Abstract

We report a case study on semi-automatic synthesis of optimal algorithms for square root problem: given the real number x and the error bound e, find the real interval such that it contains the radical of x and its width is less than e. As usual, we begin by fixing an algorithm schema, namely, iterative refining: the algorithm starts with an initial interval and repeatedly updates it by applying a refinement function, say f, on it until it becomes narrow enough. Then the synthesis amounts to finding a refinement function f that ensures that the algorithm is correct (loop invariant), terminating (contraction), and optimal. All these can be formulated as quantifier elimination over the real numbers. Hence, in principle, they can be all carried out automatically. However the computational requirement is so huge, making the automatic synthesis practically impossible with the current general quantifier elimination software. Hence, we did some hand derivations and were able to synthesize semi-automatically an optimal algorithm under suitable assumptions. We hope that the ideas behind the hand derivations could be eventually turned into an algorithm. Then we would have an automatic synthesis of optimal square-root algorithm. This is joint work with Hoon Hong.