Finite Bases with Respect to the Superposition in Classes of Elementary Recursive Functions, dissertation (1611.04843v3)

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