On weak and weighted computations over the real closure of Q

Blum, Shub and Smale [2] introduced the Blum–Shub–Smale (BSS) model of computation over the real numbers with the goal of modeling the kind of computations done in numerical analysis. Characteristic features of this model are the constant (unit) size for all real numbers and the unit cost arithmetic. The latter is in contrast with the Turing machine in which the cost of arithmetic operations grows with the size of the terms to be operated. In particular, iterated multiplications are increasingly expensive. For instance, while in the BSS model we can compute 22 n in n operations, and thus with cost n, in the Turing model the same computation will take time at least 2n. A variation on the BSS model attempting to get closer to the Turing machine in the sense above (i.e. a model in which iterated multiplication is somehow penalized) was introduced by Koiran [3]. This model, which Koiran called weak, takes inputs from R∞ (the disjoint union of Rn for n¿1) but no longer measures the cost of the computation as the number of arithmetic operations performed by the machine. Instead, the cost of each individual operation x ◦y depends on the sequences of operations which lead to the terms x and y from the input data and the machine constants. It is important to remark that, since inputs to weak machines are arbitrary real numbers, there is no modi:cation on size measuring. That is, an element in Rn has size n just as in the basic BSS model. Thus, input size is measured in the same way for the weak and the basic BSS models. Very recently, Malajovich [5] took another approach to make the BSS model closer to the Turing one. Let Ralg denote the real closure of Q i.e. the set of real numbers