A Denotational Semantics of a Probabilistic Stream-Processing Language
We extend a stream-processing programming language, whose denotational semantics is given using the domain of streams, so that it can support probabilistic action. We give a denotational semantics to the extended language (which we call SProcP). We take the domain of probability distributions over streams as the denotational domain; it constitutes a cpo according to Saheb-Djaromi .
This work is motivated by the second author’s recent hybrid-system verification work . In this work, we used nonstandard analysis  to the denotational semantics of a stream-processing language so that it accommodates infinitesimal values. The extended language can express continuous signals and computation on them. We plan to do the same extension to the current language to obtain a language for processing probabilistic continuous-time signal.
The extension is for the most part straightforward. A tricky part is the denotation of binary operations e1 + e2. Since the denotation of the expressions e1 and e2 are in general not independent from each other, we cannot write define its denotation using those of e1 and e2. Our current definition is to assume an opaque function that gives a joint distribution of the denotations of e1 and e2. If we can reason that e1 is independent from e2 (by some program analysis), the denotation of e1 + e2 can be given in more explicit way.
 N. Saheb-Djahromi. Cpo’s of measures for nondeterminism. Theor. Comput. Sci., 12:19–37, 1980.
 K. Suenaga, H. Sekine, and I. Hasuo. Hyperstream processing systems: nonstandard modeling of continuous-time signals. In POPL 2013, pages 417–430, 2013.
 A. Robinson. Non-standard analysis. Studies in logic and the foundations of mathematics. North-Holland Pub. Co., 1966. URL https://books.google.co.jp/books?id=-eJLAAAAMAAJ.