Generalized Indiscernibles as Model-complete Theories

We give an almost entirely model-theoretic account of both Ramsey classes of finite structures and of generalized indiscernibles as studied in special cases in (for example) [7], [9]. We understand "theories of indiscernibles" to be special kinds of companionable theories of finite structures, and much of the work in our arguments is carried in the context of the model-companion. Among other things, this approach allows us to prove that the companion of a theory of indiscernibles whose "base" consists of the quantifier-free formulas is necessarily the theory of the Fraisse limit of a Fraisse class of linearly ordered finite structures (where the linear order will be at least quantifier-free definable). We also provide streamlined arguments for the result of [6] identifying extremely amenable groups with the automorphism groups of limits of Ramsey classes.Comment: 21 page

Geometric Model Theory in Efficient Computability

This dissertation consists of the proof of a single main result linking geometric ideas from the first-order model theory of infinite structures with complexity-theoretic analyses of problems over classes of finite structures. More precisely, we show that for a complete finite-variable theory of finite structures, models are efficiently recoverable from elementary diagrams if and only if the theory is super-rosy. In the course of the argument, we reconstitute the machinery of \th-independence and rosiness for classes of finite-structures, as well as a characterization of rosy classes analogous to the Independence theorem for the simple theories. We show that a super-rosy theory admits a weak form of model-theoretic coordinatization, which can be converted into to an algorithm for the model-building problem mentioned above in a natural and intuitive way. Conversely, we show how to extract a model-theoretic independence relation directly from an efficient algorithm for the model-building problem