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20
A domaintheoretic account of Picard’s theorem
 In Proceedings of ICALP’04
, 2004
"... Abstract. We present a domaintheoretic version of Picard’s theorem for solving classical initial value problems in R n. For the case of vector fields that satisfy a Lipschitz condition, we construct an iterative algorithm that gives two sequences of piecewise linear maps with rational coefficients, ..."
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Cited by 18 (9 self)
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Abstract. We present a domaintheoretic version of Picard’s theorem for solving classical initial value problems in R n. For the case of vector fields that satisfy a Lipschitz condition, we construct an iterative algorithm that gives two sequences of piecewise linear maps with rational coefficients, which converge, respectively from below and above, exponentially fast to the unique solution of the initial value problem. We provide a detailed analysis of the speed of convergence and the complexity of computing the iterates. The algorithm uses proper data types based on rational arithmetic, where no rounding of real numbers is required. Thus, we obtain an implementation framework to solve initial value problems, which is sound and, in contrast to techniques based on interval analysis, also complete: the unique solution can be actually computed within any degree of required accuracy. 1
A computational model for multivariable differential calculus
 Proc. FoSSaCS 2005, LNCS
, 2005
"... Abstract. We introduce a domaintheoretic computational model for multivariable differential calculus, which for the first time gives rise to data types for differentiable functions. The model, a continuous Scott domain for differentiable functions of n variables, is built as a subdomain of the pro ..."
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Cited by 10 (6 self)
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Abstract. We introduce a domaintheoretic computational model for multivariable differential calculus, which for the first time gives rise to data types for differentiable functions. The model, a continuous Scott domain for differentiable functions of n variables, is built as a subdomain of the product of n + 1 copies of the function space on the domain of intervals by tupling together consistent information about locally Lipschitz (piecewise differentiable) functions and their differential properties (partial derivatives). The main result of the paper is to show, in two stages, that consistency is decidable on basis elements, which implies that the domain can be given an effective structure. First, a domaintheoretic notion of line integral is used to extend Green’s theorem to intervalvalued vector fields and show that integrability of the derivative information is decidable. Then, we use techniques from the theory of minimal surfaces to construct the least and the greatest piecewise linear functions that can be obtained from a tuple of n + 1 rational step functions, assuming the integrability of the ntuple of the derivative part. This provides an algorithm to check consistency on the rational basis elements of the domain, giving an effective framework for multivariable differential calculus. 1
RZ: A tool for bringing constructive and computable mathematics closer to programming practice
 CiE 2007: Computation and Logic in the Real World, volume 4497 of LNCS
, 2007
"... Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Obje ..."
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Cited by 6 (3 self)
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Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Objective Caml. The system supports a rich input language allowing descriptions of complex mathematical structures. RZ does not extract code from proofs, but allows any implementation method, from handwritten code to code extracted from proofs by other tools. 1
Denotational semantics of hybrid automata
 Proc. FoSSaCS 2006
, 2006
"... A hybrid automaton [16,2] is a digital, realtime system that interacts with an analogue environment. Hybrid automata are ubiquitous in all areas of modern ..."
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Cited by 6 (5 self)
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A hybrid automaton [16,2] is a digital, realtime system that interacts with an analogue environment. Hybrid automata are ubiquitous in all areas of modern
Inverse and implicit functions in domain theory
 Proc. 20th IEEE Symposium on Logic in Computer Science (LICS 2005
, 2005
"... C1 norm to the inverse function. A similar result holds for implicit functions. Combined with the domaintheoretic model for computationalgeometry, this provides a robust technique for construction of curves and surfaces in geometric modelling and CAD. 1. ..."
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Cited by 5 (5 self)
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C1 norm to the inverse function. A similar result holds for implicit functions. Combined with the domaintheoretic model for computationalgeometry, this provides a robust technique for construction of curves and surfaces in geometric modelling and CAD. 1.
