Quantitative semantics, topology, and possibility measures

Abstract

Given a topological space X and a complete lattice L, we study the space of L-predicates F L ( X) = [ X → L op ] op , continuous maps from X to L op in its Scott-topology. It yields a functor F L (·) from TOP- L, a full subcategory of TOP subsuming continuous domains, to SUP, the category of complete sub-lattices and maps preserving suprema. Elements of F 2( X)are continuous predicates (= closed sets), and elements of F[ 0,1]( X) may be viewed as probabilistic predicates. Alternatively, one may consider the complete sup-lattice P L ( X) = O( X) −0 L of maps μ: O( X) → L preserving suprema (= possibility measures), which results in another functor P L (·) from TOP to SUP. We show that these functors are equivalent for two restrictions. First, we leave SUP unchanged and restrict TOP- L to CONT, the category of continuous domains in their Scott-topology; second, we fix TOP but restrict L to co-continuous lattices. Conversely, if F L ( X) and P L ( X) are isomorphic for all topological spaces X then L is indeed co-continuous. Further, if X is a sober space and F L ( X) and P L ( X) are isomorphic for all complete lattices L then X is a continuous domain and its topology is the Scott-topology. Possibility measures have extensions to the upper powerdomain κ X , or to the full power set P( X), which are defined similarly to outer measures. Utilizing the notion of sup-semirings, we employ such extensions and the isomorphism F L ( X) ≅ P L ( X) to show that the sup-primes of P L ( X) are exactly scalar multiples of point valuations with sup-primes as scalars for an underlying sober space X. Combining this with classical results in the theory of continuous lattices, we restate the notion of cones in our setting and show that the space of possibility measures P L ( X) of a continuous domain X is the free L-module over X for the sup-semiring L = [0, ∞].