The Computational Complexity of Program Schemata

Abstract

Program schemata are studied in order to find general properties of programming languages. But although there seems to be an agreement about the necessity of investigating general questions of program optimization, especially the computational complexity of program schemata, until recently only very few papers deal with this or related topics. D.M. Symes [Sm 71] and also N.A. Lynch [Ly 72] present a genera­lized theory of computational complexity for computations with oracles. If the oracle is a function f, then the complexity of a computation depends on the program on the input, and on f. The possibility that there may be several realizations of f with very different intrinsic computational complexities is not considered. R.L. Constable [Cs 73] in his axiomatic approach takes the pair (f, tf) rather than f as a variable for the computational complexity of functionals, where tf is the complexity of f. Such an approach seems to be more reasonable for practical con­siderations. A.V. Aho and J.D. Ullman [AU 70] (see also N. Bracha [Br 72] ) study optimization methods for loop-free program schemata, but they don’t introduce a precise concept of computational complexity. A.K. Chandra [Cd 73] studies the lengths of computation sequences for a certain class of program schemata and proves a speedup theorem. As in the approach of Symes and Lynch the complexity of every function of the interpretation is fixed here.