## How much can analog and hybrid systems be proved (super-)Turing (2006)

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Venue: | Applied Mathematics and Computation |

Citations: | 5 - 1 self |

### BibTeX

@ARTICLE{Bournez06howmuch,

author = {Olivier Bournez},

title = {How much can analog and hybrid systems be proved (super-)Turing},

journal = {Applied Mathematics and Computation},

year = {2006},

volume = {178}

}

### OpenURL

### Abstract

Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing Machines. In a contrapositive way, they say that any model with super-Turing power must have something unreasonable. Our aim is to discuss how much theoretical computer science can quantify this, by considering several classes of continuous time dynamical systems, and by studying how much they can be proved Turing or super-Turing. 1

### Citations

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Citation Context ...y φ(t) is the variable t. Example: The trajectory of Figure 1 has discrete time 9. If we suppose that the norm of the speed vectors are 1, its continuous time is equal to its length. Recall (see e.g. =-=[7, 42]-=-) that a family of boolean circuits C = (Ci)i∈N, with Ci with i inputs and 1 output, recognizes a language L ⊂ Σ ∗ , iff for all w ∈ Σ ∗ , w ∈ L if and only if C |w| accepts w. Definition 6 (Class P) ... |

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Citation Context ... of continuous time dynamical systems, and by studying how much they can be proved Turing or superTuring. 1 Introduction One major result of the twentieth century is Kurt Gödel incompleteness theorem =-=[23]-=-, demonstrating that no proof system can capture our reasoning about natural numbers. The original arguments in [23] are based on an informal notion of deduction. A few time after Gödel’s paper, Alan ... |

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Citation Context ...have been explicitly design to solve Hilbert’s Entscheidungsproblem (whether an arbitrary formula of the predicate calculus can be decided to be a tautology). 1sthe lambda calculus from Alonzo Church =-=[15]-=-, and the recursive functions from Stephen Kleene [27]. These considerations leaded to Church-Turing’s thesis: “What is effectively calculable is computable”. In that thesis “calculable” refers to som... |

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Citation Context ...ontinuously differentiable on X, for all r ∈ N. Functions of class C ∞ include analytic functions. One other possibility is to talk about its computational properties in recursive analysis model: see =-=[56]-=- for an up-to-date monograph presentation of recursive analysis from a computability point of view, or [28] for a presentation from a complexity theory point of view. Following Ker-I Ko [28], let νQ :... |

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Citation Context ...ted by a (rational) PCD system H over [−1, 1] 3 . 2. This does not hold over [−1, 1] 2 , nor R 2 , in the general case. The trick used in [5] has already been seen is several other contexts (see e.g. =-=[29, 49]-=-): the current state of a Turing machine at some time t, given by some internal state q ∈ Q, and some tape w−mw−m+1 . . . w0w1 . . . wn, with the head in front of cell w0, is encoded into two real num... |

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Citation Context ...ts recognizable in polynomial time with a tally oracle [7]. It has been characterized as a natural class to characterize the computational power of several continuous space and time dynamical systems =-=[46, 47, 48]-=-. Class P corresponds to non-uniform polynomial time, since it consists in relaxing second condition in next characterization of polynomial time. Proposition 1 (P versus P (see e.g. [42])) A language ... |

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Citation Context ...w, where |w| stands for the length of w. 3 A Toy Example We are going to discuss the Piecewise Constant Derivative (PCD) model that has been introduced by Eugene Asarin, Oded Maler and Amir Pnueli in =-=[5]-=-, as a simple model for hybrid systems. It has later on been discussed in several papers such as [3, 4, 10]. A hybrid system is a system that combines continuous evolutions with discrete transitions. ... |

90 |
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Citation Context ...derivative systems involve discontinuous functions, and hence something non-reasonable, and hence that Theorem 1 do not deal with “realistic” functions. Actually, it can be reinforced as follows (see =-=[31]-=- for a proof) (observe that an alternate proof obtained by “smoothing” previous PCD system construction is proposed in [11]). Theorem 2 (Smooth Systems ≥ Turing [31]) Any recursively enumerable set L ... |

