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

Student Name: Amena Assem Abd-AlQader Mahmoud. Title of the Thesis: The Power of the Depth of Iteration in Defining Relations by Induction. Degree: M.Sc. (Pure Mathematics). 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 FO[f(n), g(n)] which counts the number, f(n), of ∨-symbols, and the number, g(n), of variables, in first-order formulas needed to express a given property. We prove that for f(n) ≥ log n, NSPACE[f(n)] ⊆ FO[f(n)+ ( f(n) logn )2 , f(n) logn ], and that for any f(n), g(n), FO[f(n), g(n)] ⊆ DSPACE[f(n)g(n) logn]. 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 + 2 but not expressible with quantifier rank n with any number of variables.