## Some recent developments on Shannon’s general purpose analog computer

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Venue: | Mathematical Logic Quarterly |

Citations: | 18 - 7 self |

### BibTeX

@ARTICLE{Graça_somerecent,

author = {Daniel Silva Graça},

title = {Some recent developments on Shannon’s general purpose analog computer},

journal = {Mathematical Logic Quarterly},

year = {},

pages = {473--485}

}

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### Abstract

This paper revisits one of the first models of analog computation, the General Purpose Analog Computer (GPAC). In particular, we restrict our attention to the improved model presented in [11] and we show that it can be further refined. With this we prove the following: (i) the previous model can be simplified; (ii) it admits extensions having close connections with the class of smooth continuous time dynamical systems. As a consequence, we conclude that some of these extensions achieve Turing universality. Finally, it is shown that if we introduce a new notion of computability for the GPAC, based on ideas from computable analysis, then one can compute transcendentally transcendental functions such as the Gamma function or Riemann’s Zeta function. 1

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Citation Context ... is shown that every class of IC computable functions corresponds to a specific 1 A similar philosophy can also be found, to a certain extent, in more recent models such as artificial neural networks =-=[18], [1]. As -=-Haykin says [12, p. 25]: "The design of a neural network is based directly on real-life data, with the data set being permitted to speak for itself." 2 k + i u k u v u+v v i t t 0 u(x)dv(x) ... |

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Citation Context ... performed by any human following a `rule of thumb.' This remains a highly consensual approach, with some reserves [32], [9], [31], that can be found in the computable analysis literature [25], [14], =-=[33]-=-. On the other side, a di#erent philosophy underlies analog computers. Here we don't have a notion of `algorithm' and there is no need to translate quantities into appropriate symbolic forms. Moreover... |

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Citation Context ... of `elementary steps' should be defined in a manner such that it could be performed by any human following a `rule of thumb.' This remains a highly consensual approach, with some reserves [32], [9], =-=[31]-=-, that can be found in the computable analysis literature [25], [14], [33]. On the other side, a di#erent philosophy underlies analog computers. Here we don't have a notion of `algorithm' and there is... |

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Citation Context ...own that every class of IC computable functions corresponds to a specific 1 A similar philosophy can also be found, to a certain extent, in more recent models such as artificial neural networks [18], =-=[1]. As Hayki-=-n says [12, p. 25]: "The design of a neural network is based directly on real-life data, with the data set being permitted to speak for itself." 2 k + i u k u v u+v v i t t 0 u(x)dv(x) A con... |

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Citation Context ...one uses a circuit generating tan(t), then one can approximate f(x) to any extent only by using values of t in [0, #/2). This is in some sense similar to the "compression trick" presented by=-= Moore in [19], where in-=-finite computations 7 Provided we allow the use of all reals and not only the constants -1, 0, 1. 14 can be carried out "within finite time." Therefore, one can always accelerate the computa... |

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Citation Context ...e M, there is a class F constituted by C # functions such that M can be simulated by an F-IC. Proof. This is an immediate consequence of Theorem 5.7, Corollary 5.8, and the comments following them in =-=[4]-=-, where it is stated that every Turing machine can be simulated by a system of smooth ODEs. Then using Theorem 11, one concludes the result. A review of notions of simulation can be found in [4]. In e... |

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Citation Context ... computation. This paper focuses primarily on 3 objectives. The first one is to simplify the submodel of the GPAC presented in [11]. Indeed, the general GPAC presented by Shannon (and also by Pour-El =-=[24]-=-) has some problems that can be solved by the FF-GPAC model introduced by Graca and Costa in [11]. The present paper shows that this model can be further simplified (to the PGPAC model). This point is... |

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Citation Context ...e some kind of bound a priori, it seems very hard to avoid an accumulation of errors that will compromise the computation. So, an usual procedure is to allow the use of "exact computation" (=-=eg. [30], [15]-=-, [22]). 9 ys,...,ys) 1 f(t, n g(t,ys,...,ys) 1 k-1 i y k t y 1 y n ... P t y 1 y k-1 ... Q k k Figure 4: Schema of inputs and outputs for the integrator I k in the F-FIC U . Here f, g # [C; U, F ]. y... |

51 |
Mathematical theory of the differential analyzer
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Citation Context ...on or Riemann’s Zeta function. 1 Introduction In this paper we explore a particular model of analog computation, the General Purpose Analog Computer (GPAC). The GPAC was introduced in 1941 by Shannon =-=[30]-=- as a mathematical model of an analog device, the Differential Analyzer [5]. This device was one of the most popular analog computers in the 1930s and was intended to solve numerical problems, especia... |

