PCF extended with real numbers
Theoretical Computer Science - Special issue on real numbers and computers
Foundations of programming languages
Foundations of programming languages
An abstract data type for real numbers
Theoretical Computer Science
Information and Computation - Special issue: LICS 1996—Part 1
Computable analysis: an introduction
Computable analysis: an introduction
Semantics of Exact Real Arithmetic
LICS '97 Proceedings of the 12th Annual IEEE Symposium on Logic in Computer Science
A Universal Characterization of the Closed Euclidean Interval
LICS '01 Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science
The differential Lambda-calculus
Theoretical Computer Science
Domain theory and differential calculus (functions of one variable)
Mathematical Structures in Computer Science
Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation
Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation
A Simply Typed λ-Calculus of Forward Automatic Differentiation
Electronic Notes in Theoretical Computer Science (ENTCS)
A computational model for multi-variable differential calculus
Information and Computation
A language for differentiable functions
FOSSACS'13 Proceedings of the 16th international conference on Foundations of Software Science and Computation Structures
A computational model for multi-variable differential calculus
Information and Computation
A language for differentiable functions
FOSSACS'13 Proceedings of the 16th international conference on Foundations of Software Science and Computation Structures
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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 real-valued function of a real variable in this calculus, we are able to evaluate the expression on an argument but also evaluate the L-derivative of the expression on an argument. The language is an extension of PCF with a real number data-type 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 also provides 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 L-derivative of the function.