  In this thesis we study inductive definitions over finite structures,
particularly, the depth of inductive definitions. We also study infinitary
finite variable logic which contains fixed-point logic and we introduce a new
complexity measure $\textrm{FO}_{\bigvee}[f(n),g(n)]$ which counts the number,
$f(n)$, of $\vee$-symbols, and the number, $g(n)$, of variables, in first-order
formulas needed to express a given property. We prove that for $f(n)\geq
\log{n}$, $\textrm{NSPACE}[f(n)] \subseteq
\textrm{FO}_{\bigvee}[f(n)+\left(\frac{f(n)}{\log{n}}\right)^2,\frac{f(n)}{\log{n}}]$,
and that for any $f(n),g(n)$, $\textrm{FO}_{\bigvee}[f(n),g(n)]\subseteq
\textrm{DSPACE}[f(n)g(n)\log{n}]$. Also we study the expressive power of
quantifier rank and number of variables and we prove that there is a property
of words expressible with two variables and quantifier rank $2^n+2$ but not
expressible with quantifier rank $n$ with any number of variables.
