## Computations via experiments with kinematic systems (2004)

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Citations: | 15 - 4 self |

### BibTeX

@TECHREPORT{Beggs04computationsvia,

author = {E. J. Beggs and J. V. Tucker},

title = {Computations via experiments with kinematic systems},

institution = {},

year = {2004}

}

### OpenURL

### Abstract

Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2-dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be designed to operate under (a) Newtonian mechanics or (b) Relativistic mechanics. The theorem proves that valid models of mechanical systems can compute all possible functions on discrete data. The proofs show how any information (coded by some A) can be embedded in the structure of a simple kinematic system and retrieved by simple observations of its behaviour. We reflect on this undesirable situation and argue that mechanics must be extended to include a formal theory for performing experiments, which includes the construction of systems. We conjecture that in such an extended mechanics the functions computed by experiments are precisely those computed by algorithms. We set these theorems and ideas in the context of the literature on the general problem “Is physical behaviour computable? ” and state some open problems.

### Citations

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Citation Context ...ematic theoretical investigation. In contrast, computability theory, founded by Church, Turing and Kleene in 1936, is a deep theory for the functions computable by algorithms on discrete data (Rogers =-=[46]-=-, Odifreddi [36], Griffor [26], Stoltenberg-Hansen and Tucker [54]); it is being extended to continuous data (Aberth[1], Pour-El and Richards [44], Blum et al [8], Tucker and Zucker [60, 61], Weihrauc... |

695 | Quantum Theory, the Church–Turing Principle and the Universal Quantum Computer
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Citation Context ...f quantum computation. Informal notions of quantum algorithms, computers and circuits have been developed and the the physical aspects of the Church-Turing Thesis discussed since Benioff [6], Deutsch =-=[18, 19]-=- and Yao [67]. Comparisons with classical computation have been focussed on the superior speed of quantum computation. An early rigorous definition of a quantum Turing machine is in Bernstein and Vazi... |

525 |
The Emperor’s New Mind
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Citation Context ...point of view of a theory of experimental computation, the arguments for this negative answer are not adequate, as the analysis in Davis [17] demonstrates. The strong debate of Penrose’s proposals in =-=[39]-=- is also a rejection of the positive answer. Quantum computing The experimental nature of computation, which our kinematic computers illuminate, is also the basis of the more complex field of quantum ... |

491 | Quantum complexity theory
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Citation Context ...Yao [67]. Comparisons with classical computation have been focussed on the superior speed of quantum computation. An early rigorous definition of a quantum Turing machine is in Bernstein and Vazirani =-=[7]-=- where it is shown that the quantum 13sTuring machine can be simulated by a classical one and vice versa. However, the formulation of some quantum computer models rely on classical computability theor... |

322 |
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Citation Context ...al investigation. In contrast, computability theory, founded by Church, Turing and Kleene in 1936, is a deep theory for the functions computable by algorithms on discrete data (Rogers [46], Odifreddi =-=[36]-=-, Griffor [26], Stoltenberg-Hansen and Tucker [54]); it is being extended to continuous data (Aberth[1], Pour-El and Richards [44], Blum et al [8], Tucker and Zucker [60, 61], Weihrauch [62]). Where t... |

282 | Quantum circuit complexity
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Citation Context ...tion. Informal notions of quantum algorithms, computers and circuits have been developed and the the physical aspects of the Church-Turing Thesis discussed since Benioff [6], Deutsch [18, 19] and Yao =-=[67]-=-. Comparisons with classical computation have been focussed on the superior speed of quantum computation. An early rigorous definition of a quantum Turing machine is in Bernstein and Vazirani [7] wher... |

252 |
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Citation Context ...f quantum computation. Informal notions of quantum algorithms, computers and circuits have been developed and the the physical aspects of the Church-Turing Thesis discussed since Benioff [6], Deutsch =-=[18, 19]-=- and Yao [67]. Comparisons with classical computation have been focussed on the superior speed of quantum computation. An early rigorous definition of a quantum Turing machine is in Bernstein and Vazi... |

229 |
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Citation Context ...nnected to perform successive steps of a calculation, infinitely many clock cycles could be performed in finite time. Suggestions for simulating digital logic by kinematics are in Fredkin and Toffoli =-=[20]-=- Analogue computers Analogue computation as conceived by Lord Kelvin [58], V Bush [13], and D Hartree [24], is experimental computation. The functions are of the form f : R n → R m and the physical sy... |

