Browse Reports

An honest function is one whose size honestly reflects its computation time. In 1969 Meyer and McCreight proved the "honesty theorem," which says that for every t, the t-computable functions are the same as the t''computable functions for some honest t''. Ways of constructing honest functions are considered in detail. It is shown that for any t there is an honest t'' such that the t-computable functions and the t'' computable functions are the same, and such that t'' is arbitrarily large on a dense set of argument. Moreover any construction method satisfying certain natural criteria will (almost) have this property. On the other hand it is shown that by relaxing these criteria we can guarantee that t'' t on a (weak) dense set. We can also guarantee that t'' will be bounded above by a predetermined recursive function on all but finitely many arguments. Finally, we show that in the case where t is monotone, t'' can also be made monotone. We consider the t-computable functions, and order these classes under set inclusion as t varies over the recursive functions. We show that given any total effective operator F and any recursive countable partial order R there is an r.e. sequence of machine running times T , T , ... T ...such that if iRj, then the T computable functions properly contain the F(T ) computable functions, and if i and j are incomparable, then F(T ) < T infinitely often and F(T ) < T infinitely often.