Abstract Datatypes for Real Numbers in Type Theory
نویسندگان
چکیده
Datatypes for Real Numbers in Type Theory Mart́ın Hötzel Escardó and Alex Simpson 1 School of Computer Science, University of Birmingham 2 LFCS, School of Informatics, University of Edinburgh Abstract. We propose an abstract datatype for a closed interval of real numbers to type theory, providing a representation-independent approach to programming with real numbers. The abstract datatype requires only function types and a natural numbers type for its formulation, and so can be added to any type theory that extends Gödel’s System T. Our main result establishes that programming with the abstract datatype is equivalent in power to programming intensionally with representations of real numbers. We also consider representing arbitrary real numbers using a mantissa-exponent representation in which the mantissa is taken from the abstract interval. We propose an abstract datatype for a closed interval of real numbers to type theory, providing a representation-independent approach to programming with real numbers. The abstract datatype requires only function types and a natural numbers type for its formulation, and so can be added to any type theory that extends Gödel’s System T. Our main result establishes that programming with the abstract datatype is equivalent in power to programming intensionally with representations of real numbers. We also consider representing arbitrary real numbers using a mantissa-exponent representation in which the mantissa is taken from the abstract interval.
منابع مشابه
Comparing Functional Paradigms for Exact Real-Number Computation
We compare the definability of total functionals over the reals in two functional-programming approaches to exact real-number computation: the extensional approach, in which one has an abstract datatype of real numbers; and the intensional approach, in which one encodes real numbers using ordinary datatypes. We show that the type hierarchies coincide up to second-order types, and we relate this...
متن کاملA cosmology of datatypes : reusability and dependent types
This dissertation defends the idea of a closed dependent type theory whose inductive types are encoded in a universe. Each inductive definition arises by interpreting its description – itself a firstclass citizen in the type theory. Datatype-generic programming thus becomes ordinary programming. This approach is illustrated by several generic programs. We then introduce an elaboration of induct...
متن کاملRecursion on Nested Datatypes in Dependent Type Theory
Nested datatypes are families of datatypes that are indexed over all types and where the datatype constructors relate different members of the family. This may be used to represent variable binding or to maintain certain invariants through typing. In dependent type theory, a major concern is the termination of all expressible programs, so that types that depend on object terms can still be type...
متن کاملThe λμ-calculus
Calculi with control operators have been studied as extensions of simple type theory. Real programming languages contain datatypes, so to really understand control operators, one should also include these in the calculus. As a first step in that direction, we introduce λμ, a combination of Parigot’s λμ-calculus and Gödel’s T, to extend a calculus with control operators with a datatype of natura...
متن کاملThe λμT-calculus
Calculi with control operators have been studied as extensions of simple type theory. Real programming languages contain datatypes, so to really understand control operators, one should also include these in the calculus. As a first step in that direction, we introduce λμ, a combination of Parigot’s λμ-calculus and Gödel’s T, to extend a calculus with control operators with a datatype of natura...
متن کامل