41 |
The extended analog computer
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Citation Context ...ion for the GPAC [24] still have some deficiencies. Therefore, in the next section, we restrict Shannon's model in order to avoid these problems. Another important question (already reported in [24], =-=[29]-=-) is what happens if we allow other types of black boxes beside those indicated in Fig. 1. An answer for this question will be supplied in Section 5. This approach will also enable us to present close... |

33 | Analog computers and recursive functions over the reals
- Graça, Costa
(Show Context)
Citation Context ...004 Abstract This paper revisits one of the first models of analog computation, the General Purpose Analog Computer (GPAC). In particular, we restrict our attention to the improved model presented in =-=[11]-=- and we show that it can be further refined. With this we prove the following: (i) the previous model can be simplified; (ii) it admits extensions having close connections with the class of smooth con... |

33 |
and L A Rubel, A differentially algebraic replacement theorem
- Lipshitz
(Show Context)
Citation Context ...e is a closed subinterval I # # I with non-empty interior such that y can be generated by a PGPAC on I # . The last two theorems assert a classical result on the literature about the GPAC [30], [24], =-=[17]-=-: unary functions generated by (P)GPACs are, in essence, di#erentially algebraic functions. This result indicates that a large class of functions, such as polynomials, trigonometric functions, ellipti... |

29 | An analog characterization of the Grzegorczyk hierarchy
- Campagnolo, Moore, et al.
(Show Context)
Citation Context ...nits of a GPAC are associated to computable values, it is unknown whether f should have a series expansion with computable coe#cients and be therefore computable. Another di#erent path is followed in =-=[8]-=-, [2]. Namely, Campagnolo, Costa, and Moore showed that restricted forms of integration lead to a hierarchy of continuous time systems related to the Grzegorczyk hierarchy over the naturals. In some s... |

29 | Closed-form analytic maps in one and two dimensions can simulate Turing machines - Koiran, Moore - 1999 |

29 |
The differential analyzer: a new machine for solving differential equations
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(Show Context)
Citation Context ...ticular model of analog computation, the General Purpose Analog Computer (GPAC). The GPAC was introduced in 1941 by Shannon [30] as a mathematical model of an analog device, the Differential Analyzer =-=[5]-=-. This device was one of the most popular analog computers in the 1930s and was intended to solve numerical problems, especially differential equations [3]. In short, a (mechanical) differential analy... |

29 | Iteration, inequalities, and differentiability in analog computers
- Campagnolo, Moore, et al.
(Show Context)
Citation Context ...the previous theorem does not apply, in general, to F-ICs. For the following result, consider the function θk defined by θk(x) = x k if x ≥ 0 and θk(x) = 0 if x < 0 (k ∈ N). This function can be seen =-=[7]-=- as a C k−1 version of Heaviside’s step function θ(x), where θ(x) = 1 for x ≥ 0 and θ(x) = 0 for x < 0. Because we assumed in the beginning of Section 5 that F is constituted by C 1 functions, we only... |

18 |
Turing machines can compute uncomputable functions
- Copeland, Even
- 1998
(Show Context)
Citation Context ...uence of `elementary steps' should be defined in a manner such that it could be performed by any human following a `rule of thumb.' This remains a highly consensual approach, with some reserves [32], =-=[9]-=-, [31], that can be found in the computable analysis literature [25], [14], [33]. On the other side, a di#erent philosophy underlies analog computers. Here we don't have a notion of `algorithm' and th... |

18 | A survey of transcendentally transcendental functions - Rubel - 1989 |

16 | Real recursive functions and their hierarchy
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(Show Context)
Citation Context ...t, due to the connections between PGPAC-computable functions and R-recursive functions presented in [11], if f is generated by a PGPAC via approximations, then f belongs to the class H 1 presented in =-=[20]-=-. 7 Moreover, f is also computable by Rubel's Extended Analog Computer [29]. Notice that t represents the time, that might not necessarily correspond to physical time. It is also important to remark t... |

16 |
Some mathematical limitations of the general-purpose analog computer
- Rubel
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Citation Context ...this paper is provided in Fig. 5 (some of the relations are proved in what follows). Finally, and as a third objective, we show that some of the mathematical limitations pointed out in the literature =-=[27]-=- are not inherent to the GPAC but rather on the underlying notion of computability. In particular, we will show that the Gamma function and Riemann's Zeta function can indeed be computed by a GPAC if ... |

