Complexity of Set-Valued Maps
In this dissertation, a framework for comparing the complexity of set-valued maps is introduced and applied to the study of various classes of set-valued maps, including partial orders, equivalence relations, random graphs, and so on. Rather than defining explicitly what complexity means, the approach taken in the framework is to lay down very broad axioms which must be satisfied by any notion of complexity. Thus the framework can accommodate various different measures of complexity.
Several notions of complexity, all of which fit within the framework, are studied. Some of these generalize existing notions for (single-valued) functions. In particular, the important concept of topological entropy for functions is generalized to the set-valued case. It is shown that many of the properties of topological entorpy for functions carry over to set-valued mappings. Other notions of complexity are introduced which seem to be new. the most interesting of these are two graph-based notions of complexity, in which the process of discretizing a set-valued map leads naturally to notions of complexity based on directed graphs. It is shown that these graph-based notions are related to topological entropy, in the sense of providing upper and lower bounds. Using these bounds, a general result on the entropy of partial orders can be formulated.
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