## A network model of analogue computation over metric algebras (2005)

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Venue: | Torenvliet (Eds.), Computability in Europe, 2005, Springer Lecture Notes in Computer Science |

Citations: | 3 - 1 self |

### BibTeX

@INPROCEEDINGS{Tucker05anetwork,

author = {J. V. Tucker and J. I. Zucker},

title = {A network model of analogue computation over metric algebras},

booktitle = {Torenvliet (Eds.), Computability in Europe, 2005, Springer Lecture Notes in Computer Science},

year = {2005}

}

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### Abstract

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 input-output 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 compact-open 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 custom-made concrete computation theory over C[T, A] and show that if the modules are concretely computable then so is the function Φ. 1

### Citations

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Introduction to Metamathematics
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- 1952
(Show Context)
Citation Context ...and a proof that these analogue models do not compute all algorithmically (or “digitally”) computable functions on the reals. Cristopher Moore [Moo96] defined a system of schemes rather like Kleene’s =-=[Kle52]-=-, but with primitive recursion replaced by integration. Félix Costa and his colleagues [GC03,MC04] have presented improved models extending GPAC. We present two questions related to analogue technolog... |

236 |
Outline of General Topology
- Engelking
- 1968
(Show Context)
Citation Context ...[T, A] using a custom-made contraction principle, based on the fact that C[T, A] can be locally approximated by metric spaces. This extends the well-known Banach fixed point theorem for metric spaces =-=[Eng89]-=-. We also derive continuity of Φ, assuming continuity of the module functions. This gives a mathematical model of computation by measurements on an analogue system. In Section 4 we analyse in detail a... |

152 |
Computable Analysis, An Introduction
- Weihrauch
- 2000
(Show Context)
Citation Context ...ew model, inspired by the approximation of C[T, A] by the metric spaces C[[0, k], A]. An alternative treatment of concrete computation on the space C[X, Y ] with the compact-open topology is given in =-=[Wei00]-=-, with X ⊆ IR m and Y = IR n . 5.1 Topological algebra of streams. Consider the 5-sorted topological algebra C = (A, IR, T, C[T, A], IN; d, eval) where d : A 2 → IR and eval : C[T, A] × T → A are, res... |

33 | Analog computers and recursive functions over the reals
- Graça, Costa
(Show Context)
Citation Context ...ly”) computable functions on the reals. Cristopher Moore [8] defined a system of schemes rather like Kleene’s [6], but with primitive recursion replaced by integration. Félix Costa and his colleagues =-=[3, 7]-=- have presented improved models extending GPAC. We present two questions related to analogue technology: 1. What characteristics of data, physical components, transmissions, and system architecture, m... |

29 |
The differential analyzer: a new machine for solving differential equations
- Bush
- 1931
(Show Context)
Citation Context ... such as real numbers and data streams. The systems are networks of components or modules that operate in continuous time. Historically, in analogue computation as conceived by Kelvin [TT80] and Bush =-=[Bus31]-=-, data are represented by measurable physical quantities such as length, voltage, etc., processed by networks of mechanical or electrical components. Currently, analogue computation can involve a much... |

28 |
Mathematical theory of the differential analyser
- Shannon
- 1941
(Show Context)
Citation Context ... numbers, Banach spaces, and, more generally, metric and topological spaces. The theory of analogue computation is less developed. The general purpose analog computer (GPAC) was introduced by Shannon =-=[Sha41]-=- to model Bush’s Differential Analyzer. Shannon discovered that a function can be generated by a GPAC if, and only if, it is differentially algebraic, but his proof was incomplete. Marian Pour-El [PE7... |

28 | Abstract versus concrete computation on metric partial algebras
- Tucker, Zucker
- 2004
(Show Context)
Citation Context ... For this we introduce a custom-made concrete (digital) computation theory over C[T, A]. This is an extension to the non-metric space C[T, A] of the theory of concrete computations on metric algebras =-=[TZ04]-=-. We prove a soundness theorem for analogue, relative to concrete, computation: Theorem. If the functions defined by the components of an analogue network are concretely computable, then so is the fun... |

