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Register a new userThree topological reducibilities for discontinuous functions
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We define a family of three related reducibilities, $leq_T$, $leq_{tt}$ and $leq_m$, for arbitrary functions $f,g:Xrightarrowmathbb R$, where $X$ is a compact separable metric space. The $equiv_T$-equivalence classes mostly coincide with the proper Baire classes. We show that certain $alpha$-jump functions $j_alpha:2^omegarightarrow mathbb R$ are $leq_m$-minimal in their Baire class. Within the Baire 1 functions, we completely characterize the degree structure associated to $leq_{tt}$ and $leq_m$, finding an exact match to the $alpha$ hierarchy introduced by Bourgain and analyzed by Kechris and Louveau.
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Given a cardinal $kappa$ and a sequence $left(alpha_iright)_{iinkappa}$ of ordinals, we determine the least ordinal $beta$ (when one exists) such that the topological partition relation [betarightarrowleft(top,alpha_iright)^1_{iinkappa}] holds, including an independence result for one class of cases. Here the prefix $top$ means that the homogeneous set must have the correct topology rather than the correct order type. The answer is linked to the non-topological pigeonhole principle of Milner and Rado.
We investigate interactions between Ramsey theory, topological dynamics, and model theory. We introduce various Ramsey-like properties for first order theories and characterize them in terms of the appropriate dynamical properties of the theories in question (such as [extreme] amenability of a theory or some properties of the associated Ellis semigroups). Then we relate them to profiniteness and triviality of the Ellis groups of first order theories. In particular, we find various criteria for [pro]finiteness and for triviality of the Ellis group of a given theory from which we obtain wide classes of examples of theories with [pro]finite or trivial Ellis groups. As an initial motivation, we note that profiniteness of the Ellis group of a theory implies that the Kim-Pillay Galois group of this theory is also profinite, which in turn is equivalent to the equality of the Shelah and Kim-Pillay strong types. We also find several concrete examples illustrating the lack of implications between some fundamental properties. In the appendix, we give a full computation of the Ellis group of the theory of the random hypergraph with one binary and one 4-ary relation. This example shows that the assumption of NIP in the version of Newelskis conjecture for amenable theories (proved in [16]) cannot be dropped.
We establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the standard ones. A positive consequence of the choice of morphisms is that those on the topological side are functional. Towards obtaining the topological duality, we develop a universal construction which associates to an arbitrary lattice two distributive lattice envelopes with a Galois connection between them. This is a modification of a construction of the injective hull of a semilattice by Bruns and Lakser, adjusting their concept of admissibility to the finitary case. Finally, we show that the dual spaces of the distributive envelopes of a lattice coincide with completions of quasi-uniform spaces naturally associated with the lattice, thus giving a precise spatial meaning to the distributive envelopes.
For a group $G$ definable in a first order structure $M$ we develop basic topological dynamics in the category of definable $G$-flows. In particular, we give a description of the universal definable $G$-ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification of $G$, that is to the quotient $G^*/{G^*}^{00}_M$ (where $G^*$ is the interpretation of $G$ in a monster model). More generally, we obtain these results locally, i.e. in the category of $Delta$-definable $G$-flows for any fixed set $Delta$ of formulas of an appropriate form. In particular, we define local connected components ${G^*}^{00}_{Delta,M}$ and ${G^*}^{000}_{Delta,M}$, and show that $G^*/{G^*}^{00}_{Delta,M}$ is the $Delta$-definable Bohr compactification of $G$. We also note that some deeper arguments from the topological dynamics in the category of externally definable $G$-flows can be adapted to the definable context, showing for example that our epimorphism from the Ellis group to the $Delta$-definable Bohr compactification factors naturally yielding a continuous epimorphism from the $Delta$-definable generalized Bohr compactification to the $Delta$-definable Bohr compactification of $G$. Finally, we propose to view certain topological-dynamic and model-theoretic invariants as Polish structures which leads to some observations and questions.
Temporal logics provide a formalism for expressing complex system specifications. A large body of literature has addressed the verification and the control synthesis problem for deterministic systems under such specifications. For stochastic systems or systems operating in unknown environments, however, only the probability of satisfying a specification has been considered so far, neglecting the risk of not satisfying the specification. Towards addressing this shortcoming, we consider, for the first time, risk metrics, such as (but not limited to) the Conditional Value-at-Risk, and propose risk signal temporal logic. Specifically, we compose risk metrics with stochastic predicates to consider the risk of violating certain spatial specifications. As a particular instance of such stochasticity, we consider control systems in unknown environments and present a determinization of the risk signal temporal logic specification to transform the stochastic control problem into a deterministic one. For unicycle-like dynamics, we then extend our previous work on deterministic time-varying control barrier functions.
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