The equivalence problem and correctness formulas for a simple class of programs

This paper is concerned with the semantics (or computational power) of very simple loop programs over different sets of primitive instructions. Recently, a complete and consistent Hoare axiomatics for the class of {x ← 0, x ← y, x ← x + 1, x ← x ∸ 1, do x … end} programs which contain no nested loops, was given, where the allowable assertions were those formulas in the logic of Presburger arithmetic. The class of functions computable by such programs is exactly the class of Presburger functions. Thus, the resulting class of correctness formulas has a decidable validity problem. In this paper, we present simple loop programming languages which are, computationally, strictly more powerful, i.e., which can compute more than the class of Presburger functions. Furthermore, using a logical assertion language that is also more powerful than the logic of Presburger arithmetic, we present a class of correctness formulas over such programs that also has a decidable validity problem