PCF extended with real numbers

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.

Generalized Geraghty type mappings on partial metric spaces and fixed point results

“…Other examples of partial metric spaces may be found in [11,14,16,18]. Each partial metric p on X generates a T 0 topology τ p on X which has as a base the family of open p-balls {B p (x, ε) : x ∈ X, ε > 0}, where B p (x, ε) = {y ∈ X : p(x, y) < p(x, x) + ε}, for all x ∈ X and ε > 0.…”

Comparing Functional Paradigms for Exact Real-Number Computation

“…Indeed, the abstract datatype of real numbers in RealPCF [Esc96] is specifically designed to have I as its denotational interpretation. Furthermore, Escardó and Streicher [ES99] have established a universality result with respect to the domain-theoretic semantics: every computable element in the domain interpreting a RealPCF type is definable, by a term of that type, in the language RealPCF+, which is RealPCF extended with a parallel existential operator.…”

“…This approach is intensional in the sense that one has direct access to the encodings of reals, allowing the possibility of distinguishing between different representations of the same real number. In recent years, the extensional approach has been the subject of much theoretical investigation via the study of specialist languages, such as Di Gianantonio's RL [DiG93] and Escardó's RealPCF [Esc96]. On the other hand, the intensional approach is the one that is actually used when exact real-number computation is implemented in practice-see, for example, [GL01].…”

“…For the extensional approach, we use Escardó's RealPCF+, which is RealPCF [Esc96] extended by a parallel existential operator-a language that enjoys the merit of being universal with respect to its domain-theoretic semantics [ES99]. For the intensional approach, we encode real numbers within Plotkin's PCF++, which is PCF extended by parallel-conditional and existential operators [Plo77].…”

Common Fixed Point of Maps in Complete Partial Metric Spaces

“…A basic example of a partial metric space is the pair (R + , p), where p(x, y) = max{x, y} for all x, y ∈ R + . Other examples of the partial metric spaces which are interesting from a computational point of view may be found in [5] and [10].…”