87 | Analog computation via neural networks
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Citation Context ...ts recognizable in polynomial time with a tally oracle [7]. It has been characterized as a natural class to characterize the computational power of several continuous space and time dynamical systems =-=[46, 47, 48]-=-. Class P corresponds to non-uniform polynomial time, since it consists in relaxing second condition in next characterization of polynomial time. Proposition 1 (P versus P (see e.g. [42])) A language ... |

73 | Recursion theory on the reals and continuous-time computation, Theoret
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Citation Context ...rmore the involved oracle Turing machines work in polynomial time. Other alternatives to measure the complexity of a given function exist. One of them, that we will not discuss, has been initiated by =-=[32]-=-, and consists in discussing its membership in algebraically defined classes of functions generated by a finite set of basic functions, and closed by some simple operators: see for e.g. [12, 13, 14, 2... |

67 |
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Citation Context ...idungsproblem (whether an arbitrary formula of the predicate calculus can be decided to be a tautology). 1sthe lambda calculus from Alonzo Church [15], and the recursive functions from Stephen Kleene =-=[27]-=-. These considerations leaded to Church-Turing’s thesis: “What is effectively calculable is computable”. In that thesis “calculable” refers to some intuitively given notion, whereas “computable” means... |

64 | The Undecidable - Davis - 1965 |

63 |
Computation beyond the Turing limit
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Citation Context ...ts recognizable in polynomial time with a tally oracle [7]. It has been characterized as a natural class to characterize the computational power of several continuous space and time dynamical systems =-=[46, 47, 48]-=-. Class P corresponds to non-uniform polynomial time, since it consists in relaxing second condition in next characterization of polynomial time. Proposition 1 (P versus P (see e.g. [42])) A language ... |

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Theory of Recursive Functions and Effective Computability
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Citation Context ...dexed by the constructive ordinal numbers. It is a strict hierarchy and it satisfies the strict inclusions Σα ⊂ Σβ whenever α < β. It can be related to the analytical hierarchy by ∆ 1 1 = ∪ β Σβ: see =-=[45]-=-. Class Σ1 is defined as the class of the recursively enumerable sets. When k is a constructive ordinal and when the class Σk is defined, Σk+1 is defined as the class of the languages that are recursi... |

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Citation Context ...ted by a (rational) PCD system H over [−1, 1] 3 . 2. This does not hold over [−1, 1] 2 , nor R 2 , in the general case. The trick used in [5] has already been seen is several other contexts (see e.g. =-=[29, 49]-=-): the current state of a Turing machine at some time t, given by some internal state q ∈ Q, and some tape w−mw−m+1 . . . w0w1 . . . wn, with the head in front of cell w0, is encoded into two real num... |

38 | The complexity of analog computation
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Citation Context ...methods, and hence that, potentially, continuous time dynamical systems could do things faster than Turing machines: see for example why Anastasios Vergis, Kenneth Steiglitz, and Bradley Dickinson in =-=[55]-=- avoided to take time as a natural resource in their discussion. However, Warren Smith has recently demonstrated that it is possible to prove that time can be considered as a reasonable resource under... |

37 |
The Church–Turing Thesis
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Citation Context ... is: What can be calculated by a human working mechanically with paper and pencil in a finite number of steps (in particular this covers deduction in formal systems) is computable by a Turing Machine =-=[17, 22, 54]-=-. It was soon discovered 1 that the power of Turing machines can be proved to be equal to several other formalisms that have been introduced, including 1 As observed in [16], this might be considered ... |

32 | NP-complete problems and physical reality
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Citation Context ...y advocating against hypercomputations, have argued that discussing some systems in some physical theory able of do hyper-computations helps to understand weakness of the physical models of our world =-=[2, 50, 52, 53]-=-. Our aim is in some sense a parallel computer scientist point of view: Discussing theoretical models able to do hypercomputations, helps to understand weakness of theoretical computer science models.... |

32 | Analysis of hybrid systems: An ounce of realism can save an infinity of states
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Citation Context ...at this undecidability is due to non-stability, non-robustness, sensitivity to initial values of the systems, and that it never occurs in “real systems” [3, 20]. For example, Martin Fränzle writes in =-=[21]-=- “Hence, on simple informationtheoretic grounds, the undecidability results thus obtained can be said to be artifacts of an overly idealized formalization. However, while this implies that the particu... |

