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Title: Hierarchies in first-order logic and parameterized complexity
Author: He, Y.
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2011
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Finite model theory studies the expressive power of languages over finite structures. It explores the relations between collections of finite structures and the formal languages that are used to describe them. Properties, especially expressive powers, of logics and fragments of logics were intensively studied to achieve better understanding of the problems and systems we are faced with in various applications. To date, developments in finite model theory that have been mostly motivated by applications in complexity theory and database theory. These applications study the amount of different logical resources that are needed to describe problems and give classifications of problems and languages based on such resources. In this thesis, we study a kind of natural logical resource, called quantifier structure, and give a refined classification of first-order formulas based on such resource. That is, we proved that there exists a strict hierarchy in first-order logic. The application of finite model theory in classical computational complexity gave birth to an active research area called descriptive complexity theory. We investigate similar applications in parameterized complexity theory. We define logical reductions, and study the W-hierarchy, which is defined as the closure of certain collections of Fagin-definability problems under fpt-reductions. By substituting fpt-reductions with slicewise first-order reductions in the definitions of the W-hierarchy, we obtain a hierarchy that collapses. Whereas, by substituting fpt-reductions with a kind of slicewise quantifier-free reductions we get a strict hierarchy inside the W-hierarchy. Moreover, we show that, for each t, there is a complete problem for W[t] under slicewise bounded-variable first-order reductions.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available