RETRACTED: Semantic Domains for Combining Probability and Non-Determinism

Regina Tix, Klaus Keimel, Gordon Plotkin
2005 Electronical Notes in Theoretical Computer Science  
We present domain-theoretic models that support both probabilistic and nondeterministic choice. In [36] , Morgan and McIver developed an ad hoc semantics for a simple imperative language with both probabilistic and nondeterministic choice operators over a discrete state space, using domaintheoretic tools. We present a model also using domain theory in the sense of D.S. Scott (see e.g. [15]), but built over quite general continuous domains instead of discrete state spaces. Our construction
more » ... es the well-known domains modelling nondeterminism -the lower, upper and convex powerdomains, with the probabilistic powerdomain of Jones and Plotkin [24] modelling probabilistic choice. The results are variants of the upper, lower and convex powerdomains over the probabilistic powerdomain (see Chapter 4). In order to prove the desired universal equational properties of these combined powerdomains, we develop sandwich and separation theorems of Hahn-Banach type for Scott-continuous linear, sub-and superlinear functionals on continuous directed complete partially ordered cones, endowed with their Scott topologies, in analogy to the corresponding theorems for topological vector spaces in functional analysis (see Chapter 3). In the end, we show that our semantic domains work well for the language used by Morgan and McIver. Foreword This volume is based on Regina Tix's 1999 doctoral dissertation [55], entitled Continuous D-cones: Convexity and Powerdomain Constructions and submitted to the Department of Mathematics of Technische Universität Darmstadt. Only a small part of this thesis, namely three sections of Chapter 3, has previously been published (see [56] ). Since then, the main body of the thesis, Chapter 4 on powerdomains for modelling non-determinism, has become of increasing interest: indeed the main goal of the thesis was to provide semantic domains for modelling the simultaneous occurrence of probabilistic and ordinary non-determinism. It therefore seemed appropriate to make the thesis available to a general audience. There has been a good deal of progress in the relevant domain theory since the thesis was submitted, and so Klaus Keimel has rewritten large parts of the text, while maintaining the global structure of the original dissertation. As well as making a great number of minor changes, he has incorporated some major improvements. Gordon Plotkin has proved a Strict Separation Theorem for compact sets: all of Section 3.11 is new and essentially due to him. The Strict Separation Theorem 3.11.2 enables us, in Chapter 4, to eliminate an annoying auxiliary construction used in the original thesis for both the convex upper and the biconvex powercones; one also gets rid of the requirement that the way-below relation is additive, and the whole presentation becomes simplified and shorter. Next, an annoying hypothesis of a non-equational nature is no longer required for the statement of the universal property of the biconvex powercone. Further, the hypotheses for the lower powercone have been weakened: the universal property for this powercone remains valid without requiring the base domain to be continuous. Finally, we have added Section 4.16 explicitly presenting the powerdomains combining probabilistic choice and non-determinism and their universal properties. Combining the extended probabilistic power- R. Tix et al. / Electronic Notes in Theoretical Computer Science 129 (2005) 1-104 domain with the classical convex powerdomain was not possible when Tix's thesis was submitted: it was not known then whether the extended probabilistic powerdomain over a Lawson-compact continuous domain is Lawsoncompact. Extending slightly a recent result from [3], we now know that the extended probabilistic powerdomain is Lawson-compact over any stably locally compact space. For continuous domains the converse also holds. This allows us in particular to include infinite discrete spaces. We have included these new results in section 2.2. There have also been some terminological changes. For the classical powerdomains we now speak of the lower, upper, and convex powerdomains instead of the Hoare, Smyth, and Plotkin ones. Accordingly, for the new powerdomains we speak of the convex lower, convex upper, and biconvex powercones, rather than the convex Hoare, convex Smyth, and convex Plotkin powercones. D. Varacca [57,58,59] took a related approach to combining probability and nondeterminism via indexed valuations. His equational theory is weaker; he weakens one natural equation, but the theory becomes more flexible. M. Mislove [37] has introduced an approach similar to ours for the probabilistic (not the extended probabilistic) powerdomain, his goal being a semantics for probabilistic CSP. It is quite likely that our results can be used to deduce analogous properties for the (restricted) probabilistic powerdomain. Without the 2003 Barbados Bellairs Workshop on Domain Theoretic Methods in Probabilistic Processes and the inspiring discussions there, in particular with Franck van
doi:10.1016/j.entcs.2004.06.063 fatcat:kcsyh7rsgncadlrlnlonoaraau