30 |
A computable ordinary differential equation which possesses no computable solution
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Citation Context ...ble coefficients, with computable initial conditions, that cannot be numerically solved via deterministic methods by a digital computer. One example was provided by Marian Pour-El and Ian Richards in =-=[43]-=-: there exists a polynomial-time computable function f : [0, 1] × [−1, 1] → R such that the equation dx dt = f(t, x) defined by f does not have a computable solution y on [0, δ], for any δ > 0. Same a... |

29 | Achilles and the Tortoise climbing up the arithmetical hierarchy - Asarin, Maler - 1995 |

28 | Beyond the universal Turing machine
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- 1999
(Show Context)
Citation Context ...onstraint that the machine is “intended to conform to the physical laws (if not to the resource constraints) of the actual world” [17], otherwise the thesis is known to be false: see e.g. the surveys =-=[18, 41]-=- or the examples to follow. One close variant of this thesis, also discussed in [17], is the following: “Any process that can be given a mathematical description can be simulated by a Turing machine”.... |

26 | Achilles and the Tortoise climbing up the hyperarithmetical hierarchy. Theoret
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- 1999
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Citation Context ... (x, 0) to (0, 0) has discrete time ω. Actually, the discrete time of a finite continuous time trajectory can be bounded above according to the dimension. Theorem 13 (Discrete time vs Continuous Time =-=[9, 10]-=-) Any trajectory φ of finite continuous time of a PCD system over R d , has a discrete time Td < ω d−1 for d ≥ 3. For d = 2, Td ≤ ω. Recall that the hyper-arithmetical hierarchy is an extension of the... |

25 |
Church’s thesis and principles for mechanisms. The Kleene Symposium
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Citation Context ... is: What can be calculated by a human working mechanically with paper and pencil in a finite number of steps (in particular this covers deduction in formal systems) is computable by a Turing Machine =-=[17, 22, 54]-=-. It was soon discovered 1 that the power of Turing machines can be proved to be equal to several other formalisms that have been introduced, including 1 As observed in [16], this might be considered ... |

22 | On the computational power of dynamical systems and hybrid systems, Theoret
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(Show Context)
Citation Context ...l with “realistic” functions. Actually, it can be reinforced as follows (see [31] for a proof) (observe that an alternate proof obtained by “smoothing” previous PCD system construction is proposed in =-=[11]-=-). Theorem 2 (Smooth Systems ≥ Turing [31]) Any recursively enumerable set L is computed by a C ∞ (and Rec(R)) continuous time dynamical system H over [−1, 1] 3 . It is known that there exist differen... |

22 | Relativistic computers and the Turing barrier
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Citation Context ... The three theses are independent: • first thesis has to do with computations realizable by humans working mechanically with paper and pencils [17], • second has to do with physic of the actual world =-=[17, 40, 51, 57]-=-, • third has to do with our models of the physic of the actual world [17, 40, 51, 57]. We believe that each thesis has actually to do with convictions, since none of them is truly provable, as each o... |

22 | Church’s thesis meets the N-body problem
- Smith
(Show Context)
Citation Context ...y advocating against hypercomputations, have argued that discussing some systems in some physical theory able of do hyper-computations helps to understand weakness of the physical models of our world =-=[2, 50, 52, 53]-=-. Our aim is in some sense a parallel computer scientist point of view: Discussing theoretical models able to do hypercomputations, helps to understand weakness of theoretical computer science models.... |

20 | A.: Perturbed Turing machines and hybrid systems
- Asarin, Bouajjani
- 2001
(Show Context)
Citation Context ...ant Derivative (PCD) model that has been introduced by Eugene Asarin, Oded Maler and Amir Pnueli in [5], as a simple model for hybrid systems. It has later on been discussed in several papers such as =-=[3, 4, 10]-=-. A hybrid system is a system that combines continuous evolutions with discrete transitions. Such models appear as soon as one tries to model some systems where a discrete system, such as a computer, ... |

