## PCF extended with real numbers (1996)

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### BibTeX

@MISC{Escardó96pcfextended,

author = {Martín Hötzel Escardó},

title = { PCF extended with real numbers},

year = {1996}

}

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### Abstract

We extend the programming language PCF with a type for (total and partial) real numbers. By a partial real number we mean an element of a cpo of intervals, whose subspace of maximal elements (single-point intervals) is homeomorphic to the Euclidean real line. We show that partial real numbers can be considered as “continuous words”. Concatenation of continuous words corresponds to refinement of partial information. The usual basic operations cons, head and tail used to explicitly or recursively define functions on words generalize to partial real numbers. We use this fact to give an operational semantics to the above referred extension of PCF. We prove that the operational semantics is sound and complete with respect to the denotational semantics. A program of real number type evaluates to a head-normal form iff its value is different from ⊥; if its value is different from ⊥ then it successively evaluates to head-normal forms giving better and better partial results converging to its value.