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)
- Unlimited register machines (Cutland)
- 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 Scheme instead of Turing machines to illustrate the argument, but the argument is valid no matter what language we use to describe computations (Scheme, Turing machines, BASIC, Java, 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 Scheme program that correctly tells whether other
Scheme
programs will eventually halt when given particular inputs. Let's
call our hypothetical program ** halts?**.

**
**

(define halts? (lambda (program input) (cond ...lots of complicated code... ((...more code...) #t) (else #f))))

For example, let's write a couple of simple Scheme
programs to test ** halts?**:

(define halter (lambda (input) 'done)) (define looper (lambda (input) (cond ((= input 1) (looper input)) (else 'done))))

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

returns`(halts? halter 1)``#t`returns`(halts? halter 2)``#t`

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

returns`(halts? looper 1)``#f`returns`(halts? looper 2)``#t`

So far, we have every reason to believe that ** halts?**
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 ** halts?**
exists, then I can define the following program called

(define turing (lambda (program) (cond ((halts? program program) (looper 1)) (else 'done))))

At this point, we say "Yes, so what?"

Turing laughs and says "Well, what happens when I feed
the ** turing** program to itself?"

(turing turing)

What happens indeed? Let's analyze the situation:

- The
program uses`turing`to analyze the Scheme program given to it as input:`halts?` - If the Scheme program halts when fed itself as input,
the
program loops forever.`turing` - If the Scheme program loops forever when
fed itself as input, the
program immediately halts.`turing` - But if the Scheme program happens to be
**the**, then:`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` - This is a blatant logical contradiction!
can neither halt nor loop forever; it doesn't make sense either way.`(turing turing)`

Thus our original assumption about the existence of ** halts?**
must have been invalid, since 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.
- 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)

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.