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17
Beyond The Universal Turing Machine
, 1998
"... We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of welldefined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a phi ..."
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Cited by 31 (1 self)
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We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of welldefined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a philosophical defence of its foundations.
DifferentialAlgebraic Dynamic Logic for DifferentialAlgebraic Programs
"... Abstract. We generalise dynamic logic to a logic for differentialalgebraic programs, i.e., discrete programs augmented with firstorder differentialalgebraic formulas as continuous evolution constraints in addition to firstorder discrete jump formulas. These programs characterise interacting discr ..."
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Cited by 24 (21 self)
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Abstract. We generalise dynamic logic to a logic for differentialalgebraic programs, i.e., discrete programs augmented with firstorder differentialalgebraic formulas as continuous evolution constraints in addition to firstorder discrete jump formulas. These programs characterise interacting discrete and continuous dynamics of hybrid systems elegantly and uniformly. For our logic, we introduce a calculus over real arithmetic with discrete induction and a new differential induction with which differentialalgebraic programs can be verified by exploiting their differential constraints algebraically without having to solve them. We develop the theory of differential induction and differential refinement and analyse their deductive power. As a case study, we present parametric tangential roundabout maneuvers in air traffic control and prove collision avoidance in our calculus.
Grounding Analog Computers
 Think
, 1993
"... Although analog computation was eclipsed by digital computation in the second half of the twentieth century, it is returning as an important alternative computing technology. Indeed, as explained in this report, theoretical results imply that analog computation can escape from the limitations of dig ..."
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Cited by 13 (7 self)
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Although analog computation was eclipsed by digital computation in the second half of the twentieth century, it is returning as an important alternative computing technology. Indeed, as explained in this report, theoretical results imply that analog computation can escape from the limitations of digital computation. Furthermore, analog computation has emerged as an important theoretical framework for discussing computation in the brain and other natural systems. The report (1) summarizes the fundamentals of analog computing, starting with the continuous state space and the various processes by which analog computation can be organized in time; (2) discusses analog computation in nature, which provides models and inspiration for many contemporary uses of analog computation, such as neural networks; (3) considers generalpurpose analog computing, both from a theoretical perspective and in terms of practical generalpurpose analog computers; (4) discusses the theoretical power of
The Broad Conception Of Computation
 American Behavioral Scientist
, 1997
"... A myth has arisen concerning Turing's paper of 1936, namely that Turing set forth a fundamental principle concerning the limits of what can be computed by machine  a myth that has passed into cognitive science and the philosophy of mind, to wide and pernicious effect. This supposed principle, ..."
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Cited by 12 (2 self)
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A myth has arisen concerning Turing's paper of 1936, namely that Turing set forth a fundamental principle concerning the limits of what can be computed by machine  a myth that has passed into cognitive science and the philosophy of mind, to wide and pernicious effect. This supposed principle, sometimes incorrectly termed the 'ChurchTuring thesis', is the claim that the class of functions that can be computed by machines is identical to the class of functions that can be computed by Turing machines. In point of fact Turing himself nowhere endorses, nor even states, this claim (nor does Church). I describe a number of notional machines, both analogue and digital, that can compute more than a universal Turing machine. These machines are exemplars of the class of nonclassical computing machines. Nothing known at present rules out the possibility that machines in this class will one day be built, nor that the brain itself is such a machine. These theoretical considerations undercut a numb...
"Words Lie in our Way"
, 1994
"... The central claim of computationalism is generally taken to be that the brain is a computer, and that any computer implementing the appropriate program would ipso facto have a mind. In this paper I argue for the following propositions: (1) The central claim of computationalism is not about computers ..."
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Cited by 9 (9 self)
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The central claim of computationalism is generally taken to be that the brain is a computer, and that any computer implementing the appropriate program would ipso facto have a mind. In this paper I argue for the following propositions: (1) The central claim of computationalism is not about computers, a concept too imprecise for a scientific claim of this sort, but is about physical calculi (instantiated discrete formal systems). (2) In matters of formality, interpretability, and so forth, analog computation and digital computation are not essentially different, and so arguments such as Searle's hold or not as well for one as for the other. (3) Whether or not a biological system (such as the brain) is computational is a scientific matter of fact. (4) A substantive scientific question for cognitive science is whether cognition is better modeled by discrete representations or by continuous representations. (5) Cognitive science and AI need a theoretical construct that is the continuous an...
