Effective Computation over the Real Numbers

Extended Turing Machines We will obtain our simplest model, the ETM, by adding to Turing Machines the ability to add, multiply, and compare real numbers. We allow real parameters to appear in an ETM's program. In this manner the set of real numbers becomes the underlying domain of our computations much as a finite set of symbols is the domain of computation of a Turing machine. As successively deeper levels of GLB are allowed, successively stronger computational systems result. Functionals which can be approximated iteratively at one level (such as differentiation) can be computed exactly at a somewhat higher level. By closing our computation scheme under the GLB-operation we obtain a system closed under limiting operations in general, but one for which the question of effectively choosing elements from effectively defined sets is presently unanswered. Indeed, the solution to this last question may depend upon higher axioms of set theory. More explicitly, take as the definition of an (ordinary) Turing machine a device consisting of (1) an infinite tape divided into squares on which symbols from some finite alphabet are written. (2) a read-write head capable of moving one square left or right under the control of (3) a finite-state-control (fsc). We consider an fsc to be a flowchart consisting of an allowable interconnection of boxes of the following forms: multiply number on tape with that in s and store result in s branch according to sign of number in s the real number beneath the head is copied into s the real number in s is written on square of top track currently scanned by the head (for c some real number) the constant c is put into s add number on top track square currently scanned to that in s and store result in s (j) s+T-+ s (i) c-+ s (1) Test s (k) s*T-+ s (h) Store s (g) Load s flowchart Any number of arrows may enter a HALT box, but none leave it (c) TEST This box has 1 outgoing arrow for each symbol in the Turing machine's alphabet. TEST causes a branch according to the symbol being scanned; if the ith symbol of the alphabet appears beneath the head when ~ TEST box is entered , control passGs to the flow-chart box indicated by the ith arrow leaving the TEST box (d) Write symbol i The ith symbol of the Turing machine's alphabet …