Handbook of logic in computer science (vol. 3)
Nonsmooth analysis and control theory
Nonsmooth analysis and control theory
Computable analysis: an introduction
Computable analysis: an introduction
Introduction to Implicit Surfaces
Introduction to Implicit Surfaces
Foundation of a computable solid modelling
Theoretical Computer Science
Domain theory and differential calculus (functions of one variable)
Mathematical Structures in Computer Science
A Continuous Derivative for Real-Valued Functions
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
Domain Theoretic Solutions of Initial Value Problems for Unbounded Vector Fields
Electronic Notes in Theoretical Computer Science (ENTCS)
Abstract interpretation of the physical inputs of embedded programs
VMCAI'08 Proceedings of the 9th international conference on Verification, model checking, and abstract interpretation
A hybrid denotational semantics for hybrid systems
ESOP'08/ETAPS'08 Proceedings of the Theory and practice of software, 17th European conference on Programming languages and systems
Domain-Theoretic formulation of linear boundary value problems
CiE'05 Proceedings of the First international conference on Computability in Europe: new Computational Paradigms
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We introduce a domain-theoretic computational model for multi-variable 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 sub-domain 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 domain-theoretic notion of line integral is used to extend Green's theorem to interval-valued 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 n-tuple of the derivative part. This provides an algorithm to check consistency on the rational basis elements of the domain, giving an effective framework for multi-variable differential calculus.