12 |
technological enthusiasm and british technological skepticism in the age of the analog brain
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Citation Context ...of an analog device, the Di#erential Analyzer [5]. This device was one of the most popular analog computers in the 1930s and was intended to solve numerical problems, especially di#erential equations =-=[3]-=-. In short, a (mechanical) di#erential analyzer may be seen as a set of interconnected shafts, each of which representing one of the quantities involved in the computation. Although the reader might f... |

9 | The complexity of real recursive functions - Campagnolo |

5 |
Mathematical theory of the di erential analyzer
- Shannon
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Citation Context ...on or Riemann's Zeta function. 1 Introduction In this paper we explore a particular model of analog computation, the General Purpose Analog Computer (GPAC). The GPAC was introduced in 1941 by Shannon =-=[30]-=- as a mathematical model of an analog device, the Di#erential Analyzer [5]. This device was one of the most popular analog computers in the 1930s and was intended to solve numerical problems, especial... |

4 |
Iteration, inequalities, and di#erentiability in analog computers
- Campagnolo, Moore, et al.
- 2000
(Show Context)
Citation Context ... previous theorem does not apply, in general, to F-ICs. For the following result, consider the function # k defined by # k (x) = x k if x # 0 and # k (x) = 0 if xs0 (k # N). This function can be seen =-=[7]-=- as a C k-1 version of Heaviside's step function #(x), where #(x) = 1 for x # 0 and #(x) = 0 for xs0. Because we assumed in the beginning of Section 5 that F is constituted by C 1 functions, we only c... |

4 | On the role of mathematics and mathematical knowledge in the invention of Vannevar Bush’s early analog computers - Puchta - 1996 |

3 |
The di#erential analyzer. A new machine for solving di#erential equations
- Bush
- 1931
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Citation Context ...rticular model of analog computation, the General Purpose Analog Computer (GPAC). The GPAC was introduced in 1941 by Shannon [30] as a mathematical model of an analog device, the Di#erential Analyzer =-=[5]-=-. This device was one of the most popular analog computers in the 1930s and was intended to solve numerical problems, especially di#erential equations [3]. In short, a (mechanical) di#erential analyze... |

3 |
Guest editor’s introduction
- Nyce
- 1996
(Show Context)
Citation Context ...olved in the computation. Although the reader might feel uncomfortable with this approach, based on technologically obsolete computing devices, we believe there is much to explore. Quoting James Nyce =-=[21]: &qu-=-ot;Because digital computers and computation have been so successful, they have influenced how we think about both computers as machines and computation as a process - so much so, it is di#cult today ... |

3 | Where are we going, Phil Morse? Changing agendas and the rhetoric of obviousness in the transformation of computing at MIT, 19391957 - Owens - 1996 |

2 |
A survey of continous-time computation theory
- Orponen
- 1997
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Citation Context ... kind of bound a priori, it seems very hard to avoid an accumulation of errors that will compromise the computation. So, an usual procedure is to allow the use of "exact computation" (eg. [3=-=0], [15], [22]-=-). 9 ys,...,ys) 1 f(t, n g(t,ys,...,ys) 1 k-1 i y k t y 1 y n ... P t y 1 y k-1 ... Q k k Figure 4: Schema of inputs and outputs for the integrator I k in the F-FIC U . Here f, g # [C; U, F ]. y i den... |

1 |
An analog characterization of computable functions over the real numbers. submitted for publication
- Bournez, Hainry
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Citation Context ...of a GPAC are associated to computable values, it is unknown whether f should have a series expansion with computable coe#cients and be therefore computable. Another di#erent path is followed in [8], =-=[2]-=-. Namely, Campagnolo, Costa, and Moore showed that restricted forms of integration lead to a hierarchy of continuous time systems related to the Grzegorczyk hierarchy over the naturals. In some sense,... |

1 |
Mathematical Research Today and Tomorrow, chapter Theory of computation
- Smale
- 1992
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Citation Context ...is sequence of `elementary steps' should be defined in a manner such that it could be performed by any human following a `rule of thumb.' This remains a highly consensual approach, with some reserves =-=[32]-=-, [9], [31], that can be found in the computable analysis literature [25], [14], [33]. On the other side, a di#erent philosophy underlies analog computers. Here we don't have a notion of `algorithm' a... |