Constraints on Hypercomputation

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# Constraints on Hypercomputation - PowerPoint PPT Presentation

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

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### Constraints on Hypercomputation

Greg Michaelson1 &

Paul Cockshott2

1HeriotWatt University,2 University of Glasgow

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
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
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
• 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
• 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
• 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 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
• 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
• 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
• 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 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
• 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
• 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
• 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
• 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
• 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
• 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