Abstract
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. In related work, we examine the expressive 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 doable 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←x∸1, do x ... end, if x=0 then x←c}-programs is exactly the class of Presburger functions. When the conditional instruction is changed to “if x=0 then x←y+1”, then the class of computable functions is significantly enlarged, enough so, in fact, as to render many decision problems (e.g. equivalence) undecidable.
This research was supported in part by NSF Grant MCS 83-04756.
This research was supported in part by The University Research Institute, The University of Texas at Austin and the IBM Corporation.
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Ibarra, O.H., Rosier, L.E. (1984). The equivalence problem and correctness formulas for a simple class of programs. In: Chytil, M.P., Koubek, V. (eds) Mathematical Foundations of Computer Science 1984. MFCS 1984. Lecture Notes in Computer Science, vol 176. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0030314
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DOI: https://doi.org/10.1007/BFb0030314
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