166 |
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Citation Context ...Odifreddi [36], Griffor [26], Stoltenberg-Hansen and Tucker [54]); it is being extended to continuous data (Aberth[1], Pour-El and Richards [44], Blum et al [8], Tucker and Zucker [60, 61], Weihrauch =-=[62]-=-). Where there are instruments and machines for aiding calculation one can view a computation as an experiment with a physical system. Current technologies for computing and communication, such as tho... |

150 |
Neural Networks and Analog Computation: Beyond the Turing Limit, Birkhäuser
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Citation Context ...he answer to the question ”Are neural systems computable?” is “Yes” (e.g., see Holden et al [25]). However, a convenient survey of models and a proposal that some hybrid nets are not is in Siegelmann =-=[52]-=-. From the point of view of a theory of experimental computation, the arguments for this negative answer are not adequate, as the analysis in Davis [17] demonstrates. The strong debate of Penrose’s pr... |

147 |
Hybrid Models for Motion Control Systems
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Citation Context ...tence of an ODE with computable initial conditions and no computable solution is proved in Pour El and Richards [41]. The simulation of machines by ODEs has been shown for finite automata in Brockett =-=[11]-=-, and for Turing machines in Branicky [10]. Decision problems for the differential equations of mechanics are not new. An early problem in the qualititaive theory of celestial mechanics is Poincare’s ... |

125 |
The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines
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Citation Context ...mplex field of quantum computation. Informal notions of quantum algorithms, computers and circuits have been developed and the the physical aspects of the Church-Turing Thesis discussed since Benioff =-=[6]-=-, Deutsch [18, 19] and Yao [67]. Comparisons with classical computation have been focussed on the superior speed of quantum computation. An early rigorous definition of a quantum Turing machine is in ... |

95 |
Discussion with Einstein on epistemological problems in atomic physics
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Citation Context ...f the conduct of idealised experiments is necessary. Gedanken experiments have be used since Galileo and are a complex philosophical subject in their own right, of course (see, e.g., Brown [12], Bohr =-=[9]-=-, Koyre [27], Kuhn [30]). To cite an example in kinematics, gedanken experiments related to Zeno’s paradoxes have re-surfaced in philosopical debates about infinite machines and Newtonian supertasks (... |

95 |
Unpredictability and Undecidability in Dynamical Systems
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Citation Context ...mputation. It may also yield results about non-computable behaviour of the physical system from the undecidability of the halting problem for Turing machines. A good example of this approach is Moore =-=[33]-=- on the “unpredictability” of physical systems. Moore shows how to model a Turing machine by a shift map, and, in turn, suggests a kinematic system, with a single particle guided by pin-ball “mirrors”... |

69 |
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Citation Context ...Differential Analyser of Bush [13]. Shannon discovered that a function can be generated by a GPAC if, and only if, it is differentially algebraic, but his proof was incomplete. An analysis in Pour El =-=[40]-=- yielded a new stronger model and a new proof of the equivalence (and some new gaps corrected in Lipshitz and Rubel [31]). Using the characterisation in terms of algebraic differential equations, thes... |

66 |
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Citation Context ...rgy, or information. In Beggs and Tucker [4] we give axioms for the local structures of systems and their local states. An attempt at such an axiomatistion of machines for digital processing is Gandy =-=[21]-=- in which notions of space and causality are modelled using hereditarily finite sets. The conceptual analysis is frustratingly difficult to use and has been studied in some depth by Sieg [49, 50]. Thi... |

55 |
The Wave Equation with Computable Initial Data such that its Unique Solution is not Computable
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Citation Context ...d has become more confusing, difficult and fascinating (Cooper and Odifreddi [14]). Wave mechanics A major attempt at an answer was by Pour El and Richards, who proposed “No”. In Pour El and Richards =-=[42]-=- they showed that there are solutions of the 3dimensional wave equation with computable initial values that are are not computable over unit time [0, 1]. The notion of computable was based on the unif... |

54 | Parallel quantum computation
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Citation Context ...ny models for quantum computation are being developed and our understanding of this complex notion of computation is at an early stage. It seems not to be known if quantum cellular automata (Margolus =-=[32]-=-) can be simulated by a classical Turing machine. Classical versus quantum systems Penrose’s study and reflections on computability, physical laws and consciousness have stimulated a great deal of tho... |

