One possible approach to exact real arithmetic is to use linear fractional transformations to represent real numbers and computations on real numbers. We show how to determine the digits that can be emitted from a transformation, and present a criterion which ensures that it is possible to emit a digit. Using these results, we prove that the obvious algorithm to compute n digits from the application of a transformation to a real number has complexity O(n 2 ), and present a method to reduce this complexity to that of multiplying two n bit integers. 1 Introduction Linear Fractional Transformations (LFT's) provide an elegant approach to real number arithmetic [5,14,9,12,10,4]. One-dimensional LFT's x 7! ax+c bx+d are used as digits and to implement basic unary functions, while two-dimensional LFT's (x; y) 7! axy+cx+ey+g bxy+dx+fy+h provide binary operations such as addition and multiplication, and can be combined to obtain infinite expression trees denoting transcendental functions...

Numerical computations form an essential part of almost any real-world program. Traditional approaches to termination of logic programs are restricted to domains isomorphic to N , more recent works study termination of integer computations. Termination of computations involving real numbers is cumbersome and counter-intuitive due to rounding errors and implementation conventions. We present a novel technique that allows us to prove termination of such computations.

Mathematical programs (MPs) are a class of constrained optimization problems that include linear, mixed-integer, and disjunctive programs. Strategies for solving MPs rely heavily on various transformations between these subclasses, but most are not automated because MP theory does not presently treat programs as syntactic objects. In this work, we present the first syntactic definition of MP and of some widely used MP transformations, most notably the big-M and convex hull methods for converting disjunctive constraints. We use an embedded OCaml DSL on problems from chemical process engineering and operations research to compare our automated transformations to existing technology—finding that no one technique is always best—and also to manual reformulations—finding that our mechanizations are comparable to human experts. This work enables higher-level solution strategies that can use these transformations as subroutines.

We present two algorithms for computing the root, or equivalently the fixed point, of a function in exact real arithmetic. The first algorithm uses the iteration of the expression tree representing the function in real arithmetic based on linear fractional transformations and exact floating point. The second and more general algorithm is based on a trisection of intervals and can be compared with the well-known bisection method in numerical analysis. It can be applied to any representation for exact real numbers; here it is described for the sign binary system in [\Gamma1; 1] which is equivalent to the exact floating point with linear fractional transformations. Keywords : Shrinking intervals, Normal products, Exact floating point, Expression trees, Sign Binary System, Iterative method, Trisection. 1 Introduction In the past few years, continued fractions and linear fractional transformations (lft), also called homographies or Mobius transformations, have been used to develop various...

In [Sch00] a bijection has been established, for the case of semilattices, between invariant partial metrics and semivaluations. Semivaluations are a natural generalization of valuations on lattices to the context of semilattices and arise in many di#erent contexts in Quantitative Domain Theory ([Sch00]). Examples of well known spaces which are semivaluation spaces are the Baire quasi-metric spaces of [Mat95], the complexity spaces of [Sch95] and the interval domain ([EEP97]).

Introduction One approach to exact real number arithmetic is based on linear fractional transformations (LFT's). Specifically, there is a framework and an implementation put forward by the group around Edalat and Potts at Imperial College [Pot96, PE97, PEE97, EP97]. The introduction of tensors and the absorption rules in this framework seem rather ad hoc. It is the purpose of this note to shed some light on this subject. The reader should be familiar with the definitions of tensors and the operations and absorption rules from [PE97]. Specifically, we are looking for answers to the following questions: 1. Why do tensors describe 2-dimensional LFT's? 2. Is there an explanation for the absorption rules? 3. Why do left- and right-absorption commute? The framework of [PE97] provides a "semantic" answer to this questions saying: 1. Every 2-dimensional LFT is described by 8 parameters, and we can arrange these parameters in a 2 \Theta 4-matrix. 2. The

The aim of this work is to characterize constructive real numbers through a minimal axiomatization. We introduce, discuss and justify 16 constructive axioms. Then we address their expressivity considering the alternative axiomatizations. 1 Overview of the work Real numbers are classically dened as a complete ordered eld, but this is not the case in a constructive logic setting: actually the totality of order is not a constructive property. This work tries to understand (again) constructive real numbers. Our main contribution is a new system of axioms, synthesized with the aim of being minimal, i.e. of assuming the least number of primitive notions and properties. Such a system is consistent with respect to reference models we have in mind | (equivalence classes of) Cauchy sequences [TvD88] and co-inductive streams of digits [CDG00]| and will be compared to other proposals in the literature [Bri99, GPWZ00]. The expressive power of the axioms will be addressed in order to guara...

Digital computers permeate our physical world. This phenomenon creates a pressing need for tools that help us understand a priori how digital computers can affect their physical environment. In principle, simulation can be a powerful tool for animating models of the world. Today, however, there is not a single simulation environment that comes with a guarantee that the results of the simulation are determined purely by a real-valued model and not by artifacts of the digitized implementation. As such, simulation with guaranteed fidelity does not yet exist. Towards addressing this problem, we offer an expository account of what is known about exact real arithmetic. We argue that this technology, which has roots that are over 200 years old, bears significant promise as offering exactly the right technology to build simulation environments with guaranteed fidelity. And while it has only been sparsely studied in this large span of time, there are reasons to believe that the time is right to accelerate research in this direction. ∗ This research was sponsored by the NSF under Award 0439017,

When solving machine learning problems, there is currently little automated support for easily experimenting with alternative statistical models or solution strategies. This is because this activity often requires expertise from several different fields (e.g., statistics, optimization, linear algebra), and the level of formalism required for automation is much higher than for a human solving problems on paper. We present a system toward addressing these issues, which we achieve by (1) formalizing a type theory for probability and optimization, and (2) providing an interactive rewrite system for applying problem reformulation theorems. Automating solution strategies this way enables not only manual experimentation but also higher-level, automated activities, such as autotuning. Keywords: machine learning, algorithm derivation, interactive modeling, type theory

We show that, in a fairly general setting including higher-types, may, must and probabilistic testing are semidecidable. The case of must testing is perhaps surprising, as its mathematical definition involves universal quantification over the infinity of possible outcomes of a non-deterministic program. The other two involve existential quantification and integration. We also perform first steps towards the semi-decidability of similar tests under the simultaneous presence of non-deterministic and probabilistic choice. Keywords: Non-deterministic and probabilistic computation, higher-type computability theory and exhaustible sets, may and must testing, operational and denotational semantics, powerdomains. 1