On simple programs with primitive conditional statements

This paper is concerned with the expressive power (or computational power) of loop programs over different sets of primitive instructions. In particular, we show that an {x ← 0, x ← y, x ← x + 1, do x … end, if x = 0 then y ← z}-program which contains no nested loops can be transformed into an equivalent {x ← 0, x ← y, x ← x + 1, do x … end}-program (also without nested loops) in exponential time and space. This translation was earlier claimed, in the literature, to be obtainable in polynomial time, but then this was subsequently shown to imply that PSPACE = PTIME. Consequently, the question of translatability was left unanswered. Also, we show that the class of functions computable by {x ← 0, x ← y, x ← x + 1, x − 1, do x … end, if x = 0 then x ← c}-programs is exactly the class of Presburger functions.