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Cristopher Moore – One of the best experts on this subject based on the ideXlab platform.
iteration inequalities and differentiability in Analog ComputersJournal of Complexity, 2000Co-Authors: Manuel L Campagnolo, Cristopher Moore, Jose Felix CostaAbstract:
Shannon’s general purpose Analog computer (GPAC) is an elegant model of Analog computation in continuous time. In this paper, we consider whether the set G of GPAC-computable functions is closed under iteration, that is, whether for any function f(x)?G there is a function F(x, t)?G such that F(x, t)=ft(x) for nonnegative integers t. We show that G is not closed under iteration, but a simple extension of it is. In particular, if we relax the definition of the GPAC slightly to include unique solutions to boundary value problems, or equivalently if we allow functions xk?(x) that sense inequalities in a differentiable way, the resulting class, which we call G+?k, is closed under iteration. Furthermore, G+?k includes all primitive recursive functions and has the additional closure property that if T(x) is in G+?k, then any function ofx computable by a Turing machine in T(x) time is also.
Analog Computers and the iteration functional, 1998Co-Authors: Manuel L Campagnolo, Cristopher Moore, Jose Felix CostaAbstract:
In this paper we extend the class of differentially algebraic functions computed by Shannon’s General Purpose Analog Computer (GPAC). We relax Pour-El’s definition of GPAC to obtain new operators and we use recursion theory on the reals to define a new class of Analog computable functions. We show that a function F (t, x) which simulates t time-steps of a Turing machine on input x, and more generally a functional that allows us to define the t’th iterate of a definable function, are definable in this system. Therefore, functions like Γ which are not generable by GPAC become computable in this extension.
dynamical recognizers real time language recognition by Analog ComputersTheoretical Computer Science, 1998Co-Authors: Cristopher MooreAbstract:
Abstract We consider a model of Analog computation which can recognize various languages in real time. We encode an input word as a point in R d by composing iterated maps, and then apply inequalities to the resulting point to test for membership in the language. Each class of maps and inequalities, such as quadratic functions with rational coefficients, is capable of recognizing a particular class of languages. For instance, linear and quadratic maps can have both stack-like and queue-like memories. We use methods equivalent to the Vapnik-Chervonenkis dimension to separate some of our classes from each other: linear maps are less powerful than quadratic or piecewise-linear ones, polynomials are less powerful than elementary (trigonometric and exponential) maps, and deterministic polynomials of each degree are less powerful than their non-deterministic counterparts. Comparing these dynamical classes with various discrete language classes helps illuminate how iterated maps can store and retrieve information in the continuum, the extent to which computation can be hidden in the encoding from symbol sequences into continuous spaces, and the relationship between Analog and digital computation in general. We relate this model to other models of Analog computation; in particular, it can be seen as a real-time, constant-space, off-line version of Blum, Shub and Smale’s real-valued machines.
Leo Corry – One of the best experts on this subject based on the ideXlab platform.
turing s pre war Analog Computers the fatherhood of the modern computer revisitedCommunications of The ACM, 2017Co-Authors: Leo CorryAbstract:
Turing’s machines of 1936 were a purely mathematical notion, not an exploration of possible blueprints for physical calculators.
G Korn – One of the best experts on this subject based on the ideXlab platform.
continuous system simulation and Analog Computers from op amp design to aerospace applicationsIEEE Control Systems Magazine, 2005Co-Authors: G KornAbstract:
Analog computer simulation has been completely replaced by more accurate, cheaper, and convenient digital techniques. Analog Computers provided three decades of real simulation experience well before digital Computers were fast enough and have left a significant legacy of mathematical modeling techniques. Almost none of the machines produced in the course of the 30-year rise and fall of a small industry are left. We were in a well-motivated hurry to develop Computers and guided weapons and to get into space first. We lacked the time, funds, and floor space to preserve aging artifacts. ASTRAC II and LOCUST joined more important Analog simulator landmarks on the trash heap: MIT’s great differential analyzer, Grumman Aircraft’s impressive lunar-lander mockup, and NASA/Huntsville’s lunar-excursion simulator are all gone for good. The same fate overtook almost all historically significant digital Computers. Each machine would have needed over 2000 square feet of museum space. Museums find that much space only for dinosaur skeletons.