20 | Classical physics and the Church-Turing thesis
- Yao
(Show Context)
Citation Context ... The three theses are independent: • first thesis has to do with computations realizable by humans working mechanically with paper and pencils [17], • second has to do with physic of the actual world =-=[17, 40, 51, 57]-=-, • third has to do with our models of the physic of the actual world [17, 40, 51, 57]. We believe that each thesis has actually to do with convictions, since none of them is truly provable, as each o... |

19 | Three counterexamples refuting Kieu’s plan for quantum adiabatic hypercomputation and some uncomputable quantum mechanical tasks
- Smith
- 2006
(Show Context)
Citation Context ...y advocating against hypercomputations, have argued that discussing some systems in some physical theory able of do hyper-computations helps to understand weakness of the physical models of our world =-=[2, 50, 52, 53]-=-. Our aim is in some sense a parallel computer scientist point of view: Discussing theoretical models able to do hypercomputations, helps to understand weakness of theoretical computer science models.... |

18 | Robust simulations of Turing machines with analytic maps and flows
- Graça, Campagnolo, et al.
- 2005
(Show Context)
Citation Context ...nded space is relaxed, it has been recently obtained by Daniel Graça, Manuel Campagnolo and Jorge Buescu that Turing machines can be simulated by analytic maps (furthermore in an error-robust manner) =-=[24]-=-. Theorem 3 (Non-Compact Analytic Systems ≥ Turing [24]) Any recursively enumerable set L is computed by an analytic (and Rec(R)) continuous time dynamical system H over R 7 . 5 On Relaxing Rationalit... |

18 | Some recent developments on Shannon’s General Purpose Analog
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Citation Context ...tiated by [32], and consists in discussing its membership in algebraically defined classes of functions generated by a finite set of basic functions, and closed by some simple operators: see for e.g. =-=[12, 13, 14, 25, 35, 36]-=-. 6 Many other natural representations of rational numbers can be chosen and provide the same class of computable functions: see [28, 56]. 4sIn this paper, we will consider dynamical systems as recogn... |

17 |
Complexité algorithmique des systèmes dynamiques continus et hybrides
- Bournez
- 1999
(Show Context)
Citation Context ...ant Derivative (PCD) model that has been introduced by Eugene Asarin, Oded Maler and Amir Pnueli in [5], as a simple model for hybrid systems. It has later on been discussed in several papers such as =-=[3, 4, 10]-=-. A hybrid system is a system that combines continuous evolutions with discrete transitions. Such models appear as soon as one tries to model some systems where a discrete system, such as a computer, ... |

16 |
Finite-dimensional analog computers: Flows, maps, and recurrent neural networks
- Moore
- 1998
(Show Context)
Citation Context ...tems = Turing) Continuous time dynamical systems of C ∞ ∩ Rec(R) over [−1, 1] d have precisely the power of Turing machines: they recognize precisely recursively enumerable sets. It is conjectured in =-=[33]-=- that no analytic map on a compact, and finitedimensional space, can simulate a Turing machine, through a reasonable input and output encoding. The question whether we can suppose the continuous time ... |

16 | The many forms of hypercomputation
- Ord
(Show Context)
Citation Context ...hurch-Turing’s thesis: “What is effectively calculable is computable”. In that thesis “calculable” refers to some intuitively given notion, whereas “computable” means “computable by a Turing machine” =-=[17, 22, 41]-=-. Following Jack Copeland [17], the original thesis refers to a notion of calculation, where calculation is intended in the sense that it can be 2 realized by a human computing mechanically with paper... |

13 | Elementarily computable functions over the real numbers and R-subrecursive functions, Theoretical Computer Science 348
- Bournez, Hainry
- 2005
(Show Context)
Citation Context ...tiated by [32], and consists in discussing its membership in algebraically defined classes of functions generated by a finite set of basic functions, and closed by some simple operators: see for e.g. =-=[12, 13, 14, 25, 35, 36]-=-. 6 Many other natural representations of rational numbers can be chosen and provide the same class of computable functions: see [28, 56]. 4sIn this paper, we will consider dynamical systems as recogn... |

