The Church-Turing Thesis: still valid after all these years?

1
The Church-Turing Thesisstill valid after all
these years?

• Antony Galton
• Department of Computer Science
• University of Exeter, UK

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What is the Church-Turing thesis?

3
Alonzo Church Alan Turing 1903-1995
1912-1954

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Church

• Alonzo Church, 1936, An unsolvable problem of
  elementary number theory.
• Introduced recursive functions and l-definable
  functions and proved these classes equivalent.

• We define the notion of an effectively
  calculable function of positive integers by
  identifying it with the notion of a recursive
  function of positive integers.

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Turing

• Alan Turing, 1936, On computable numbers, with an
  application to the Entscheidungs-problem.
• Introduced the idea of a Turing machine
  computable number

• The Turing machine computable numbers include
  all numbers which could naturally be regarded as
  computable.

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Turing Machines

• A mechanism for performing calculations by
  reading and writing symbols on an unbounded
  linear tape divided into discrete squares.
• Finite set of states Q, finite alphabet A.
• Instructions of the form When in state q reading
  symbol a, change to state q, write symbol a,
  and read next symbol to right or left.
• Computation completed when the halting state is
  reached.

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Origin of the Turing Machine

• Turing arrived at his conception through an
  analysis of the essential features of computation
  as performed by humans.
• TM states correspond to states of mind
• Finite alphabet of discriminable symbols
• Linear tape is idealisation of working surface
• Displacement corresponds to shift of attention
  from one part of the surface to another

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Limitations of Turing Machines

• The Halting Problem Given a TM T and input i, to
  determine whether T, when run with i, will
  eventually halt.
• Turing showed that there is no TM which can solve
  this problem
• and that the Halting Problem is equivalent to
  the Entscheidungsproblem for first-order logic.

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Computable numbers and functions

• Turings computable numbers are real numbers
  whose expressions as a decimal are calculable by
  finite means.
• This is not a serious limitation, since it is
  almost equally easy to define and investigate
  computable functions
• Church discussed effectively calculable functions
  of the positive integers, and Turing proved these
  are equivalent to the Turing machine computable
  functions.

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A formulation of the Church-Turing thesis

• A function over the natural numbers is computable
  (i.e., effectively calculable) if and only if
  there is a Turing machine which computes it.
• The restriction to natural numbers is less
  serious than it sounds any determinate
  input/output relation defined on finite strings
  over a finite alphabet is equivalent to a
  function on the natural numbers.

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Interpreting the C-T thesis

• C-T states Computable TM-computable
• What exactly does computable mean?
• Computable by a human following a fixed
  finitely-specifiable routine?
• Computable using any physically possible
  computing device?
• Computable using any logically possible computing
  device?

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The Historical Context

• In 1936, there were no electronic computers.
• The word computer referred to a human who
  performed calculations following fixed (but
  potentially highly complex) routines.
• Robin Gandys interpretation (1988)
• Turings Theorem Any function which is
  effectively calculable by an abstract human being
  following a fixed routine is effectively
  calculable by a Turing machine and conversely.

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Infinity and Idealisation

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An abstract human being?

• C-T can only refer to humans in an idealised way
• The human computer never makes a mistake, either
  in reading or writing symbols or in following the
  prescribed instructions.
• The human computer has unlimited time and space
  available (but only ever uses a finite quantity
  of both).

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Three versions of the thesis

• The TM-computable functions are precisely
• the functions computable by an idealised human
  computer Human C-T thesis
• the functions computable using any physically
  possible means Physical C-T thesis
• the functions computable using any logically
  possible means The Logical C-T thesis

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More idealisation

• An adding machine with 10-digit registers, used
  in the normal way, cannot be used to compute
• 888888888888 999999999999.
• Thus to assert that this machine computes the
  addition function on the natural numbers involves
  a further element of idealisation
• That the machine embodies (as closely as possible
  given certain limitations of size, convenience,
  etc) an idealised abstract machine which does
  compute precisely the addition function.

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What idealisations are acceptable?

• An acceptable idealisation No limit is placed on
  how much tape is used.
• An unacceptable idealisation A Turing machine
  computation which uses the whole of an infinite
  tape (e.g., the input is required to be the
  complete decimal expansion of p).
• The difference here is between potential infinity
  and actual infinity.

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Two Kinds of Infinity

• Euclid showed that, for any positive integer n,
  there are more than n primes.

• This is often expressed as there are infinitely
  many primes.

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The Quantifier-Shift Fallacy

• The shift from For every n there exists P to
  There exists P such that for every n is an
  example of the Quantifier-shift Fallacy (Geach,
  1972). It is not logically warranted.
• From the fact that there is no limit to the
  number of primes that can potentially be
  discovered, it does not logically follow that
  there is an actual infinite totally of primes.

