Subrecursive program schemata I & II(I. Undecidable equivalence problems, II. Decidable equivalence problems)

Abstract

The study of program schemata and the study of subrecursive programming languages are both concerned with limiting program structure in order to permit a more complete analysis of algorithms while retaining sufficiently rich computing power to allow interesting algorithms. In this paper we combine these approaches by defining classes of subrecursive program schemata and investigating their equivalence problems. Since the languages are all subrecursive, any scheme written in any one of them must halt (as long as we assume the basic functions and predicates are all total). Hence equivalence of schemes is the first question of interest we can ask about these languages. We consider schematic versions of various subrecursive programming languages similar to the Loop language. We distinguish between Pre-Loop and Post-Loop languages on the basis of whether the exit condition in an iteration loop is tested before iteration, as in Algol (Pre-), or after iteration, as in FORTRAN (Post-). We show that at the program level all these languages have the same computing power (the primitive recursive functions) and all have unsolvable equivalence problems (of arithmetic degree &pgr;01). But at the level of schemes, Pre-Loop has an unsolvable equivalence problem, while at least one formulation of Post-Loop has a solvable equivalence problem.