PCF extended with real numbers

• Authors: Martín Hötzel Escardó
• Venue: Theoretical Computer Science - Special issue on real numbers and computers
• Year: 1996

We extend the programming language PCF with a type for (total andpartial) real numbers. By a partial realnumber we mean an element of a cpo ofintervals, whose subspace of maximal elements (single-point intervals)is homeomorphic to the Euclidean real line. We show that partial realnumbers can be considered as “continuous words”.Concatenation of continuous words corresponds to refinement of partialinformation. The usual basic operations cons, head and tail used toexplicitly or recursively define functions on words generalize topartial real numbers. We use this fact to give an operational semanticsto the above referred extension of PCF. We prove that the operationalsemantics is sound and complete with respect to the denotationalsemantics. A program of real number type evaluates to a head-normal formiff its value is different from ⊥; if its value is different from ⊥ then it successively evaluates to head-normal formsgiving better and better partial results converging to its value.—Author's Abstract