Partially Wrong Completions ' Charles B . Dunham Department of Computer Scienc e The University of Western Ontario London, Canada N6A 5B 7 The completion of a partial recursive function is a total computable function that agree s with where it is defined . If we could complete the function corresponding to a Universal Turin g machine, then there would be completions for all partial recursive functions : not only that, bu t if we knew that master completion, we would have an effective completion of all partial recursiv e functions . But the author (1968) and Rogers (1967, p .37) independently came up with relativel y elementary partial recursive functions which are not completable . Rogers has a challenging problem , 2-33, on completions . This year the author started considering a variation which could have applicability. There are a lot of problems, which range from difficult to impossible to solve, but for which potential solution s can be checked very easily. This is true for the author's former major interest, alternating minima x approximation, for integer factorization, and some NP-complete problems . The variation, actuall y a generalization, is that we permit to be wrong some of the time . Now if i, were only wrong a finit e number of times, attaching a pre-processor before 0 to pick out the offending argument and insert the right answer would produce the classical completion, which we have just shown is impossibl e in general . So we will have to permit which is wrong a countable number of times. It doesn' t seen of any practical value or theoretical interest to have which is only right a finite number of times, since this could be implemented by a pre-processor which picks up the target argument s and outputs from a table, while using any total function for the rest . So the only interesting O's are those for which right answers and wrong answers are both countable . I am asking the question , do such 5 always exist? but the real question is, would they be of any value? (as the theory o f partial recursive functions is purely formal, this would come from outside the theory) . For the universal Turing machine expressed in the Busy Beaver formulism, we could run all configuration s a computable number of times, with a countable block of configurations stopping very early wit h the right answer and a countable number of cases cut off with wrong answers (by uncomputabilit y results) . This at first doesn't seem to be of value or even interest . But actual succesful approaches to difficult to impossible problems seem often to try for a while to get the correct answer or on e very close, but if results don't appear after a computable time, a fast heuristic or just a departur e occurs, which isn't much different from the disparged approach to the universal Turing machin e above . REFERENCE S C .DUNHAM (1968), An uncompletable function, Amer . Math . Monthly 75, 1104-1105 . H .ROGERS (1967), Theory of Recursive Functions and Effective Computability, McGraw-Hill . 'Copyright ©Charles B . Dunham, 2000

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