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:1. some C-T semi-decidable problem is now

decidable

2. some C-T undecidable problem is now semi-decidable

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 1-35.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