17 |
Algebraic specification of synchronous concurrent algorithms and architectures
- Thompson, Tucker
- 1991
(Show Context)
Citation Context ...eness of analogue with respect to digital computation, would be of great importance. We have studied computation on discrete time streams in [TZ94], and networks that process discrete time streams in =-=[TT91]-=-. 2 Analogue networks An analogue network N consists of a number of modules and channels computing and communicating with data from a topological algebra A. 2.1 Data and time. Assume we are working wi... |

16 | Real recursive functions and their hierarchy
- Mycka, Costa
(Show Context)
Citation Context ...ly”) computable functions on the reals. Cristopher Moore [8] defined a system of schemes rather like Kleene’s [6], but with primitive recursion replaced by integration. Félix Costa and his colleagues =-=[3, 7]-=- have presented improved models extending GPAC. We present two questions related to analogue technology: 1. What characteristics of data, physical components, transmissions, and system architecture, m... |

11 | Computable functions on stream algebras
- Tucker, Zucker
- 1993
(Show Context)
Citation Context ...the whole network. Settling a converse result, i.e. completeness of analogue with respect to digital computation, would be of great importance. We have studied computation on discrete time streams in =-=[TZ94]-=-, and networks that process discrete time streams in [TT91]. 2 Analogue networks An analogue network N consists of a number of modules and channels computing and communicating with data from a topolog... |

10 |
editor. Handbook of Computability Theory
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- 1999
(Show Context)
Citation Context ... and L. Torenvliet, LNCS, Springer-Verlag. ⋆⋆ The research of the second author was supported by a grant from the Natural Sciences and Engineering Research Council of Canadasdevelop in new directions =-=[Gri99]-=-. Of particular relevance is Computable Analysis, where it is applied to computable functions on real numbers, Banach spaces, and, more generally, metric and topological spaces. The theory of analogue... |

6 |
Recursion theory on the reals and continuous time computation. Theoretical Computer Science 162
- Moore
- 1996
(Show Context)
Citation Context ...ators. This yielded a new proof of Shannon’s equivalence and a proof that these analogue models do not compute all algorithmically (or “digitally”) computable functions on the reals. Cristopher Moore =-=[Moo96]-=- defined a system of schemes rather like Kleene’s [Kle52], but with primitive recursion replaced by integration. Félix Costa and his colleagues [GC03,MC04] have presented improved models extending GPA... |

6 |
Signals and Systems (2nd edition
- Oppenheim, Willsky
- 1997
(Show Context)
Citation Context ...t turned out that we did not need the assumption of time invariance (satisfied by the standard module functions in §5.6) which, like causality (which we did need) is common in dynamical system theory =-=[OW97]-=-. What is the significance of this assumption — or its absence? 2. What if we allow partial or many-valued module functionsor partial streams? 3. Find reasonable conditions, other than the contraction... |

5 |
Abstract computability and its relation to the general-purpose analog computer
- Pour-El
- 1974
(Show Context)
Citation Context ...a41] to model Bush’s Differential Analyzer. Shannon discovered that a function can be generated by a GPAC if, and only if, it is differentially algebraic, but his proof was incomplete. Marian Pour-El =-=[PE74]-=- gave a characterisation of the analogue computable functions, focusing on the classic analogue systems built from adders, scalar multipliers and integrators. This yielded a new proof of Shannon’s equ... |

3 |
Analog and Hybrid Computing
- Hyndman
- 1970
(Show Context)
Citation Context ... points in U at which κ can be defined continuously. Part (a) is immediate from Theorem 1. We omit the proof of (b). 4 A case study We apply the theory of Section 3 to an example from a standard text =-=[Hyn70]-=-. 4.1 The physical system. (See Figure 2.) A mass M is suspended by a spring with stiffness K and damping coefficient D. A force f (varying with time t) is applied to M. We want to compute its displac... |

1 |
Treatise on Natural Philosophy (2nd edition) Part I
- Thompson, Tait
(Show Context)
Citation Context ...continuous data, such as real numbers and data streams. The systems are networks of components or modules that operate in continuous time. Historically, in analogue computation as conceived by Kelvin =-=[TT80]-=- and Bush [Bus31], data are represented by measurable physical quantities such as length, voltage, etc., processed by networks of mechanical or electrical components. Currently, analogue computation c... |