Concurrency and Approximate Reasoning

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The project deals with two subjects, first involves `concurrency', the second `approximate reasoning'. Concurrent systems abound in human experience but their fully adequate conceptualization as yet eludes us. Our increasing dependence on ever more complex systems in the management and control of human affairs and activities increases the urgency for developing more adequate and preferable more formal concepts to maintain reliable control over systems we have created. The solution of the problem of correct specification of the design and verification of its behaviour becomes crucial, and a satisfactory conceptual apparatus for rigorous specification and verification becomes essential. The project will explore theoretical issues involved in the formal reasoning about concurrent systems and approximate reasoning. Our general aim is to develop a framework, based on the concept of `generalized causality' modelled by `discrete relational structures' and `generalized traces' (both proposed by Janicki and Koutny), to support the design and the verification of sophisticated concurrent systems. When empirical data are represented as some formal structures, the outcome may not have all required properties. For example any ranking relations should at least be a partial order, but a relation obtained by various subjective judgments may include cycles. The problem is how to find the `best approximation' of a given structure by a structure with specific properties. While `rough sets' handle lower and upper approximations exceptionally well, `optimal' and `structural' (i.e. having desired special properties) approximations are much less developed. We plan to develop a family of abstract metrics that can be used for various classes of sets, relational structures and properties. Moreover, we want to simulate the concept of a metric by a sequence of `lower' and `upper' approximations that preserve the desired properties (in both the standard theory of relations and rough sets settings). In both cases finding some efficient algorithms and supporting software is a long term ultimate goal.