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Recursion Theory on the Reals and Continuoustime Computation
 Theoretical Computer Science
, 1995
"... We define a class of recursive functions on the reals analogous to the classical recursive functions on the natural numbers, corresponding to a conceptual analog computer that operates in continuous time. This class turns out to be surprisingly large, and includes many functions which are uncomp ..."
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Cited by 73 (4 self)
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We define a class of recursive functions on the reals analogous to the classical recursive functions on the natural numbers, corresponding to a conceptual analog computer that operates in continuous time. This class turns out to be surprisingly large, and includes many functions which are uncomputable in the traditional sense.
C.: Empty space computes: The evolution of an unconventional supercomputer
 ACM International Conference on Computing Frontiers
, 2006
"... Lee A. Rubel defined the extended analog computer to remove the limitations of Shannon’s general purpose analog computer. Partial differential equation solvers were a “quintessential ” part of Rubel’s theoretical machine. These components have been implemented with “empty space ” (VLSI circuits with ..."
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Cited by 4 (0 self)
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Lee A. Rubel defined the extended analog computer to remove the limitations of Shannon’s general purpose analog computer. Partial differential equation solvers were a “quintessential ” part of Rubel’s theoretical machine. These components have been implemented with “empty space ” (VLSI circuits without transistors), as well as conductive plastic. For the past decade research at Indiana University has explored the design and applications of extended analog computers. The machines have become increasingly sophisticated and flexible. The “empty ” computational area solves partial differential equations. The rest of the space includes fuzzy logic elements, configuration memory and input/output channels. This paper describes the theoretical definition, architecture and implementation of these unconventional computers. Two applications are described in detail. Rubel’s model can be viewed as an abstract specification for a distributed supercomputer. We close with a description of this inexpensive 64node supercomputer that is based on our current single processor, and which has been placed into the public domain. The next step is to implement the improved architecture in VLSI, and seek computation speeds approaching trillions of partial differential equations per second.
A network model of analogue computation over metric algebras
 Torenvliet (Eds.), Computability in Europe, 2005, Springer Lecture Notes in Computer Science
, 2005
"... Abstract. We define a general concept of a network of analogue modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock T. The inputs and outputs of the network are continuous streams u: T → A, and the inputoutput behaviour of the ..."
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Cited by 3 (1 self)
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Abstract. We define a general concept of a network of analogue modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock T. The inputs and outputs of the network are continuous streams u: T → A, and the inputoutput behaviour of the network with system parameters from A is modelled by a function Φ: C[T,A] p ×A r →C[T,A] q (p, q> 0,r ≥ 0), where C[T,A] is the set of all continuous streams equipped with the compactopen topology. We give an equational specification of the network, and a semantics which involves solving a fixed point equation over C[T,A] using a contraction principle. We analyse a case study involving a mechanical system. Finally, we introduce a custommade concrete computation theory over C[T,A] and show that if the modules are concretely computable then so is the function Φ. 1
Computability of analogue networks
"... We define a general concept of a network of analogue modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock. The inputs and outputs of the network are continuous streams u: → A, and the inputoutput behaviour of the network with ..."
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We define a general concept of a network of analogue modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock. The inputs and outputs of the network are continuous streams u: → A, and the inputoutput behaviour of the network with system parameters from A is modelled by a function Φ: C [ , A] p ×A r → C [ , A] q (p, q> 0, r ≥ 0), where C [ , A] is the set of all continuous streams equipped with the compactopen topology. We give an equational specification of the network, and a semantics which involves solving a fixed point equation over C [ , A] using a contraction principle. We analyse two case studies involving mechanical systems. Finally, we introduce a custommade concrete computation theory over C [ , A] and show that if the modules are concretely computable then so is the function Φ. Key words and phrases: analogue computing, analogue network, concrete computation, continuous tine streams, compactopen topology