35 |
Computable Analysis
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Citation Context ...deep theory for the functions computable by algorithms on discrete data (Rogers [46], Odifreddi [36], Griffor [26], Stoltenberg-Hansen and Tucker [54]); it is being extended to continuous data (Aberth=-=[1]-=-, Pour-El and Richards [44], Blum et al [8], Tucker and Zucker [60, 61], Weihrauch [62]). Where there are instruments and machines for aiding calculation one can view a computation as an experiment wi... |

35 |
A differentially algebraic replacement theorem, and analog computability
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Citation Context ... differentially algebraic, but his proof was incomplete. An analysis in Pour El [40] yielded a new stronger model and a new proof of the equivalence (and some new gaps corrected in Lipshitz and Rubel =-=[31]-=-). Using the characterisation in terms of algebraic differential equations, these analogue models were shown not to compute all computable functions on R (Pour El [40]). These models are close to the ... |

35 |
Some undecidable problems involving elementary functions of a real variable
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Citation Context ...ecent example of undecidability is da Costa and Doria [15] on the integration of Hamiltonians using quadratures, proved using the undecidability of the integration of elementary functions (Richardson =-=[45]-=-). 6.2 The search for a conceptual analysis The search for non-computable aspects of particular physical examples, or of whole classes of mathematical models, must be complemented by a search for the ... |

31 |
A computable ordinary differential equation which possesses no computable solution
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(Show Context)
Citation Context ...rom how physical ideas can be use to perform computation. Some examples follow. The existence of an ODE with computable initial conditions and no computable solution is proved in Pour El and Richards =-=[41]-=-. The simulation of machines by ODEs has been shown for finite automata in Brockett [11], and for Turing machines in Branicky [10]. Decision problems for the differential equations of mechanics are no... |

31 | Concrete models of computation for topological algebras, Theoretical Computer Science 219 - Stoltenberg-Hansen, Tucker - 1999 |

30 |
Mathematical theory of the differential analyser
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Citation Context ... R m and the physical systems are made from mechanical or electro-mechanical components. The theory of analogue computers is modest. A general purpose analog computer (GPAC) was introduced in Shannon =-=[48]-=- as a model of the Differential Analyser of Bush [13]. Shannon discovered that a function can be generated by a GPAC if, and only if, it is differentially algebraic, but his proof was incomplete. An a... |

29 | Abstract versus concrete computation on metric partial algebras
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(Show Context)
Citation Context ... data (Rogers [46], Odifreddi [36], Griffor [26], Stoltenberg-Hansen and Tucker [54]); it is being extended to continuous data (Aberth[1], Pour-El and Richards [44], Blum et al [8], Tucker and Zucker =-=[60, 61]-=-, Weihrauch [62]). Where there are instruments and machines for aiding calculation one can view a computation as an experiment with a physical system. Current technologies for computing and communicat... |

18 |
A notion of mechanistic theory, Synthese 29
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(Show Context)
Citation Context ...iate the study of mechanical examples of the kind given here. 6.1 The search for non-computability The question “Is physical behaviour computable?” was asked in computability theory in, e.g., Kreisel =-=[28]-=-. The problem is unresolved, it will not go away, and has become more confusing, difficult and fascinating (Cooper and Odifreddi [14]). Wave mechanics A major attempt at an answer was by Pour El and R... |

16 |
On effective topological spaces, The
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(Show Context)
Citation Context ... worth noting that on data types with metric space structures Ceitin’s Theorem says, roughly, that computablity implies continuity. Studies of generalisations and converses of this theorem are Spreen =-=[53]-=- and Stoltenberg-Hansen and Tucker [57]. Thus, in designing physical systems for computation, one proviso is to avoid singular cases which give rise to discontinuities. In the mechanics of point parti... |

15 |
Analog computation with continuous ODEs, in
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- 1994
(Show Context)
Citation Context ...nditions and no computable solution is proved in Pour El and Richards [41]. The simulation of machines by ODEs has been shown for finite automata in Brockett [11], and for Turing machines in Branicky =-=[10]-=-. Decision problems for the differential equations of mechanics are not new. An early problem in the qualititaive theory of celestial mechanics is Poincare’s Centre Problem (see, e.g., Seigel and Mose... |

15 |
A function for thought experiments
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(Show Context)
Citation Context ...ised experiments is necessary. Gedanken experiments have be used since Galileo and are a complex philosophical subject in their own right, of course (see, e.g., Brown [12], Bohr [9], Koyre [27], Kuhn =-=[30]-=-). To cite an example in kinematics, gedanken experiments related to Zeno’s paradoxes have re-surfaced in philosopical debates about infinite machines and Newtonian supertasks (Perez Laraudogoitia [37... |

