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Continuous Real-valued functions on Domains and Related Spaces

Drew Moshier Department of Computer Science, Mathematics & Physics Chapman University 4pm Tuesday 12 October 2004 Room 2511, JCMB, King's Buildings

Real-valued functions on domains are of interest for a variety of reasons. As they assign numerical values (weights) to data, they can be a source of information about things like rates of convergence. They also, obviously, play a role in domain theoretic models of probabilistic processes. In spite of the fact that R, the classical space of reals, is not a domain, one would like to have techniques to reason about real-valued functions (as opposed, say to functions into the reals extended with infinity). One cannot simply take the function space from a domain D to R and treat that as another domain, for example. One must consider other methods.

In this talk, we will emphasize the motivations for generalizing domains to stably compact spaces specifically with real-valued functions in mind, and for reconsidering classical results about real-valued functions, such as Stone-Gelfand duality, as applied to computationally useful spaces.

The main technical result of the talk extends Stone-Gelfand duality to stably compact spaces, and thus to a large class of continuous domains (including all FS domains) as a special case. The result is a characterization of algebras constructed from these spaces as semi-rings of positive, bi-continuous functions into the reals. With the technical result in hand, we will return to our motivations, and consider how to reason with (bi-continuous) weights on domains.

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