MoL-2013-03: Dependence Logic in Algebra and Model Theory

MoL-2013-03: Paolini, Gianluca (2013) Dependence Logic in Algebra and Model Theory. [Report]

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The aim of this thesis is to frame the dependence and independence notions formulated in mathematical contexts in the more general theory of dependence logic. In particular, the open question that we faced was whether the kind of dependence and independence relations studied in dependence logic arise also in algebra and geometric model theory. Dependence logic is the study of a family of logical formalisms obtained extending the language of first-order logic with dependence and independence atoms. These atoms are characterized via the use of the so-called team semantics, a semantics that is based on sets of assignments instead of single assignments. In the thesis we considered the following five atoms: =(x,y), \bot(x), x \perp y, \perp_z(x) and x \perp_z y. To each one of these atoms corresponds a notion of dependence or independence that is abstractly characterized in terms of teams, i.e. sets of assignments. Each atom gives rise to an atomic language and for each atomic language a sound and complete (or partially complete) deductive system has been elaborated. In our analysis the chosen strategy was to interpret the dependence and independence atoms in algebraic and model-theoretic contexts in which relevant dependence and independence notions have been formulated, and then to verify if these interpretations are sound and complete with respect to the deductive systems that characterize the behavior of the atoms in abstract terms, i.e. with respect to team semantics. We addressed the issue in increasing order of generality. Firstly, we considered the linear and algebraic dependence and independence notions of linear algebra and field theory. Secondly, we considered the notions of dependence and independence definable in function of the model-theoretic operator of algebraic closure. Then, we considered the dependence and independence notions definable in a pregeometry. Finally, we studied the forking independence relation in ω-stable theories. In all these cases we have been able to prove a soundness and completeness result answering positively the motivating question of the thesis. Apart from their mathematical interest, these results support the claims of V ̈a ̈an ̈anen and Galliani, putting the exact and authoritative concepts of dependence and independence occurring in mathematics and formal mathematics under the wide wing of dependence logic.

Item Type: Report
Report Nr: MoL-2013-03
Series Name: Master of Logic Thesis (MoL) Series
Year: 2013
Uncontrolled Keywords: Logic; Mathematics
Date Deposited: 12 Oct 2016 14:38
Last Modified: 12 Oct 2016 14:38

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