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1987: Impossible for computers

impossible for computers philip J Erdelsky Dr. Dobb Journal May 1987

1987: Impossible for computers

 

Things Computers Can Never Do

Philip J. Erdelsky

First Published in Dr. Dobb’s Journal May 1987

original article: efgh.com/math/impossible
french translation by Anna Chekovsky

Anyone who has witnessed the enormous improvements in computers in the last 40 years may get the impression that computers will eventually be able to solve every well-defined problem. Progress in language understanding and other forms of artificial intelligence has been disappointing, but human language is full of ambiguities, so that’s not a well-defined problem. Chess, on the other hand, is very well defined. Although it was once considered the epitome of intelligent activity, computers can now play chess better than all but a few human players.

Some problems, although well defined, are too large to be solved in a reasonable time even on our largest computers. But surely, if a computer could be freed from all limitations on time and memory, couldn’t it solve any well-defined problem?

The surprising answer to this question, which was known to mathematicians even before the first real computers were constructed, is no. There are some things no computer can ever do because it can be proved that there are no algorithms to do them — just as there is no way to square a circle with a compass and straightedge.

These things are not mere mathematical curiosities. They are things that programmers would like to have their computers do for them and things that the suppliers of software development tools would like to incorporate into their debuggers. Computer science curricula usually include the subject of uncomputable functions, but programmers who are not computer science majors sometimes ask for the impossible without realizing it.

Alan Turing in 1935 asked whether there is a method by which a computer program can determine whether any other computer program will halt. This is the famous “halting problem.” Turing showed that it has no solution.

A debugger with this ability would certainly be useful. Failure to halt normally is a common form of program failure. Moreover, the debugger could be applied successively to parts of the failed program to isolate the part that is hanging up.

It is not obvious that such a debugger is impossible. Of course, the debugger can’t just single-step the program to see if it halts. If the program doesn’t halt, the debugger could run forever without determining that this is the case. Or it might give up just as the program is about to terminate, as human programmers sometimes do. At some point, the debugger would have to be able to say, “Aha! This loop is infinite!” It seems as though a cleverly written debugger, having all the tools of modern high-level languages at its disposal, might be able to do that.

The impossibility proof is based on the following argument. If you have a debugger that can solve the halting problem, given unlimited time and memory, then you can use the same code to make the debugger do other things, some of which are self-contradictory and hence impossible.

The particular computer language is not important. If you can solve the halting problem for one language, you can solve it for another. Just use a compiler or other translation program before solving the halting problem. Notice that translating an assembly-language program to a higher level language is quite easy, although the object program is bound to be inefficient. The goal, however, is to show that a solution to the halting problem is impossible, not merely inefficient.

Turing himself proposed a minimal machine that has come to be called the Turing Machine. Its memory was supposed to be infinitely long but only one bit wide, and the machine had only sequential access to it, as with a tape. The programming language was essentially a flowchart with only a few basic commands. Nevertheless, Turing showed that his machine was able to emulate any other machine, given enough time and a suitable program. Such a construction is not necessary for our purposes-you can imagine that the computer is programmed in some familiar high-level language.

Now consider the problem of determining whether a program can print out a specified string S (with or without other output). If you can solve the halting problem, you can solve this problem. Just replace every print statement in the program with a routine that does not send the output to the printer but keeps track of the output and halts when the string S appears. Then, to keep the program from halting for any other reason, replace all the halt statements in the program with endless loops. Then solve the halting problem for the result.

Such a program would be …
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