A powerdomain of possibility measures

We provide a domain-theoretic framework for possibility theory by studying possibility measures on the lattice of opens of a topological space. The powerspaces Poss(X) and Poss 0;1] (X) of all such maps extend to functors in the natural way. We may think of possibility measures as continuous valuations by replacing`+' with`_' in their modular law. The functors above send continuous maps to sup-maps and continuous domains to completely distributive lattices; in the latter case they are locally continuous. Finite suprema of scalar multiples of point valuations form a basis of the powerdomains above if O(X) is the Scott-topology of a continuous domain. The notion of 0; 1]-and 0; 1]-modules corresponds to that of continuous cones if addition on the reals and on the module is replaced by suprema. The pow-erdomain Poss(D) is the free 0; 1]-module and Poss 0;1] (D) the free 0; 1]-module over a continuous domain D. 1 Possibility Measures This extended abstract attempts to recast some of the work done in quantitative domain theory within the traditional domain theory of continuous domains and lattices. We illustrate our approach with the two prime targets of quantitative analysis, the completely distributive lattices 0; 1] and 0; 1]. Given a dcpo D, there are numerous semantic scenarios in which we are interested in the function space D ! 0; 1]], or D ! 0; 1]]. For example, the former is a natural carrier of meaning for Markov chain processes 1], or quantitative model checks 2], whereas the second space could be the carrier of meaning for a cost, or running time analysis. Note that we stipulated that these functions are continuous. That way we ensure that the process of approximating total 1 Reinhold Heckmann, Klaus Keimel and Philipp S underhauf make valuable suggestions when I presented preliminary parts of this material at the