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A suggestion

• When considering computation and computability,
• Idealisations which involve the introduction of a
  potentially infinite (i.e., unbounded) quantity
  are generally acceptable.
• Idealisations which involve the introduction of
  an actually infinite quantity are prima facie
  unacceptable.

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The Human C-T Thesis

• The supposition that every TM-computable function
  can be computed by an idealised human computer
  appears to assume only acceptable forms of
  idealisation.
• Thus there are good reasons to suppose that the
  Human C-T thesis is true, as has been asserted by
  many commentators.

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The Physical C-T Thesis

• If the Human thesis is true, then for the
  Physical thesis to be true as well we require
  that an idealised human computer can compute any
  function computable by any physically possible
  means.
• If this is false, then there are physically
  possible computations that are not humanly
  possible this is called hypercomputation
  (Copeland and Proudfoot, 1999)

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Hypercomputation

• Questioning the Physical Church-Turing thesis
  Accelerating Turing Machines and Infinite
  Computation

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An Accelerating Turing Machine (ATM)

• This is just like an ordinary Turing machine,
  except that each computation step takes half the
  time of the preceding step.
• If the first step takes half a second, then
  infinitely many steps can be completed in one
  second.
• This would enable us to compute non-recursive
  functions (Copeland, 2002).

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Goldbachs Conjecture

• Goldbachs Conjecture (GC) states that every even
  number greater than 2 can be expressed as the sum
  of two primes.
• As of now, this has been neither proved nor
  disproved.
• It is straightforward to check whether any
  particular even number is the sum of two primes.

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A Goldbach machine G

• Start with a blank tape.
• Initially write T on square 0.
• Beginning with 4, take each even number in turn
  and check whether it can be expressed as the sum
  of two primes.
• If it is, proceed to the next even number, but if
  not, halt and replace the symbol on square 0 by
  F.

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Using G to resolve GC

• The TM G halts if and only if GC is false.
• If GC is false, then we can use G to discover
  this fact.
• If GC is true, we cannot discover this using G in
  the ordinary way.
• But if we run G as an ATM then after one second
  the symbol on square 0 will be T if GC is true,
  and F if GC is false.

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The Power of ATMs

• ATMs could be used to solve any problem of the
  form nP(n), where P is a TM-computable property
  of natural numbers.
• The Halting Problem is of this kind, since for
  any m, i, the predicate H(m,i,n), meaning TM m,
  when run with input i, has halted after n steps,
  is a TM-computable function of n.

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Are ATMs acceptable?

• Machine FAn accelerates for the first n steps but
  thereafter runs at constant speed.
• If the FAn are considered acceptable, then
• For each positive integer n, we can construct a
  TM which can perform n steps in under a second.
• This does not warrant the inference to
• We can construct a TM which, for each positive
  integer n, can perform n steps in under a second
• which is what is required for an ATM.

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Physical limits to acceleration

• In any case, there are physical obstacles to
  constructing ATMs.
• No operation can take less than 10-43 seconds
  (the Planck time), regarded as the smallest
  interval over which physical change can occur.
• But almost all of the computation steps of an ATM
  must take less time than this!

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Infinite Computations

• An infinite computation has the following
  properties
• An infinite sequence of computations is initiated
  at a certain time t
• A result which may depend on all the computations
  in this sequence is made available at some later
  time td (where d is finite)
• An ATM would be one way of performing infinite
  computations. Is it the only way?

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Relativistic Computations

• A number of authors have investigated the
  possibility of realising infinite computations in
  some form of space-time sanctioned by General
  Relativity.
• Pitowsky, 1990
• Hogarth, 1992, 1994
• Earman and Norton, 1993
• Shagrir and Pitowky, 2003

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Malament-Hogarth Spacetimes

• A Malament-Hogarth point in a relativistic
  space-time is a point p whose past light-cone
  includes the whole of some future-directed
  half-infinite timeline l.
• Such points are compatible with General
  Relativity, but they can only occur in certain
  special models of the theory called
  Malament-Hogarth Spacetimes.

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Anti-de Sitter Spacetime

• Point p is a Malament-Hogarth point since the
  whole of half-timeline l is contained in the past
  light-cone of p (the unshaded portion).

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Infinite computations in Malament-Hogarth
spacetime

• At s, we send a TM along timeline l, programmed
  to determine whether P(n) holds for n0,1,2,3,
• Meanwhile we follow timeline L to point p, which
  is in the future of the entire timeline l.
• Along l, if ever an n is found for which P(n)
  holds, the TM sends a signal to p.
• At p, if a signal is received from the TM then
  the solution to nP(n) is yes, otherwise no.

