This Week in Logic at CUNY

Computational Logic Seminar
Tuesday, April 9, 2013 2:00 pm
Speaker: Marek Krótkiewicz and Krystian Wojtkiewicz Opole University, Poland
Title: Semantic Knowledge Base – system build in Association Oriented Database Model

Semantic Knowledge Base is a new approach towards building systems capable of storing and processing complex information also in the means of inference over it. The main assumption is to distinguish concepts (meanings) from terms (natural language identifiers). The information is held in two modules: ontological core and semantic network module. Ontological core is used for building a hypergraph, where each node is a concept and edges are links between them. Semantic networks are used to store more complex information (rules and facts) in a structure using predicate-object structures build over previously defined concepts. The system provides also support for inference in the means of modal and hybrid logic through custom build system of quantifiers and possibility of describing dimensions and spaces, where each and every part of information is valid or invalid (extended Kripke’s model of possible world theory).

The system has been designed in association oriented database model (AODB). In this model we assume two main categories: collections and associations. Collections are used to store information in objects holding the attributes values, while associations store information about n-ary relation between collections. Associations are consider to be containers for roles and each role is considered as the permission given to the collection to be part of an association. The model assumes that both, collections and association may build inheritance structures, where virtual, private and real inheritance of accordingly attributes and roles occur. Each of the structure elements of AODB has its own unique role therefore their semantic is strictly connected to the grammar of the model. Moreover such construction made it possible to unify conceptual and physical model of the database into one structure. AODB has its native object storage model. There have also been two languages designed and implemented – Association Modeling Language (AML) and Association Query Language (AQL). The latter is using graph templates as queries and returns list of graph.


Set theory seminar
Friday, April 12, 2013 11:00 am, room 6494 (not 6417)
Speaker: Norman Perlmutter The CUNY Graduate Center
Title: Dissertation Defense: Inverse limits of models of set theory and the large cardinal hierarchy near a high-jump cardinal

This dissertation consists of two chapters, each of which investigates a topic in set theory, more specifically in the research area of forcing and large cardinals. The two chapters are independent of each other.

The first chapter analyzes the existence, structure, and preservation by forcing of inverse limits of inverse-directed systems in the category of elementary embeddings and models of set theory. Although direct limits of directed systems in this category are pervasive in the set-theoretic literature, the inverse limits in this same category have seen less study. I have made progress towards fully characterizing the existence and structure of these inverse limits. Some of the most important results are as follows. If the inverse limit exists, then it is given by either the entire thread class or a rank-initial segment of the thread class. Given sufficient large cardinal hypotheses, there are systems with no inverse limit, systems with inverse limit given by the entire thread class, and systems with inverse limit given by a proper subset of the thread class. Inverse limits are preserved in both directions by forcing under fairly general assumptions. Prikry forcing and iterated Prikry forcing are important techniques for constructing some of the examples in this chapter.

The second chapter analyzes the hierarchy of the large cardinals between a supercompact cardinal and an almost-huge cardinal, including in particular high-jump cardinals. I organize the large cardinals in this region by consistency strength and implicational strength. I also prove some results relating high-jump cardinals to forcing.  A high-jump cardinal is the critical point of an elementary embedding $j: V to M$ such that $M$ is closed under sequences of length $supset{j(f)(kappa) st f: kappa to kappa}$.  Two of the most important results in the chapter are as follows. A Vopenka cardinal is equivalent to an Woodin-for-supercompactness cardinal. The existence of an excessively hypercompact cardinal is inconsistent.

Model theory seminar, CUNY Logic Workshop
Friday, April 12, 2013 12:30 pm
Speaker: Patrick Speissegger McMaster University
Title: Pfaffian functions vs. Rolle leaves

In the early 1980s, after Khovanskii’s ICM lecture, van den Dries formulated the conjecture that the expansion P of the real field by all pfaffian functions was model complete. Thinking about the problem led him to discover o-minimality; but while P has been known to be o-minimal since Wilkie’s groundbreaking work in 1996, van den Dries’s conjecture is still open today. Recently, Lion and I proved a variant of this conjecture, in which “pfaffian functions” are replaced with “nested Rolle leaves”, which in essence correspond to the objects originally studied by Khovanskii. The mystery lies in how these two expansions are related. I will explain each of them and exhibit a third related notion, found recently in joint work with Jones, which might clarify this relationship.

CUNY Logic Workshop
Friday, April 12, 2013 2:00 pm
Speaker: Karen Lange Wellesley College
Title: An algebraic characterization of recursively saturated real closed fields
Link: (with D’Aquino and Kuhlmann) give a valuation theoretic characterization for a real closed field to be recursively saturated. Previously, Kuhlmann, Kuhlmann, Marshall, and Zekavat gave such a characterization for kappa-saturation, for all infinite cardinals kappa. Our result extends the characterization for a divisible ordered abelian group to be recursively saturated found in some unpublished work of Harnik and Ressayre.

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