Towards computing distances between programs via Scott domains

Abstract

This paper introduces an approach to defining and computing distances between programs via continuous generalized distance functions ρ: A×A→D, where A and D are directed complete partial orders with the induced Scott topology, A is a semantic domain, and D is a domain representing distances (usually, some version of interval numbers). A continuous distance function ρ can define a T _o topology on a nontrivial domain A only if the axiom ∃0 ε D.∀x ε A.ρ(x,x)=0 does not hold. Hence, the notion of relaxed metric is introduced for domains — the axiom ρ(x,x)=0 is eliminated, but the axiom ρ(x,y)=ρ(y,x) and a version of the triangle inequality tailored for the domain D remain.

The paper constructs continuous relaxed metrics yielding the Scott topology for all continuous Scott domains with countable bases. This construction is closely related to partial metrics of Matthews and valuation spaces of O'Neill, but it describes a wider class of domains in a more intuitive way from the computational point of view.

Partially supported by NSF Grant CCR-9216185 and Office of Naval Research Grant ONR N00014-93-1-1015.