf the human mind is really a kind of computer, it must be equivalent in power to a Turing machine |

Is there anything a Turing machine **cannot** do, even in principle?
*YES!*

Example: Looper TM eventually halts on input **0000bbb...** but loops
forever on input **0000111bbb...**

No Turing machine can infallibly tell if another Turing machine will get stuck in an infinite loop on some given input.

In other words, no computer program can infallibly tell if another computer program will ever halt on some given input.

Put another way, no computer program can infallibly tell if another program is completely free of bugs.

How did Turing prove that such a program is in principle impossible?

We'll use JavaScript instead of Turing machines to illustrate the argument, but the argument is valid no matter what language we use to describe computations (JavaScript, BASIC, Turing machines, Excel, etc.)

Turing's approach was to **assume** that a loop-detector program could be
written.

He then showed that this leads directly to a **logical contradiction**!

So, following in Turing's steps, let's just assume that it's possible to
write a JavaScript program that correctly tells whether other JavaScript programs will
eventually halt when given particular inputs. Let's call our hypothetical
program ** WillHalt**.

function WillHalt(program, data) { . . . ...lots of complicated code... . . . if ( ...more code... ) { return true; } else { return false; } }

For example, let's write a couple of simple JavaScript programs to test
** WillHalt**:

function Halter(input) { alert('done'); } function Looper(input) { while (input == 1) { input = 1; } }

** Halter** always halts, no matter what input
we feed it:

returns`WillHalt("function Halter(input){alert('done');}", 1)``true`returns`WillHalt("function Halter(input){alert('done');}", 2)``true`

** Looper** loops forever if we happen to feed
it the value 1. Any other value will cause it to halt:

returns`WillHalt("function Looper(input){while(input==1){input=1;}}", 1)``false`returns`WillHalt("function Looper(input){while(input==1){input=1;}}", 2)``true`

So far, we have every reason to believe that ** WillHalt** could exist, at least in principle, even though it
might be a rather hard program to write.

At this point, Turing says "OH YEAH? If ** WillHalt**
exists, then I can define the following program called

`function Turing(program)
{`` var n;`` if (WillHalt(program,
program) == true) {`` while
(2 + 2 == 4) {``
n = 0;`` }`` } else {`` alert('done');`` }``}`

"Yes," we say, "so what?"

Turing laughs and says "Well what happens when I feed
the ** Turing** program to

What happens indeed? Let's analyze the situation:

- The
program uses`Turing`to analyze the JavaScript program given to it as input:`WillHalt`- If the JavaScript program halts when fed itself as input, the
program loops forever.`Turing` - If the JavaScript program loops forever when fed itself as input, the
program immediately halts.`Turing`

- If the JavaScript program halts when fed itself as input, the
- But if the JavaScript program happens to be
**the**, then we have:`Turing`program itself- If the
program halts when fed itself as input, the`Turing`program loops forever.`Turing` - If the
program loops forever when fed itself as input, the`Turing`program halts.`Turing`

- If the
- This is a blatant logical contradiction!
can neither halt nor loop forever; it doesn't make sense either way.`Turing`

Thus our original assumption about the existence of ** WillHalt** must have been invalid, since as we've seen, it's easy to define
the logically-impossible

*Q.E.D.*

ConclusionThe task of deciding if an arbitrary computation will ever terminate cannot be described computationally. |

Turing discovered another amazing fact about Turing machines:

A single Turing machine, properly programmed, can simulate any other Turing machine.

Such a machine is called a Universal Turing Machine (UTM)

- The UTM accepts a
**coded description**of a Turing machine and simulates the behavior of the machine on the input data. - The coded description acts as a
**program**that the UTM executes. - The UTM's own internal program is fixed (and rather complicated).
- The existence of the UTM is what makes computers fundamentally different from other machines such as telephones, CD players, VCRs, refrigerators, toaster-ovens, or cars.
- Computers are the only machines that can simulate any other machine to an arbitrary degree of accuracy. (Example: a JavaScript calculator)
- This is the real reason why computers have taken over the world.

How can we "encode" a Turing machine? Here's one way:

Example: Looper TM

States: 1, 2 `--> 0`,

Symbols:

Moves: L, R

Rule 1: (1, **0**, 1, **0**, R) `-->`
`0101010100`

Rule 2: (1, **1**, 1, **1**, L)
`-->` `01001010010`

Rule 3: (1,
**b**, 2, **b**, R) `-->`
`010001001000100`

`1110101010100110100101001011010001001000100111`

We could run our hypothetical Loop-detector Turing machine on the above
encoding with the input ** 0000111**:

Eventually the machine would halt with a single ** 1 **as output,
meaning an infinite loop was detected:

But, alas, we know that such a loop-detecting Turing machine is impossible, as Turing showed.