User talk:Likebox/Incompletness

slight emendation
0. Halting problem: Given any computer program, there does not exist a computer testing-program which tells you whether or not the computer program under test eventually halts.
 * Turing's original expression of the same notion: "...there can be no machine E which, when supplied with the S.D. of an arbitrary machine M, will determine whether M ever printes a given symbol (0 say)'" (italics in original, Turing 1936-7 reprinted in Davis 1965:134)
 * The key here is the use of the ∀ in the in the formalization of this statement: ~(∃E(∀M...))

1. Kleene's separation lemma: there does not exist s a program program which that tells you whether another computer program R returns a yes or a no.
 * Proof: construct SPITE to print its code, calculate the expected output, and do the opposite.

In recursion theory, Kleene's separation lemma is usually stated this way: "There is a subset S of a recursively enumerable set R which cannot be recursively separated from its complement R-S". R is the set of programs that return yes or no. S is the set of programs that return yes, and R-S is the set of programs that return no.

2. There does not exist a program which tells you whether any other computer program finishes in polynomial or exponential time.
 * Proof: construct SPITE(N) to ignore the input N, print its code, calculate the expected running time, then run an algorithm of the opposite running time on N. The first part does not depend on the input, and runs in fixed time.

3. word problem for groups: There does not exist a program which determines whether two elements in a finitely presented group s are equivalent. The decision problem was first posed by Max Dehn in 1911. Alan Turing proved that the problem for finitely presented monoids (no inverses) was equivalent to the halting problem and therefore undecidable, and the problem for groups was shown to be equivalent to the halting problem in 1952.

4. Behavior of Cellular automata: There does not exist a program which can determine whether an arbitrary starting configuration of Conway's Game of Life will eventually reach the empty configuration.

All the practical decision problems which have been shown to be undecidable, including all the ones above, are equivalent to the halting problem. Turing showed that there are much stronger degrees of undecidablity (Turing 1939), while Post showed that there are weaker ones. It remains an open question to find natural mathematical forms for these types of undecidable problems.

Reply
I can't cite a source for most of the stuff, because I am lazy. But I'll do some research tomorrow and try to fix it up. Thanks for reading it over. Likebox 06:32, 15 October 2007 (UTC)

counter-machines, register-machines, RAMs and RASPS
I have a lot of the references re counter machine, RAM, and RASP in .pdf so if you see any that you want to read, lemme know at my talk page. Bill Wvbailey 19:28, 15 October 2007 (UTC)