Results 1  10
of
11
On The Computational Power Of Neural Nets
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1995
"... This paper deals with finite size networks which consist of interconnections of synchronously evolving processors. Each processor updates its state by applying a "sigmoidal" function to a linear combination of the previous states of all units. We prove that one may simulate all Turing Machines by su ..."
Abstract

Cited by 156 (26 self)
 Add to MetaCart
This paper deals with finite size networks which consist of interconnections of synchronously evolving processors. Each processor updates its state by applying a "sigmoidal" function to a linear combination of the previous states of all units. We prove that one may simulate all Turing Machines by such nets. In particular, one can simulate any multistack Turing Machine in real time, and there is a net made up of 886 processors which computes a universal partialrecursive function. Products (high order nets) are not required, contrary to what had been stated in the literature. Nondeterministic Turing Machines can be simulated by nondeterministic rational nets, also in real time. The simulation result has many consequences regarding the decidability, or more generally the complexity, of questions about recursive nets.
On some Relations between Dynamical Systems and Transition Systems
 In Proceedings of ICALP
, 1994
"... . In this paper we define a precise notion of abstraction relation between continuous dynamical systems and discrete statetransition systems. Our main result states that every Turing Machine can be realized by a dynamical system with piecewiseconstant derivatives in a 3dimensional space and thus ..."
Abstract

Cited by 32 (4 self)
 Add to MetaCart
. In this paper we define a precise notion of abstraction relation between continuous dynamical systems and discrete statetransition systems. Our main result states that every Turing Machine can be realized by a dynamical system with piecewiseconstant derivatives in a 3dimensional space and thus the reachability problem for such systems is undecidable for 3 dimensions. A decision procedure for 2dimensional systems has been recently reported by Maler and Pnueli. On the other hand we show that some nondeterministic finite automata cannot be realized by any continuous dynamical system with less than 3 dimensions. 1 Introduction There has been recently an increasing interest in models of hybrid systems, i.e., systems that combine intercommunicating discrete and continuous components (see [9], [12], [3]). The introduction of these models is motivated by a real practical concern: more and more computers (discrete transition systems) are nowadays embedded within realworld control loops...
Closedform Analytic Maps in One and Two Dimensions Can Simulate Turing Machines
, 1996
"... We show closedform analytic functions consisting of a finite number of trigonometric terms can simulate Turing machines, with exponential slowdown in one dimension or in real time in two or more. 1 A part of this author's work was done when he was visiting DIMACS at Rutgers University. 1 Introduc ..."
Abstract

Cited by 31 (4 self)
 Add to MetaCart
We show closedform analytic functions consisting of a finite number of trigonometric terms can simulate Turing machines, with exponential slowdown in one dimension or in real time in two or more. 1 A part of this author's work was done when he was visiting DIMACS at Rutgers University. 1 Introduction Various authors have independently shown [9, 12, 4, 14, 1] that finitedimensional piecewiselinear maps and flows can simulate Turing machines. The construction is simple: associate the digits of the x and y coordinates of a point with the left and right halves of a Turing machine's tape. Then we can shift the tape head by halving or doubling x and y, and write on the tape by adding constants to them. Thus two dimensions suffice for a map, or three for a continuoustime flow. These systems can be thought of as billiards or optical ray tracing in three dimensions, recurrent neural networks, or hybrid systems. However, piecewiselinear functions are not very realistic from a physical p...
A Survey of ContinuousTime Computation Theory
 Advances in Algorithms, Languages, and Complexity
, 1997
"... Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuoustime computation. However, while specialcase algorithms and devices are being developed, relatively little work exists o ..."
Abstract

