Constraints on Hypercomputation

1
Constraints on Hypercomputation

• Greg Michaelson1
• Paul Cockshott2
• 1HeriotWatt University,2 University of Glasgow

2
Church-Turing Thesis

• effective calculability
• A function is said to be effectively
  calculable'' if its values can be found by some
  purely mechanical process ... (Turing 1939)
• Church-Turing Thesis
• all formalisations of effective calculability are
  equivalent
• e.g. Turing Machines (TM), ? calculus, recursive
  function theory

3
Hypercomputation

• are there computations that are not effectively
  calculable?
• Wegner Eberbach (2004) assert that
• TM model is too weak to describe e.g. the
  Internet, evolution or robotics
• superTuring computations (sTC) are a superset of
  TM computations
• interaction machines, ? calculus -calculus
  capture sTC

4
Challenging Church-Turing 1

• a successful challenge to the Church-Turing
  Thesis should show that
• all terms of some C-T system can be reduced to
  terms of the new system,
• there are terms of the new system which cannot be
  reduced to terms of that C-T system

5
Challenging Church-Turing 2

• might demonstrate
• some C-T semi-decidable problem is now decidable
• some C-T undecidable problem is now
  semi-decidable
• some C-T undecidable problem is now decidable
• characterisations of classes 1-3
• canonical exemplars for classes 1-3

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C-T Physical Realism 1

• new system must encompass effective computation
• physically realisable in some concrete machine
• potentially unbounded resources not problematic
• e.g. unlimited TM tape

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C-T Physical Realism 2

• reject system if
• its material realisation conflicts with the laws
  of physics
• it requires actualised infinities as steps in the
  calculation process.

8
C-T Physical Realism 2

• infinite computation?
• accelerating TMs (Copeland 2002)
• relativistic limits to function of machine
• analogue computation over reals? (Copeland review
  1999)
• finite limits on accuracy with which a physical
  system can approximate real numbers

9
Interaction Machines 1

• Wegner Eberbach allege that
• all TM inputs must appear on the tape prior to
  the start of computation
• interaction machines (IM) perform I/O to the
  environment.
• IM canonical model is the Persistent Turing
  Machine(PTM) (Goldin 2004)
• not limited to a pre-given finite input tape
• can handle potentially infinite input streams.

10
Interaction Machines 2

• Turing conceived of TMs as interacting open
  endedly with environment
• e.g. Turing test formulation is based on
  computer explicitily with same properties as TM
  (Turing 1950)
• TM interacting with tape is equivalent to TM
  interacting with environment e.g. via teletype
• by construction see paper

11
Interaction Machines 3

• IMs, PTMs TMs are equivalent
• by construction see paper
• PTM is a classic but non-terminating TM
• PTM's, and thus Interaction Machines, are a
  sub-class of TM programs

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? Calculus 1

• ? calculus is not a model of computation in the
  same sense as the TM
• TM is a specification of a buildable material
  apparatus
• calculi are rules for the manipulation of strings
  of symbols
• rules will not do any calculations unless there
  is some material apparatus to interpret them

13
? Calculus 2

• program can apply ? calculus re-write rules of
  the to character strings for terms
• ? calculus has no more power than underlying von
  Neumann computer
• language used to describe ? calculus
• channels, processes, evolution
• implies physically separate but communicating
  entities evolving in space/time
• does the ? calculus imply a physically realisable
  distributed computing apparatus?

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? Calculus 3

• cannot build a reliable parallel/ distributed
  mechanism to implement arbitrary ? calculus
  process composition
• synchronisation implies instantaneous
  transmission of information
• i.e. faster than light communication if processes
  are physically separated
• for processors in relative motion, unambiguous
  synchronisation shared by different moving
  processes is not possible
• processors can not be physically mobile for 3 way
  synchronisation (Einstein 1920)

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? Calculus 4

• Wegner Eberbach require implied infinities of
  channels and processes
• could only be realised by an actual infinity of
  fixed link computers
• finite resource but of unspecified size like a TM
  tape
• for any actual calculation a finite resource is
  used, but the size of this is not specified in
  advance

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? Calculus 5

• Wegner Eberbach interpret as many times as is
  needed' as meaning an actual infinity of
  replication
• deduce that the calculus could implement infinite
  arrays of cellular automata (CA)
• cite Garzon (1995) to the effect that they are
  more powerful than TMs.
• CAs require a completed infinity of cells
• cannot be an effective means of computation.

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Conclusion 1

• Wegner Eberbach do not demonstrate for IM or ?
  calculus
• some C-T semi-decidable problem which is now
  decidable
• some C-T undecidable problem which is now
  semi-decidable
• some C-T undecidable problem which is now
  decidable
• characterisations of classes 1-3
• canonical exemplars for classes 1-3

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Conclusion 2

• Wegner Eberbach do not demonstrate physical
  realisability of IM or ? calculus
• longer paper submitted to Computer Journal (2005)
  includes
• fuller details of constructions
• critique of -calculus