Effective Computability of Solutions of Differential Inclusions The Ten Thousand Monkeys Approach
"... Abstract: In this paper we consider the computability of the solution of the initialvalue problem for differential inclusions with semicontinuous righthand side. We present algorithms for the computation of the solution using the “ten thousand monkeys” approach, in which we generate all possible so ..."
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Cited by 5 (2 self)
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Abstract: In this paper we consider the computability of the solution of the initialvalue problem for differential inclusions with semicontinuous righthand side. We present algorithms for the computation of the solution using the “ten thousand monkeys” approach, in which we generate all possible solution tubes, and then check which are valid. In this way, we show that the solution of an uppersemicontinuous differential inclusion is uppersemicomputable, and the solution of a differential inclusion defined by a onesided locally Lipschitz function is lowersemicomputable computable. We show that the solution of a locally Lipschitz differential equation is computable even if the function is not effectively locally Lipschitz. We also recover a result of Ruohonen, in which it is shown that if the solution is unique, then it is computable, even if the righthand side is not locally Lipschitz. We also prove that the maximal interval of existence for the solution must be effectively enumerable open, and give an example of a computable locally Lipschitz function which is not effectively locally Lipschitz.
Computational bounds on polynomial differential equations
, 2008
"... In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations. We consider elementary (in the sense of Analysis) discretetime dynamical systems satisfying certain criteria of robustness. We show that those systems can be simul ..."
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Cited by 4 (3 self)
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In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations. We consider elementary (in the sense of Analysis) discretetime dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuoustime dynamical systems which can be expanded into fully polynomial ordinary differential equations in Q[π]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines. We also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initialvalue problem defined with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56. Combined with earlier results on the computability of solutions of polynomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines.
History of "Church's theses" and a manifesto on converting physics into a rigorous algorithmic discipline
, 1999
"... Church's thesis claims that any "reasonable computer" may be simulated by a Turing machine. The "strong" thesis says that the simulation may be performed with only polynomial slowdown. This document is both a history of "Church's thesis"  and particularly of ..."
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Cited by 4 (0 self)
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Church's thesis claims that any "reasonable computer" may be simulated by a Turing machine. The "strong" thesis says that the simulation may be performed with only polynomial slowdown. This document is both a history of "Church's thesis"  and particularly of the notion that it is a statement about physics  and an opinionated philosophical statement.
N.: Topological complexity of blowup problems
 Journal of Universal Computer Science
, 2009
"... Abstract: Consider the initial value problem of the firstorder ordinary differential equation d x(t) =f(t, x(t)), x(t0) =x0 dt where the locally Lipschitz continuous function f: R l+1 → R l with open domain and the initial datum (t0,x0) ∈ R l+1 are given. It is shown that the solution operator pro ..."
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Cited by 3 (1 self)
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Abstract: Consider the initial value problem of the firstorder ordinary differential equation d x(t) =f(t, x(t)), x(t0) =x0 dt where the locally Lipschitz continuous function f: R l+1 → R l with open domain and the initial datum (t0,x0) ∈ R l+1 are given. It is shown that the solution operator producing the maximal “time ” interval of existence and the solution on it is computable. Furthermore, the topological complexity of the blowup problem is studied for functions f defined on the whole space. For each such function f the set Z of initial conditions (t0,x0) for which the positive solution does not blow up in finite time is a Gδset. There is even a computable operator determining Z from f. Forl ≥ 2 this upper Gδcomplexity bound is sharp. For l = 1 the blowup problem is simpler.
A Computationally Universal Field Computer That is Purely Linear
, 1997
"... As defined in MacLennan (1987), a field computer is a (spatial) continuumlimit neural net. This paper investigates field computers whose temporal dynamics is also continuumlimit, being governed by an integrodifferential equation. Such systems are motivated both as a means of studying neural nets ..."
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Cited by 2 (0 self)
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As defined in MacLennan (1987), a field computer is a (spatial) continuumlimit neural net. This paper investigates field computers whose temporal dynamics is also continuumlimit, being governed by an integrodifferential equation. Such systems are motivated both as a means of studying neural nets and as a model for cognitive processing. As this paper proves, even when they are purely linear. such systems are computationally universal. The "trick" used to get such universal nonlinear behavior from a purely linear system is quite similar to the way nonlinear macroscopic physics arises from the purely linear microscopic physics of Schrödinger's equation. More precisely, the "trick" involves interpreting the system in a nonlinear way. That is, the meaning of the system's output is determined by which neurons have an activation exceeding a threshold (which in this paper is taken to be 0), rather than by the actual activation values of the neurons. (This