MoL-2013-03:
Paolini, Gianluca
(2013)
*Dependence Logic in Algebra and Model Theory.*
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## Abstract

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 |

URI: | https://eprints.illc.uva.nl/id/eprint/892 |

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