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Formalising exact arithmetic in type theory
 New Computational Paradigms: First Conference on Computability in Europe, CiE 2005
, 2005
"... Abstract. In this work we focus on a formalisation of the algorithms of lazy exact arithmetic à la Potts and Edalat [1]. We choose the constructive type theory as our formal verification tool. We discuss an extension of the constructive type theory with coinductive types that enables one to formalis ..."
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Abstract. In this work we focus on a formalisation of the algorithms of lazy exact arithmetic à la Potts and Edalat [1]. We choose the constructive type theory as our formal verification tool. We discuss an extension of the constructive type theory with coinductive types that enables one to formalise and reason about the infinite objects. We show examples of how infinite objects such as streams and expression trees can be formalised as coinductive types. We study the type theoretic notion of productivity which ensures the infiniteness of the outcome of the algorithms on infinite objects. Syntactical methods are not always strong enough to ensure the productivity. However, if some information about the complexity of a function is provided, one may be able to show the productivity of that function. In the case of the normalisation algorithm we show that such information can be obtained from the choice of real number representation that is used to represent the input and the output. 1
Two Algorithms for Root Finding in Exact Real Arithmetic
, 1998
"... 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. T ..."
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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 wellknown 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...
A note on tensors in the LFT approach to exact real arithmetic
, 1997
"... 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 ..."
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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 2dimensional LFT's? 2. Is there an explanation for the absorption rules? 3. Why do left and rightabsorption commute? The framework of [PE97] provides a "semantic" answer to this questions saying: 1. Every 2dimensional LFT is described by 8 parameters, and we can arrange these parameters in a 2 \Theta 4matrix. 2. The