15 |
A recursive function defined on a compact interval and having a continuous derivative that is not recursive
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- 1971
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Citation Context ...fter a certain time we make a measurement of the wave, and a ‘noncomputable’ result emerges. The tool shapes the surface according to a computable function, whose derivative is not computable (Myhill =-=[35]-=-). By a computable function, we mean that its value at a given point can be calculated to a given precision given enough time (i.e. clock cycles). If we insist that the construction be performed in a ... |

15 |
Computable and continuous partial homomorphisms on metric partial algebras
- Stoltenberg-Hansen, Tucker
- 2003
(Show Context)
Citation Context ...the computability of operators on Banach spaces in Pour El and Richards [44], and in the general setting of partial homomorphisms of arbitrary metric partial algebras in Stoltenberg-Hansen and Tucker =-=[57]-=-. Because the wave equation is a fundamental model of physical phenomena, Pour El and Richards suggested that the result indicated that there exist physical systems that could show non-computable beha... |

12 | Computation by ‘while’ programs on topological partial algebras, Theoretical Computer Science 219 - Tucker, Zucker - 1999 |

11 |
Calculating Instruments and Machines
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(Show Context)
Citation Context ...inite time. Suggestions for simulating digital logic by kinematics are in Fredkin and Toffoli [20] Analogue computers Analogue computation as conceived by Lord Kelvin [58], V Bush [13], and D Hartree =-=[24]-=-, is experimental computation. The functions are of the form f : R n → R m and the physical systems are made from mechanical or electro-mechanical components. The theory of analogue computers is modes... |

11 |
Coupled map lattices as computational systems
- Holden, Tucker, et al.
- 1992
(Show Context)
Citation Context ...al tissue can be modelled as hybrid logical/algorithmic networks has led to many results that confirm that the answer to the question ”Are neural systems computable?” is “Yes” (e.g., see Holden et al =-=[25]-=-). However, a convenient survey of models and a proposal that some hybrid nets are not is in Siegelmann [52]. From the point of view of a theory of experimental computation, the arguments for this neg... |

11 |
Handbook of Computability Theory
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- 1999
(Show Context)
Citation Context ...on. In contrast, computability theory, founded by Church, Turing and Kleene in 1936, is a deep theory for the functions computable by algorithms on discrete data (Rogers [46], Odifreddi [36], Griffor =-=[26]-=-, Stoltenberg-Hansen and Tucker [54]); it is being extended to continuous data (Aberth[1], Pour-El and Richards [44], Blum et al [8], Tucker and Zucker [60, 61], Weihrauch [62]). Where there are instr... |

10 | Computability and noncomputability in classical analysis - Richards - 1983 |

8 |
Zwei Unentscheitbare Probleme der Analysis. Zeitschrift für mathematische Logik und Grundlagen der Mathematik
- Scarpellini
- 1963
(Show Context)
Citation Context ...actice of analogue computing until the 1960s. That an undecidable predicate of a computable function on R might be experimentally computable by a suitable analogue machine was observed in Scarpellini =-=[47]-=-. Recently, the theory of analogue computing has been restarted by C Moore with very general mathematical models (Moore [34]). These models define functions by schemes rather like Kleene’s, but with p... |

8 |
Quantum computational networks. Proceedings of the Royal Society of London
- Deutsch
- 1989
(Show Context)
Citation Context ...f quantum computation. Informal notions of quantum algorithms, computers and circuits have been developed and the the physical aspects of the Church-Turing Thesis discussed since Benioff [6], Deutsch =-=[18, 19]-=- and Yao [67]. Comparisons with classical computation have been focussed on the superior speed of quantum computation. An early rigorous definition of a quantum Turing machine is in Bernstein and Vazi... |

7 |
Is wave propogation computable or can wave computers beat the Turing machine
- Weihrauch, Zhong
- 2002
(Show Context)
Citation Context ...systems that could show non-computable behaviour. However, the experimental basis of the proposal was too weak to support the suggestion, as pointed out in Kreisel [29]. Recently, Weihrauch and Zhong =-=[65]-=- have re-visited the wave equation, claiming that by using the appropriate norms, solving the wave equation is a computable problem. They have shown that using the C 1 norm on the differentiable funct... |

