Consistency for counting quantifiers.

We apply the algebraic approach for Constraint Satisfaction Problems (CSPs) with counting quantifiers, developed by Bulatov and Hedayaty, for the first time to obtain classifications for computational complexity. We develop the consistency approach for expanding polymorphisms to deduce that, if H has an expanding majority polymorphism, then the corresponding CSP with counting quantifiers is tractable. We elaborate some applications of our result, in particular deriving a complexity classification for partially reflexive graphs endowed with all unary relations. For each such structure, either the corresponding CSP with counting quantifiers is in P, or it is NP-hard

Combinatorial Properties of Finite Models

We study countable embedding-universal and homomorphism-universal structures and unify results related to both of these notions. We show that many universal and ultrahomogeneous structures allow a concise description (called here a finite presentation). Extending classical work of Rado (for the random graph), we find a finite presentation for each of the following classes: homogeneous undirected graphs, homogeneous tournaments and homogeneous partially ordered sets. We also give a finite presentation of the rational Urysohn metric space and some homogeneous directed graphs. We survey well known structures that are finitely presented. We focus on structures endowed with natural partial orders and prove their universality. These partial orders include partial orders on sets of words, partial orders formed by geometric objects, grammars, polynomials and homomorphism orders for various combinatorial objects. We give a new combinatorial proof of the existence of embedding-universal objects for homomorphism-defined classes of structures. This relates countable embedding-universal structures to homomorphism dualities (finite homomorphism-universal structures) and Urysohn metric spaces. Our explicit construction also allows us to show several properties of these structures.Comment: PhD thesis, unofficial version (missing apple font

Homological stability, characteristic classes and the minimal genus problem

The purpose of this thesis is to study the (co-)homological properties of the classifying space of subsurface bundles in a trivial background bundle with fiber a manifold M. We will investigate homological stability pheonomena of this moduli space if M is simply-connected and at least 5-dimensional and the subsurfaces are equipped with tangential structures. Additionally we will investigate the representability of second homology classes by surfaces in general topological spaces. In the case of manifolds this yields a measure for the failure of homological stability if M is not simply-connected. In the introduction we will also briefly touch on how to proceed from these homological stability results to determining the stable characteristic classes of subsurface bundles

How to Deal with Unbelievable Assertions

We tackle the problem that arises when an agent receives unbelievable information. Information is unbelievable if it conflicts with the agent’s convictions, that is, what the agent considers knowledge. We propose two solutions based on modifying the information so that it is no longer unbelievable. In one solution, the source and the receiver of the information cooperatively resolve the conflict. For this purpose we introduce a dialogue protocol in which the receiver explains what is wrong with the information by using logical interpolation, and the source produces a new assertion accordingly. If such cooperation is not possible, we propose an alternative solution in which the receiver revises the new piece of information by its own convictions to make it acceptable.Peer reviewe