Nonexistence of Program Optimizers in Several Abstract Settings

For various settings and various dynamic criteria for gauging optimality of programs, there does not exist a master program (optimizer) 9 ~ such that if P is any program which computes a partial function possessing an optimal program, then :'~, operating on the program P as input, halts eventually and outputs an optimal program P ' for computing that partial function. Optimality can be gauged by a criterion suggested by a variant of M. Blum's compression theorem for an arbitrary complexity measure, by optimality except for a linear factor for amount of memory used by a Turing machine, or by optimality within E on a RASP. Thus, our techniques are compatible with techniques for producing optimal programs which are as diverse as upward diagonalization, downward diagonalization, and the size arguments of Hartmanis. Our nonexistence results continue to hold even if we only ask that an optimizer behave properly when the input program P satisfies certain convergence properties (e.g., when P computes a total function) and possesses an equivalent optimal program which is neither too hard nor too easy to compute.