Irreducibility and Irreducibility and Unpredictability in Unpredictability in NatureNature

Computer Science Department SJSUComputer Science Department SJSU

CS240CS240

Harry FuHarry Fu

OverviewOverview

IntroductionIntroduction Using CA as Model for NatureUsing CA as Model for Nature IrreducibilityIrreducibility UnpredictabilityUnpredictability Determinism and Free WillDeterminism and Free Will ConclusionsConclusions

IntroductionIntroduction

Scientists prefer using models to describe Scientists prefer using models to describe nature.nature.– Mathematic Formula Mathematic Formula – Physics LawsPhysics Laws

Some formulas and laws are: Some formulas and laws are: – Complex Complex – IncomprehensibleIncomprehensible– Only describe some phenomena in natureOnly describe some phenomena in nature

Looking for better model to describe Looking for better model to describe nature nature – Use illustrative pattern to describe behavior of Use illustrative pattern to describe behavior of

nature will fit human perception and analysisnature will fit human perception and analysis

Apply Possible Models Apply Possible Models

Use Mathematic formulas or Use Mathematic formulas or Physics Laws.Physics Laws.

Use Philosophy, or human Use Philosophy, or human intuition.intuition.

Or use a simple yet meaningful Or use a simple yet meaningful representation. representation.

Use Cellular Automata to start our Use Cellular Automata to start our science exploration.science exploration.

Cellular Automata as Cellular Automata as ModelModel Large array of cells with synchronous update Large array of cells with synchronous update

in parallel. in parallel. The parallelism resembled some features in The parallelism resembled some features in

physical world, such as space and time physical world, such as space and time relation.relation.

The evolutionary progression behavior in CA The evolutionary progression behavior in CA also move through space and time.also move through space and time.

CA can be setup in 1D, 2D, or 3D that CA can be setup in 1D, 2D, or 3D that emulates fundamental features of our emulates fundamental features of our universe.universe.

Give an initial condition with simple rules, the Give an initial condition with simple rules, the universe start evolving. universe start evolving.

Emergence in NatureEmergence in Nature

Emergence emerge in nature.Emergence emerge in nature. Property present in some behaviors we Property present in some behaviors we

have seen in nature.have seen in nature.– Irreducibility Irreducibility – Unpredictability Unpredictability

In Cellular Automata, these properties In Cellular Automata, these properties are inevitably exist.are inevitably exist.

Some characteristics can be seen in Some characteristics can be seen in these properties. these properties.

IrreducibilityIrreducibility

Cohesive Relation.Cohesive Relation. Computational Irreducibility.Computational Irreducibility. Entropy Increase.Entropy Increase.

– In Second Law of Thermodynamics.In Second Law of Thermodynamics. Irreducibility led to notion of Irreducibility led to notion of

unpredictability.unpredictability.

Cohesive RelationCohesive Relation

Cohesion Cohesion – Concept for irreducibilityConcept for irreducibility– Creates stability to prevent object from fluctuate or Creates stability to prevent object from fluctuate or

change in its formchange in its form Cohesive objects are moving throughCohesive objects are moving through

– Space and TimeSpace and Time CA also exhibit this cohesive relationCA also exhibit this cohesive relation

– Cells are progressively updated through space and timeCells are progressively updated through space and time– In Class IV CA, information are communicated over long In Class IV CA, information are communicated over long

rangerange Organisms are cohesive. Organisms are cohesive. (e.g. Birds with Broken Wing)(e.g. Birds with Broken Wing)

– structural connectionstructural connection– functional connectionfunctional connection

Cohesive Broken Wing Cohesive Broken Wing BehaviorBehavior

Killdeer Thick Knee

Computational Computational IrreducibilityIrreducibility Some mathematics are merely Some mathematics are merely

symbolic system that represent symbolic system that represent solution that we really want to find. solution that we really want to find.

