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), λ 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

6. 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

7. 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

12.  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?

14.  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)

15.  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

16.  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.

17. 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

18. 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