The Power of the Depth of Iteration in Defining Relations by Induction

Abstract

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.