Complexity

(aconstructor-theore5cview)ChiaraMarle:oWolfsonCollege

UniversityofOxford

January2017

Impossible (bythe2ndLaw)

Possible (?)

Scale-dependent

Impossible (bytheConservationofEnergy)

Scale-independent

Everylawmustbeexpressedintermsofpredictionsgiventheboundaryconditionsinspace-timeandthedynamicallaws

TheTraditionalConceptionofFundamentalPhysics

ConstructorTheory

Alllawsofphysicsareexpressedasstatementsaboutwhichtasksarepossible,

whichareimpossibleandwhy

Dynamicallawsandtheoriesoftheinitialconditionscanberetrievedasemergentconsequencesoftheprinciples

ConstructorTheory’slawsareprinciples,constrainingsubsidiarytheories(e.g.QuantumTheory,GeneralRelativity,...)

Input Attributes of Substrates ⇒ Output Attributes of Substrates

ThefundamentalobjectsofConstructorTheoryaretasks,not

constructors!

‘attribute’=setofallstatesinwhichthesubstratehasacertainproperty,accordingtothesubsidiarytheory.

Input Attributes of Substrates ⇒Constructor

Output Attributes of Substrates

Aconstructorforataskisanobjectthat,wheneverpresentedwiththesubstratesinoneoftheinputattributes,deliversthemin(oneof)thecorrespondingoutputattributes,andretainsthepropertyofdoingthisagain.

Ataskisimpossibleifthereisalawofphysicsforbiddingitsbeingperformedtoarbitrarilyhighaccuracy,possibleotherwise

Interoperabilityofinformation Anexampleofascale-independentlaw

Informationmedium:asubstratewithasetofattributesX(aninformationvariable)thatcanbepermutedinallpossiblewaysandthatcanbe‘cloned’

!!

(x,x0)→ (x,x){ }x∈X∪

⎛⎝⎜

⎞⎠⎟✓

x →Π(x)

x∈X∪

⎛⎝⎜

⎞⎠⎟✓

,∀Π

thismeansthatthetaskispossible

InteroperabilityPrincipleforInformation ThecombinationoftwoinformationmediawithinformationvariablesX1andX2isaninformationmediumwithinformationvariableX1xX2.

ProgrammableConstructor

VS1

S2

V[Pk]S2

isaconstructorforthetaskTPk.

informationmediumwithinformationvariable{P1,P2,...,Pn},withthepropertythat:

Example:CNot

Constructor

‘No-design’laws

No-designlawsaresuchthatgenericresourcesareallowedtocontainonlyapproximateconstructorsforelementarytasks.

Genericresources:substratesavailableineffectivelyunlimitednumbers.

“thetaskofconstructinganobjectXfromgenericresourcesispossible”

isfundamentallydifferentfrom

“theobjectXoccursatsometimet”

1)Goodapproximationstoconstructorsmustbeprogrammable

Thatataskispossible,underno-designlaws,impliesthat:

2)Theprogramconsistsofasequenceofelementarytasks

3)Von-Neumann’s‘Replicator-Vehiclelogic’isnecessaryforself-reproduction.

Irreversibilityinconstructortheoryisexpressedinscale-independentterms,as: “AtaskTispossibleanditstransposeis

impossible”

Thisirreversibilityiscompatiblewithtime-reversal-symmetriclaws.

Comments:

1) Noneedforprobability

2) Composition(SWAPandunittask)areassumedtobeelementary...but,arethey?

Insummary: Whatemergentlawsaredependsonthemodeofexplanation. Therecanbeexactlawsabout‘complex’systems