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Is Infinite Computation possible in our Universe?

• It is not known whether our universe contains
  Malament-Hogarth points.
• Or, if it does, whether we can in principle
  exploit them for the purposes of infinite
  computation.
• Or, if we can, whether this would work in
  practice.

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What the experts say

• Earman and Norton (1993)
• It is not clear that any M-H spacetime
  qualifies as physically possible and physically
  realistic.
• Hogarth (1994)
• The physically possible computing limit is
  firmly tied to some contingent and as yet unknown
  facts about the world.

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Yet more idealisation

• The M-H infinite computation involves another
  kind of idealisation that a computer can
  function correctly for an infinite period of
  time.
• This in turn depends on a more basic
  idealisation that there are infinite timelines
  in the universe.
• Neither of these idealisations is needed for the
  standard Church-Turing thesis. Nor are they
  acceptable, since they involve actual infinities.

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Could actual infinities become acceptable?

• Only if it can be demonstrated that actually
  infinite phenomena exist in our universe.
• One way in which this could happen would be to
  show that an M-H based infinite computation can
  be conducted in practice.
• But how could we verify that the phenomena
  thereby revealed really do involve actual
  infinities as claimed?

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Other routes to hypercomputation?

• A number of different computational paradigms
  have been suggested as possible sources of
  hypercomputation. I shall consider
• Quantum computation
• Neural networks
• Analogue computation

41
Quantum Computation

• Standard QC is based on qubits, systems with
  two states existing in superposition. Deutsch
  (1985) showed such QC promises efficiency gains
  but no extra functionality.
• Kieu (2001) proposed an alternative model,
  Quantum Adiabatic Computation, and presented an
  algorithm for Hilberts 10th Problem (known to to
  be equivalent to the TM Halting Problem).
• However, this requires physical implementation of
  systems with infinitely many energy levels. Is
  this another unacceptable idealisation?

42
Neural Networks

• Networks of simple processors linked by weighted
  connections. Computational powers depend on the
  architecture and the connection weights.
• Siegelmann (1995) showed that NNs with
  infinite-precision real-number weights can
  compute non-TM-computable functions.
• Infinite precision requires actual, not merely
  potential infinity, and cannot be realised
  physically. An unacceptable idealisation.

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Analogue Computation

• Whereas digital computation uses physical systems
  with discrete state-spaces, analogue computation
  can exploit the (apparent) continuity in many
  physical processes.
• Moore (1996) developed a recursion theory on real
  numbers to provide the theoretical underpinning
  for continuous-time analogue computation.
• But again, non-Turing-computation can only be
  achieved under the physically unrealistic
  assumption of infinite-precision representations
  of real numbers.

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Would hypercomputation invalidate C-T?

• The Human C-T remains valid even in the face of
  hypercomputation, unless the hypercomputation can
  be realised as an effective procedure carried out
  by humans.
• But the Physical C-T would indeed be invalidated
  but only so long as the idealisation involved
  in hypercomputation is regarded as acceptable.

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A conservative conclusion

• We should be very wary of concluding that C-T is
  invalidated on the strength of existing proposals
  for hypercomputation.
• As yet there exists no evidence whatever that any
  form of hypercomputation is feasible in practice.
• So the C-T lives to see another day!

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A few afterthoughts

• Is the human brain as a computer?
• Are there computational processes in nature?

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Mind and nature

• Does C-T imply that the human mind and other
  natural phenomena are nothing but
  Turing-equivalent computations?
• No! There is no reason to think that these are
  computations of any sort. At best, certain
  aspects of them can be simulated by means of
  computation.

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Artificial Intelligence

• C-T is not concerned with the powers of human
  beings except insofar as they are consciously
  following effective procedures to compute
  specified input-output relations.
• Normal human behaviour does not come under this
  heading, and therefore there is no reason to
  suppose that it can be described in the form of
  computation, Turing-equivalent or otherwise.

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Computation in nature?

• Computation is intentional it consists of
  processes initiated by intelligent agents with
  the purpose of deriving outputs bearing specified
  relations to some inputs.
• Intelligent agents can exploit physical processes
  in nature as means to perform computations, but
  this does not mean that nature itself ever
  performs computations.

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Biomolecular Computation?

• Biomolecular processes such as DNA replication
  can be thought of as processing information in
  the (relatively impoverished) Shannon-Weaver
  sense but not in the sense of being
  informative.
• Such processes are things that happen, but
  computations are actions.
• But we can exploit these processes as tools to
  help us perform computations. And this may lead
  to many exciting developments in the future.