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Constraints on Hypercomputation. Greg Michaelson 1 & Paul Cockshott 2 1 HeriotWatt University, 2 University of Glasgow. Church-Turing Thesis. effective calculability

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constraints on hypercomputation

Constraints on Hypercomputation

Greg Michaelson1 &

Paul Cockshott2

1HeriotWatt University,2 University of Glasgow

church turing thesis
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), λ calculus, recursive function theory
  • 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
challenging church turing 1
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
challenging church turing 2
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
c t physical realism 1
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
c t physical realism 2
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.
c t physical realism 21
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
interaction machines 1
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.
interaction machines 2
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
interaction machines 3
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
calculus 1
 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
calculus 2
 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?
calculus 3
 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)
calculus 4
 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
calculus 5
 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.
conclusion 1
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
conclusion 2
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