An Analog Characterization of the Grzegorczyk Hierarchy

Abstract

We study a restricted version of Shannon's general purpose analog computer in which we only allow the machine to solve linear differential equations. We show that if this computer is allowed to sense inequalities in a differentiable way, then it can compute exactly the elementary functions, the smallest known recursive class closed under time and space complexity. Furthermore, we show that if the machine has access to a function f ðxÞ with a suitable growth as x goes to infinity, then it can compute functions on any given level of the Grzegorczyk hierarchy. More precisely, we show that the model contains exactly the nth level of the Grzegorczyk hierarchy if it is allowed to solve n À 3 non-linear differential equations of a certain kind. Therefore, we claim that, at least in this region of the complexity hierarchy, there is a close connection between analog complexity classes, the dynamical systems that compute them, and classical sets of subrecursive functions. # 2002 Elsevier Science (USA)

An Analog Characterization of Elementarily Computable Functions over the Real Numbers

“…As one may expect, this direction of the proof has many similarities with the proof L ⊂ E in [9,8]: main differences lie in the presence of non-total functions and of schema LIM. A structural induction shows:…”

“…Class L is related to functions elementarily computable over integers in classical recursion theory and functions elementarily computable over the real numbers in recursive analysis (discussed in [30]): any function of class L is elementarily computable in the sense of recursive analysis, and conversely, any function over the integers computable in the sense of classical recursion theory is the restriction to integers of a function that belongs to L [9,8].…”

A survey of recursive analysis and Moore’s notion of real computation

“…In analogy with the function algebra capturing E, Campagnolo, Moore, and Costa proposed following class of real functions [6,8,5].…”

“…The proof of E ⊆ dp(L) can be done by induction on the construction tree of functions in E using the operations of L at each step. For the other direction dp(L) ⊆ E the proof given in [8] applies induction on the construction tree of functions in L. However, instead of the intuitive appeal of using the operations of the function algebra E at each step the fact that L is elementarily computable is used to build Turing machines that would output appropriately close approximations. The tricky part here is that given a function f ∈ L that preserves N, it is not necessary that the functions used to construct f also preserve N. Another proof for this latter direction is given in [10] which is purely algebraic.…”

“…Some recent developments towards that direction can be found in [15,14,12,2]. There have also been work relating analog models to discrete recursive classes (see [5,6,8,7]). In this article we give a detailed survey on two theories of continuous computation: recursive analysis and Moore's function algebra, and the relationships between them.…”