Logic at GU
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A game that characterizes definability of classes of structures by first-order sentences containing a given number of quantifiers was introduced by Immerman in 1981. In this talk I describe two other games that are equivalent with the Immerman game in the sense that they characterize definability by a given number of quantifiers.
In the Immerman game, Duplicator has a canonical optimal strategy, and hence Duplicator can be completely removed from the game by replacing her moves with default moves given by this optimal strategy. On the other hand, in the other two games there is no such optimal strategy for Duplicator. Thus, the Immerman game can be regarded as a one-player game, but the other two games are genuine two-player games.
The talk is based on joint work with Kerkko Luosto.
There is a mature body of work in logic aiming to characterize logical formalisms in terms of their structural or model-theoretic properties. The origins of this work can be traced to Alfred Tarski’s program to characterize metamathematical notions in “purely mathematical terms” and to Per Lindström’s abstract characterizations of first-order logic. For the past forty years, rule-based logical languages have been widely used in databases and in related areas of computer science to express integrity constraints and to specify transformations in data management tasks, such as data exchange and ontology-based data access. The aim of this talk is to present an overview of more recent results that characterize various classes of rule-based logical languages in terms of their structural or model-theoretic properties.
A classical result by Lovász asserts that two graphs G and H are isomorphic if and only if they have the same left profile, that is, for every graph F, the number of homomorphisms from F to G coincides with the number of homomorphisms from F to H. A similar result is also known to hold for right profiles, that is, two graphs G and H are isomorphic if and only if for every graph F, the number of homomorphisms from G to F coincides with the number of homomorphisms from H to F. During the past several years, there has been a study of equivalence relations that are relaxations of isomorphism obtained by restricting the left profile or the right profile to suitably restricted classes of graphs, instead of the class of all graphs. Furthermore, a notion of a query algorithm based on homomorphism counts was recently introduced and investigated. The aim of this talk is to present an overview of some of the main results in this area with emphasis on the differences between left homomorphism counts and right homomorphism counts.