40,241 research outputs found
Geometric Aspects of Multiagent Systems
Recent advances in Multiagent Systems (MAS) and Epistemic Logic within
Distributed Systems Theory, have used various combinatorial structures that
model both the geometry of the systems and the Kripke model structure of models
for the logic. Examining one of the simpler versions of these models,
interpreted systems, and the related Kripke semantics of the logic (an
epistemic logic with -agents), the similarities with the geometric /
homotopy theoretic structure of groupoid atlases is striking. These latter
objects arise in problems within algebraic K-theory, an area of algebra linked
to the study of decomposition and normal form theorems in linear algebra. They
have a natural well structured notion of path and constructions of path
objects, etc., that yield a rich homotopy theory.Comment: 14 pages, 1 eps figure, prepared for GETCO200
A Classification Approach for Automated Reasoning Systems--A Case Study in Graph Theory
Reasoning systems which create classifications of structured objects face the problem of how object descriptions can be used to reflect their components as well as relations among these components. Current reasoning systems on graph theory do not adequately provide models to discover complex relations among mathematical concepts (eg: relations involving subgraphs) mainly due to the inability to solve this problem. This thesis presents an approach to construct a knowledge-based system, GC (Graph Classification), which overcomes this difficulty in performing automated reasoning in graph theory. We describe graph concepts based on an attribute called Linear Recursive Constructivity (LRC). LRC defines classes by an algebraic formula supported by background knowledge of graph types. We use subsumption checking on decomposed algebraic expressions of graph classes as a major proof method. The search is guided by case-split-based inferencing. Using the approach GC has generated proofs for many theorems such as any two distinct cycles (closed paths) having a common edge e contain a cycle not traversing e , if cycle C1 contains edges e1, e2, and cycle C2 contains edges e2, e3, then there exists a cycle that contains e1 and e3 and the union of a tree and a path is a tree if they have only a single common vertex.
The main contributions of this thesis are: (1) Development of a classification-based knowledge representation and a reasoning approach for graph concepts, thus providing a simple model for structured mathematical objects. (2) Development of an algebraic theory for simplifying and decomposing graph concepts. (3) Development of a proof search and a case-splitting technique with the guidance of graph type knowledge. (4) Development of a proving mechanism that can be generate constructive proofs by manipulating only simple linear formalization of theorems
The algebraic chromatic splitting conjecture for Noetherian ring spectra
We formulate a version of Hopkins' chromatic splitting conjecture for an
arbitrary structured ring spectrum , and prove it whenever is
Noetherian. As an application, these results provide a new local-to-global
principle in the modular representation theory of finite groups.Comment: Final version to appear in Mathematische Zeitschrif
Logic in Opposition
It is claimed hereby that, against a current view of logic as a theory of consequence, opposition is a basic logical concept that can be used to define consequence itself. This requires some substantial changes in the underlying framework, including: a non-Fregean semantics of questions and answers, instead of the usual truth-conditional semantics; an extension of opposition as a relation between any structured objects; a definition of oppositions in terms of basic negation. Objections to this claim will be reviewed
Local duality for structured ring spectra
We use the abstract framework constructed in our earlier paper to study local
duality for Noetherian -ring spectra. In particular, we
compute the local cohomology of relative dualizing modules for finite morphisms
of ring spectra, thereby generalizing the local duality theorem of Benson and
Greenlees. We then explain how our results apply to the modular representation
theory of compact Lie groups and finite group schemes, which recovers the
theory previously developed by Benson, Iyengar, Krause, and Pevtsova.Comment: Revised version, to appear in Journal of Pure and Applied Algebr
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