Some corollaries of the correspondence between partial metrics and multivalued equalities

Abstract

Multivalued equalities originally arose in the context of the theory of sheaves, while partial metrics originally arose in the context of the theory of domains for denotational semantics. It turns out that multivalued equalities and partial metrics are closely related: one can either consider them equivalent up to the choice of dual notation, or one can formally capture the duality between logical and metric viewpoints by explicitly requiring logical values and distances to be represented by dual structures. Not only does this allow the transfer of the ideas and results between these two fields, but the most interesting situations are those of interplay between logical and metric considerations. The computational understanding of partial metrics as upper bounds and of multivalued equalities as lower bounds is discussed, together with the lower bound counterparts to partial metrics and the corresponding upper bound counterparts to multivalued equalities. It is shown that separated pre-sheaves of sets and functions over complete Heyting algebras can be understood as the pre-sheaves of ultrametrics valued in the corresponding Brouwerian algebras and non-expansive maps of those ultrametrics. It is proposed that the natural logical counterparts for partial metrics valued in non-negative reals are multivalued equalities valued in the quantale of non-positive reals. The issue of canonical partial metrics on higher-order and reflexive Scott domains is revisited. The intuition behind strong triangularity (“Vickers form”), the computational version of strong triangularity, and weighted quasi-metrics are discussed in the context of quantaloid enrichment.