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

TL;DR

This thesis explores the depth of inductive definitions over finite structures, introduces a new complexity measure for logical formulas, and analyzes their expressive power and computational complexity.

Contribution

It introduces the complexity measure FO_{∨}[f(n),g(n)] and establishes its relationships with space complexity classes, along with results on the expressive power of logical formulas.

Findings

FO_{∨}[f(n),g(n)] relates to space complexity classes.

Logical formulas with bounded variables and quantifier rank have limitations.

New bounds on the expressive power of logical properties.

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.