Can humans solve the halting problem?
It might seem like humans could solve the halting problem. After all, a programmer can often look at a program and tell whether it will halt. It is useful to understand why this cannot be true. For simplicity, we will consider the halting problem for programs with no input, which is also undecidable.
To "solve" the halting problem means to be able to look at ''any'' program and tell whether it halts. It is not enough to be able to look at ''some'' programs and decide. Even for simple programs, it isn't clear that humans can always tell whether they halt. For example, we might ask if this pseudocode function, which corresponds to a particular Turing machine, ever halts:
function searchForOddPerfectNumber()
var int n = 1 // arbitrary-precision integer
loop {
var int sumOfFactors = 0
for factor from 1 to n - 1 {
if factor is a factor of n then
sumOfFactors = sumOfFactors + factor
}
if sumOfFactors = n then
exit loop
n = n + 2
}
return
This program searches until it finds an odd perfect number, then halts. It halts if and only if such a number exists, which is a major open question in mathematics. So, after centuries of work, mathematicians have yet to discover whether a simple, ten-line program halts. This makes it difficult to see how humans could solve the halting problem.
More generally, it's usually easy to see how to write a simple brute-force search program that looks for counterexamples to any particular conjecture in number theory; if the program finds a counterexample, it stops and prints out the counterexample, and otherwise it keeps searching forever. For example, consider the famous (and still unsolved) twin prime conjecture. This asks whether there are arbitrarily large prime numbers p and q with p+2 = q. Now consider the following program, which accepts an input N:
function findTwinPrimeAbove(int N)
int p = N
loop
if p is prime and p + 2 is prime then
return
else
p = p + 1
This program searches for twin primes p and p+2 both at least as large as N. If there are arbitrarily large twin primes, it will halt for all possible inputs. But if there is a largest pair of twin primes P and P+2, then the program will never halt if it is given an input N larger than P. Thus if we could answer the question of whether this program halts on all inputs, we would have the long-sought answer to the twin prime conjecture. It's similarly straightforward to write programs which halt depending on the truth or falsehood for many other conjectures of number theory.
Yes, of course you are right. That is why the Church-Turing Thesis is just that, a thesis, not a theorem. The argument above is for those that think humans are "clearly" smarter/better than computers. The point here is that this is in no way "clearly" true.
ReplyDeleteI thought the same thing after learning about the halting problem, and your article is the first satisfactory answer I've come across. Very good examples.
ReplyDelete(Coming reeeeaaaally late to this but better late than never :)
ReplyDeleteThe issue I find with your argument is that it involves human non-omniscient.
However the halting problem is deeper than that in the sense that even if we were to grant someone omniscience (e.g. suppose we are talking about Laplace's demon) he/she would still be unable to decide all problems.
So even if one knows the answer to any conjecture conceivable they would still be unable to, say, enumerate the real numbers.
The halting problem has nothing to do with human limitations but rather with the nature of nature itself.
Of course your argument works quite nicely to demonstrate to someone who does not want to bother with diagonilization arguments and so forth and they just claim that humans are more clever than computers (without assuming that humans are all omniscient though)
My question to such a person though would be "how on earth an animal such as the homo sapiens differs from a Turing machine."
-Nick