Constraints on Hypercomputation Greg Michaelson 1 Paul Cockshott

Constraints on Hypercomputation Greg Michaelson 1 & Paul Cockshott 2 1 Heriot. Watt 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 – super. Turing computations (s. TC) are a superset of TM computations – interaction machines, calculus & $-calculus capture s. TC

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: 1. some C-T semi-decidable problem is now decidable 2. some C-T undecidable problem is now semidecidable 3. some C-T undecidable problem is now decidable 4. characterisations of classes 1 -3 5. 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: 1. some C-T semi-decidable problem which is now decidable 2. some C-T undecidable problem which is now semi-decidable 3. some C-T undecidable problem which is now decidable 4. characterisations of classes 1 -3 5. 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