MODEL THEORY AND COMPLEXITY THEORY

MODEL THEORY AND COMPLEXITY THEORY

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##### Date

2007-07-06

##### Authors

Gomaa, Walid

##### Advisor

Gasarch, William

Kueker, David

Kueker, David

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##### Abstract

Descriptive complexity theory is a branch of complexity theory that
views the hardness of a problem in terms of the complexity of
expressing it in some logical formalism; among the resources
considered are the number of object variables, quantifier depth,
type, and alternation, sentences length (finite/infinite), etc.
In this field we have studied two problems: (i) expressibility in
$\exists SO$ and (ii) the descriptive complexity of finite abelian
groups. Inspired by Fagin's result that $ NP=\exists SO$, we have
developed a partial framework to investigate expressibility inside
$\exists SO$ so as to have a finer look into $NP$. The framework
uses combinatorics derived from second-order
Ehrenfeucht-Fra\"{\i}ss\'{e} games and the notion of game types.
Among the results obtained is that for any $k$, divisibility by $k$
is not expressible by an $\exists SO$ sentence where (1) each
second-order variable has arity at most $2$, (2) the first-order
part has at most $2$ first-order variables, and (3) the first-order
part has quantifier depth at most $3$.
In the second project we have investigated the descriptive
complexity of finite abelian groups. Using
Ehrenfeucht-Fra\"{\i}ss\'e games we find upper and lower bounds on
quantifier depth, quantifier alternations, and number of variables
of a first-order sentence that distinguishes two finite abelian
groups. Our main results are the following. Let $G_1$ and $G_2$ be a
pair of non-isomorphic finite abelian groups, and let $m$ be a
number that divides one of the two groups' orders. Then the
following hold: (1) there exists a first-order sentence $\varphi$
that distinguishes $G_1$ and $G_2$ such that $\varphi$ is
existential, has quantifier depth $O(\log{m})$, and has at most 5
variables and (2) if $\varphi$ is a sentence that distinguishes
$G_1$ and $G_2$ then $\varphi$ must have quantifier depth
$\Omega(\log m)$.
In infinitary model theory we have studied abstract elementary
classes. We have defined Galois types over arbitrary subsets of the
monster (large enough homogeneous model), have defined a simple
notion of splitting, and have proved some properties of this notion
such as invariance under isomorphism, monotonicity, reflexivity,
existence of non-splitting extensions.