Real Computation

`Standard' models of computation—Turing machines, Random Access Machines (RAMs), and even parallel variants like the PRAM—all operate on discrete objects: bits, Booleans, integers; at most, rational numbers (fractions) are considered in the form of numerator/denominator pairs of integers. These are the entities to be read, stored, processed, and output. A large amount of scientific computation, however, evolves around continuum rather than discrete problems. Fields like fluid dynamics, computational material science, and space mission design are just a few of them. In fact, most of applied mathematics amounts to solving various classes of ordinary or partial differential equations over the reals. This raises the need for a formal model to describe and analyze the prospects and limits of computations over $\mathbb{R}$. Leaving aside continuous-time (so-called analog, as opposed to clocked) computers, most models are either of an algebraic nature (such as the real-RAM aka Blum-Shub-Smale Machine) or produce approximations (like in Domain Theory, interval arithmetic, Recursive Analysis and Weihrauch's TTE); some even both (Hotz' analytic machines).