12 |
Some structural complexity aspects of neural computation
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Citation Context ...at previous arguments can be generalized to yield a whole structural complexity of the power of PCD systems according to their constants, similar 9 to the one that was obtained for neural networks in =-=[6]-=-. 9 But different, since the model here is not really equivalent, and is more problematic. Mostly linear precision does not suffice here. 10s6 Imposing Smoothness Following the constructions from [11]... |

7 |
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Citation Context ...plexity bounds: For any recursive real number a between 0 and 1, there exists a polynomial-time computable function f : [0, 1] × [−1, 1] such that y(x) = ax2 is the unique solution of dx dt = f(t, x) =-=[28, 30]-=-. From these considerations, we get. Corollary 1 (Smooth and Computable Systems = Turing) Continuous time dynamical systems of C ∞ ∩ Rec(R) over [−1, 1] d have precisely the power of Turing machines: ... |

6 |
and Nachum Dershowitz, A formalization of the Church-Turing Thesis, submitted
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(Show Context)
Citation Context ... or to the actual world of which we do not have a model 3 . There have been however several tentatives of proofs in literature, relying on more “basic” hypotheses about the involved notions: see e.g. =-=[8, 22]-=-. If we take each thesis in a contrapositive way, they mean that any system that computes something not computable by a Turing machine involves something, call it a “resource”, that is either non-calc... |

4 | Continuous time computation with restricted integration capabilities
- Campagnolo
(Show Context)
Citation Context ...tiated by [32], and consists in discussing its membership in algebraically defined classes of functions generated by a finite set of basic functions, and closed by some simple operators: see for e.g. =-=[12, 13, 14, 25, 35, 36]-=-. 6 Many other natural representations of rational numbers can be chosen and provide the same class of computable functions: see [28, 56]. 4sIn this paper, we will consider dynamical systems as recogn... |

4 |
Analog computation beyond the Turing limit
- Mycka
(Show Context)
Citation Context ...measuring the complexity of involved resources is very close 4 to one of the motivation of José Félix Costa and Jerzy Mycka for studying analog computations in their series of papers (see for example =-=[34, 35, 36, 37, 38]-=-), expressed explicitly in [39]. We also add that we do not claim that the considered models have any physical reality. We mainly focus on these models, since they are models that have already been co... |

4 | History of “Church’s theses” and a manifesto on converting physics into a rigorous algorithmic discipline - Smith - 1999 |

3 | The concept of computability
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Citation Context ...y a Turing Machine [17, 22, 54]. It was soon discovered 1 that the power of Turing machines can be proved to be equal to several other formalisms that have been introduced, including 1 As observed in =-=[16]-=-, this might be considered as not so surprising, since both formalisms have been explicitly design to solve Hilbert’s Entscheidungsproblem (whether an arbitrary formula of the predicate calculus can b... |

3 |
A dynamical system which must be stable whose stability cannot be proved
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Citation Context ...unity, a folklore conjecture appeared saying that this undecidability is due to non-stability, non-robustness, sensitivity to initial values of the systems, and that it never occurs in “real systems” =-=[3, 20]-=-. For example, Martin Fränzle writes in [21] “Hence, on simple informationtheoretic grounds, the undecidability results thus obtained can be said to be artifacts of an overly idealized formalization. ... |

2 | recursive functions and real extentions of recursive functions - Real - 2004 |

2 |
Robust undecidability of timed and hybrid systems. Hybrid systems: computation and control
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Citation Context ...em is indeed decidable”. There were several attempts to formalize and prove (or to disprove) this conjecture: it has been proved that small perturbations of the trajectory still yields undecidability =-=[26]-=-. Infinitesimal perturbations of the dynamics for a certain model of hybrid systems has shown to rise to decidability [21]. This has been extended to several models by [3]. Let us look at this latter ... |

2 |
The wave equation with computable inital data such that its unique solution is not computable
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Citation Context ... any δ > 0. Same authors later on expanded their result to show that wave equation (which is a partial equation), even with computable initial data, can have a unique solution which is not computable =-=[44]-=-. However, if an ordinary differential equation over a compact has a unique solution, then it must be computable: see e.g. [28]. This holds has soon as f is twice continuously differentiable. Remark: ... |

2 |
On the uncomputability of hydrodynamics
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Citation Context |