7 |
Is the linear Schrödinger propagator Turing computable
- Weihrauch, Zhong
- 2001
(Show Context)
Citation Context ... × H s−1 [R 3 , R] was computable, where H s is a Sobolev space 11sof functions. Hence they propose that the answer is, in fact, “Yes” in the case of ndimensional wave systems. In Weihrauch and Zhong =-=[66]-=- is a proof that the Schrödinger equation has computable solutions. Let us try to convert the Pour El and Richards method to a ‘real’ experiment. We start with a flat sheet of ice, and use a computer ... |

6 |
The Laboratory of the Mind.: Thought Experiments
- Brown
- 1991
(Show Context)
Citation Context ...r account of the conduct of idealised experiments is necessary. Gedanken experiments have be used since Galileo and are a complex philosophical subject in their own right, of course (see, e.g., Brown =-=[12]-=-, Bohr [9], Koyre [27], Kuhn [30]). To cite an example in kinematics, gedanken experiments related to Zeno’s paradoxes have re-surfaced in philosopical debates about infinite machines and Newtonian su... |

6 |
A da Costa and F A Doria, Undecidability and incompleteness in classical mechanics
- C
- 1991
(Show Context)
Citation Context ...cesses that we can use but not at the level of classical physics. In da Costa and Doria [16] this idea is formulated as Penrose’s Thesis and a “counter-example” suggested, based on da Costa and Doria =-=[15]-=-, that shows classical mechanics can produce non-computable behaviour. We consider the counterexample is not convincing, not least because it complicated and fails to allow a robust experiment. Our ba... |

6 |
A da Costa and F A Doria, Classical physics and
- C
- 1991
(Show Context)
Citation Context ... systems and the mind (Penrose [39]). Relevant here is his conjecture that nature can produce non-computable processes that we can use but not at the level of classical physics. In da Costa and Doria =-=[16]-=- this idea is formulated as Penrose’s Thesis and a “counter-example” suggested, based on da Costa and Doria [15], that shows classical mechanics can produce non-computable behaviour. We consider the c... |

6 |
Recursion theory on the reals and continuous time computation. Theoretical Computer Science 162
- Moore
- 1996
(Show Context)
Citation Context ...tally computable by a suitable analogue machine was observed in Scarpellini [47]. Recently, the theory of analogue computing has been restarted by C Moore with very general mathematical models (Moore =-=[34]-=-). These models define functions by schemes rather like Kleene’s, but with primitive recursion replaced by integration and others added, but can define functions beyond the class of computable functio... |

6 |
Calculation by Man and Machine: Conceptual Analysis
- Sieg
- 2001
(Show Context)
Citation Context ...is Gandy [21] in which notions of space and causality are modelled using hereditarily finite sets. The conceptual analysis is frustratingly difficult to use and has been studied in some depth by Sieg =-=[49, 50]-=-. This analysis, although focussed on refining the idea of mechanical computation as portayed in Turing machines, is relevant to our problem. Digital computation 15sby machines is also an example of p... |

4 |
Review of Pour El and Richards
- Kreisel
- 1974
(Show Context)
Citation Context ...indicated that there exist physical systems that could show non-computable behaviour. However, the experimental basis of the proposal was too weak to support the suggestion, as pointed out in Kreisel =-=[29]-=-. Recently, Weihrauch and Zhong [65] have re-visited the wave equation, claiming that by using the appropriate norms, solving the wave equation is a computable problem. They have shown that using the ... |

3 |
The differential analyser. A new machine for solving differential equations
- Bush
- 1931
(Show Context)
Citation Context ...ld be performed in finite time. Suggestions for simulating digital logic by kinematics are in Fredkin and Toffoli [20] Analogue computers Analogue computation as conceived by Lord Kelvin [58], V Bush =-=[13]-=-, and D Hartree [24], is experimental computation. The functions are of the form f : R n → R m and the physical systems are made from mechanical or electro-mechanical components. The theory of analogu... |

3 |
The myth of hypercomputation, in Turing Festschrift
- Davis
(Show Context)
Citation Context ... that some hybrid nets are not is in Siegelmann [52]. From the point of view of a theory of experimental computation, the arguments for this negative answer are not adequate, as the analysis in Davis =-=[17]-=- demonstrates. The strong debate of Penrose’s proposals in [39] is also a rejection of the positive answer. Quantum computing The experimental nature of computation, which our kinematic computers illu... |

3 | Embedding metric spaces into cpo’s, Theoretical Computer Science 16 - Weihrauch, Schreiber - 1981 |

3 | The Wave Propagator is Turing Computable - Weihrauch, Zhong - 1999 |