Finite Bases with Respect to the Superposition in Classes of Elementary Recursive Functions, dissertation

TL;DR

This thesis characterizes the structure of elementary recursive functions, identifies finite bases for certain classes, and explores permutation groups generated by two permutations within computational complexity classes.

Contribution

It provides explicit finite bases for classes of elementary functions and describes the permutation groups generated by two permutations in classes like FP.

Findings

Lower elementary functions are characterized by specific compositions with bounded floors.

Finite composition bases are identified for classes in the uniform TC^0 and Grzegorczyk hierarchy.

Permutation groups of polynomial-time functions are generated by two permutations.

Abstract

This is a thesis that was defended in 2009 at Lomonosov Moscow State University. In Chapter 1: 1. It is proved that that the class of lower (Skolem) elementary functions is the set of all polynomial-bounded functions that can be obtained by a composition of $x+1$, $xy$, $\max(x-y,0)$, $x\wedge y$, $\lfloor x/y \rfloor$, and one exponential function ($2^x$ or $x^y$) using formulas that have no more than 2 floors with respect to an exponent (for example, $(x+y)^{xy+z}+1$ has 2 floors, $2^{2^x}$ has 3 floors). Here $x\wedge y$ is a bitwise AND of $x$ and $y$. 2. It is proved that $\{x+y,\ \max(x-y,0),\ x\wedge y,\ \lfloor x/y \rfloor,\ 2^{\lfloor \log_2 x \rfloor^2}\}$ and $\{x+y,\ \max(x-y,0),\ x\wedge y,\ \lfloor x/y \rfloor,\ x^{\lfloor \log_2 y \rfloor}\}$ are composition bases in the functional version of the uniform $\mathrm{TC}^0$ (also known as $\mathrm{FOM}$). 3. The hierarchy of classes exhausting the class of elementary functions is described in terms of compositions with restrictions on a number of floors in a formula. The results of Chapter 1 are published in: 1) Volkov S.A. An exponential expansion of the Skolem-elementary functions, and bounded superpositions of simple arithmetic functions (in Russian), Mathematical Problems of Cybernetics, Moscow, Fizmatlit, 2007, vol. 16, pp. 163-190 2) doi:10.1134/S1064562407040217 In Chapter 2 a simple composition basis in the class ${\cal E}^2$ of Grzegorczyk hierarchy is described. This result is published in DOI: 10.1515/156939206779238436 In Chapter 3 it is proved that the group of permutations $\mathrm{Gr}(Q)=\{f:\ f,f^{-1}\in Q\}$ is generated by two permutations for many classes $Q$. For example, this is proved for $Q=\mathrm{FP}$, where $\mathrm{FP}$ is the class of all polynomial-time computable functions (of the length of input). The results of chapter 3 are published in DOI: 10.1515/DMA.2008.046