Cited by 29 (6 self)
 Add to MetaCart
Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuoustime computation. However, while specialcase algorithms and devices are being developed, relatively little work exists on the general theory of continuoustime models of computation. In this paper, we survey the existing models and results in this area, and point to some of the open research questions. 1 Introduction After a long period of oblivion, interest in analog computation is again on the rise. The immediate cause for this new wave of activity is surely the success of the neural networks "revolution", which has provided hardware designers with several new numerically based, computationally interesting models that are structurally sufficiently simple to be implemented directly in silicon. (For designs and actual implementations of neural models in VLSI, see e.g. [30, 45]). However, the more fundamental...
NBody Spacetime Constraints
, 1995
"... Animators frequently choreograph complex motions for multiple objects that interact through collision and obstruction. In such situations, the use of physically based dynamics to confer visual realism creates challenging computational problems. Typically forward simulation is well understood, but th ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Animators frequently choreograph complex motions for multiple objects that interact through collision and obstruction. In such situations, the use of physically based dynamics to confer visual realism creates challenging computational problems. Typically forward simulation is well understood, but the inverse problem of motion synthesisthat of synthesizing motions consistent both with physical law and with the animator's requirementsis generally tedious and sometimes intractable. We show how Nbody inverse problems can be formulated as optimization tasks. We present a simply stated, but combinatorially formidable example that exhibits all of the essential sources of complexity common to Nbody motion synthesis, and show how it can be solved approximately using heuristic methods based on evolutionary computation. Key Words: Animation, motion synthesis, heuristic methods, stochastic optimization, evolutionary computation, billiardball problems. Introduction Simulating the dynamic...
The traveling beam: optical solution for bounded NPcomplete problems
 The fourth international conference on fun with algorithms (FUN
, 2007
"... Architectures for optical processors designed to solve bounded instances of NPComplete problems are suggested. One approach mimics the traveling salesman by traveling beams that simultaneously examine the different possible paths. The other approach uses a preprocessing stage in which O(n 2) masks ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Architectures for optical processors designed to solve bounded instances of NPComplete problems are suggested. One approach mimics the traveling salesman by traveling beams that simultaneously examine the different possible paths. The other approach uses a preprocessing stage in which O(n 2) masks are constructed, each representing a different edge in the graph. The choice and combination of the appropriate (small) subset of these masks yields the solution. The solution is rejected in cases where the combination of these masks totally blocks the light and accepted otherwise. We present detailed designs for basic primitives of the optical processor. We propose
Optical computing
, 2008
"... We consider optical computers that encode data using images and compute by transforming such images. We give an overview of a number of such optical computing architectures, including descriptions of the type of hardware commonly used in optical computing, as well as some of the computational effici ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We consider optical computers that encode data using images and compute by transforming such images. We give an overview of a number of such optical computing architectures, including descriptions of the type of hardware commonly used in optical computing, as well as some of the computational efficiencies of optical devices. We go on to discuss optical computing from the point of view of computational complexity theory, with the aim of putting some old, and some very recent, results in context. Finally, we focus on a particular optical model of computation called the continuous space machine. We describe some results for this model including characterisations in terms of wellknown complexity classes.
Large Scale Simulations of Complex Systems Part I: Conceptual Framework
, 1997
"... In this working document, we report on a new approach to high performance simulation. The main inspiration to this approach is the concept of complex systems: disparate elements with well defined interactions rules and non nonlinear emergent macroscopic behavior. We provide arguments and mechanisms ..."
Abstract
 Add to MetaCart
In this working document, we report on a new approach to high performance simulation. The main inspiration to this approach is the concept of complex systems: disparate elements with well defined interactions rules and non nonlinear emergent macroscopic behavior. We provide arguments and mechanisms to abstract temporal and spatial locality from the application and to incorporate this locality into the complete design cycle of modeling and simulation on parallel architectures. Although the main application area discussed here is physics, the presented Virtual Particle (VIP) paradigm in the context of Dynamic Complex Systems (DCS), is applicable to other areas of compute intensive applications. Part I deals with the concepts behind the VIP and DCS models. A formal approach to the mapping of application taskgraphs to machine taskgraphs is presented. The major part of section 3 has recently (July 1997) been accepted for publication in Complexity. In Part II we will elaborate on the execution behavior of