Dependence Logic in Algebra and Model Theory
Gianluca Paolini
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.
Keywords: Logic; Mathematics