A continuous derivative for realvalued functions
 New Computational Paradigms, Changing Conceptions of What is Computable
, 2008
"... We develop a notion of derivative of a realvalued function on a Banach space, called the Lderivative, which is constructed by introducing a generalization of Lipschitz constant of a map. As with the Clarke gradient, the values of the Lderivative of a function are nonempty weak * compact and conv ..."
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Cited by 3 (3 self)
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We develop a notion of derivative of a realvalued function on a Banach space, called the Lderivative, which is constructed by introducing a generalization of Lipschitz constant of a map. As with the Clarke gradient, the values of the Lderivative of a function are nonempty weak * compact and convex subsets of the dual of the Banach space. The Lderivative, however, is shown to be upper semi continuous, a result which is not known to hold for the Clarke gradient. We also formulate the notion of primitive maps dual to the Lderivative, an extension of Fundamental Theorem of Calculus for the Lderivative and a domain for computation of realvalued functions on a Banach space with a corresponding notion of effectivity. For realvalued functions on finite dimensional Euclidean spaces, the Lderivative can be obtained within an effectively given continuous domain. We also show that in finite dimensions the Lderivative and the Clarke gradient coincide thus providing a computable representation for the latter in this case. This paper is dedicated to the historical memory of Sharaf aldin Tusi (d. 1213), the Iranian mathematician who was the first to use the derivative systematically to solve for roots of cubic polynomials and find their maxima. 1
A Language for Differentiable Functions
"... Abstract—We introduce a typed lambda calculus in which real numbers, real functions, and in particular continuously differentiable and more generally Lipschitz functions can be defined. Given an expression representing a realvalued function of a real variable in this calculus, we are able to evalua ..."
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Cited by 1 (0 self)
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Abstract—We introduce a typed lambda calculus in which real numbers, real functions, and in particular continuously differentiable and more generally Lipschitz functions can be defined. Given an expression representing a realvalued function of a real variable in this calculus, we are able to evaluate the expression on an argument but also evaluate the generalised derivative, i.e., the Lderivative, equivalently the Clarke gradient, of the expression on an argument. The language is an extension of PCF with a real number datatype, similar to Real PCF and RL, but is equipped with primitives for min and weighted average to capture computable continuously differentiable or Lipschitz functions on real numbers. We present an operational semantics and a denotational semantics based on continuous Scott domains and several logical relations on these domains. We then prove an adequacy result for the two semantics. The denotational semantics is closely linked with Automatic Differentiation also called Algorithmic Differentiation, which has been an active area of research in numerical analysis for decades, and our framework can also be considered as providing denotational semantics for Automatic Differentiation. We derive a definability result showing that for any computable Lipschitz function there is a closed term in the language whose evaluation on any real number coincides with the value of the function and whose derivative expression also evaluates on the argument to the value of the generalised derivative of the function.
Differential operator and weak topology for Lipschitz maps
, 2009
"... We show that the Scott topology induces a topology for realvalued Lipschitz maps on Banach spaces which we call the Ltopology. It is the weakest topology with respect to which the Lderivative operator, as a second order functional which maps the space of Lipschitz functions into the function spac ..."
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Cited by 1 (1 self)
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We show that the Scott topology induces a topology for realvalued Lipschitz maps on Banach spaces which we call the Ltopology. It is the weakest topology with respect to which the Lderivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of nonempty weak * compact and convex valued maps equipped with the Scott topology, is continuous. For finite dimensional Euclidean spaces, where the Lderivative and the Clarke gradient coincide, we provide a simple characterisation of the basic open subsets of the Ltopology in terms of ties or primitive maps of functions. We use this to verify that the Ltopology is strictly coarser than the wellknown Lipschitz norm topology. A complete metric on Lipschitz maps is constructed that is induced by the Hausdorff distance, providing a topology that is strictly finer than the Ltopology but strictly coarser than the Lipschitz norm topology. We then develop a fundamental theorem of calculus of second order in finite dimensions showing that the continuous integral operator from the continuous Scott domain of nonempty convex and compact valued functions to the continuous Scott domain of ties is inverse to the continuous operator induced by the Lderivative. We finally show that in dimension one the Lderivative operator is onto and that it is a computable functional.