Fundamental Mathematics:Fundamental Mathematics:– 1/131/13 = 0. = 0.076923076923076923076923076923076923… …

(Geometric Series)(Geometric Series)– √√33 = 1.7320508075688772935… = 1.7320508075688772935…

(Random sequence of decimal digits)(Random sequence of decimal digits)– ππ = 3.14159… (The exact value remains = 3.14159… (The exact value remains

mystery)mystery)

Computational Computational Irreducibility Irreducibility – Random – Random Sequence of Sequence of ππ 3.1415926535897932384626433833.141592653589793238462643383

2795028841971693993751058209727950288419716939937510582097494459230781640628620899862804944592307816406286208998628034825342117067982148086513282348253421170679821480865132823066470938446095505822317253530664709384460955058223172535940812848111745028410270193859408128481117450284102701938521105559644622948954930381962110555964462294895493038196……

[Wolfram, p 137][Wolfram, p 137]

Computational Computational Irreducibility Irreducibility – In CA– In CA

Code 98111117

Class I

Code 91111177Class II

Computational Computational IrreducibilityIrreducibility – In CA – In CA

Code 93111117Class III

Code 91151117Class IV

Butterfly Growth Cycle Butterfly Growth Cycle in Biological Processin Biological Process

Eggs Larva orCaterpillar

Butterfly Growth Cycle Butterfly Growth Cycle in Biological Processin Biological Process cont.cont.

Pupa or Cocoon

Adult Monarch Butterfly Emerged

Entropy IncreaseEntropy Increase

Entropy Entropy – led to notion of irreducibility led to notion of irreducibility

Information tend to increase in a system Information tend to increase in a system – – Inferred by Second Law of ThermodynamicsInferred by Second Law of Thermodynamics

More information More information – Creates disorderCreates disorder– Randomness seem to emergeRandomness seem to emerge

The patterns generated in CA The patterns generated in CA – Neither die out Neither die out – Nor conform to any regularityNor conform to any regularity

Entropy IncreaseEntropy Increase cont.cont.

When entropy is exhibited in a system, When entropy is exhibited in a system, reducing its computation is nearly reducing its computation is nearly impossible. impossible.

Second Law of Thermodynamics Second Law of Thermodynamics – If one repeats the same measurements at If one repeats the same measurements at

different times, then the entropy deduced from different times, then the entropy deduced from the system would tend to increase over time. the system would tend to increase over time.

Amount of irreducible information increaseAmount of irreducible information increase– It becomes computationally irreducibleIt becomes computationally irreducible– Probability of accurate prediction diminishesProbability of accurate prediction diminishes

Entropy IncreaseEntropy Increase – 3 – 3 State CA ExampleState CA Example

Code 93111117Simple Nesting Pattern

Code 93111117Random Pattern

UnpredictabilityUnpredictability

Defining Randomness.Defining Randomness. Perception of Complexity.Perception of Complexity. A notion of Uncertainty. A notion of Uncertainty. Can multiple histories exist in our Can multiple histories exist in our

universe?universe?

UnpredictabilityUnpredictability – – Defining RandomnessDefining Randomness Defining a true randomnessDefining a true randomness

– Human perception Human perception – AnalysisAnalysis

Most intuitively, randomness is described Most intuitively, randomness is described with behavior from a system without with behavior from a system without apparent regularity.apparent regularity.

Reversible CA Rule 37R exhibits behaviorReversible CA Rule 37R exhibits behavior– Order Order – DisorderDisorder– It does not obey Second Law of ThermodynamicsIt does not obey Second Law of Thermodynamics

The behavior of 37R seems unpredictable. The behavior of 37R seems unpredictable. – This kind of behavior can also be seen natureThis kind of behavior can also be seen nature

UnpredictabilityUnpredictability – – Defining RandomnessDefining Randomness

Rule 37RUnpredictable Behavior

[Wolfram, p 440]

UnpredictabilityUnpredictability – – Defining RandomnessDefining Randomness

Rule 30Unpredictable Behavior and Pattern

[Wolfram]

Edge PatternEdge Pattern

- Maximum Slope

- Absolute Upper Limit

UnpredictabilityUnpredictability – – Perception of ComplexityPerception of Complexity

Code 9111177Complex Pattern

Code 9111177One Cell Initial

Condition

UnpredictabilityUnpredictability – A – A notion of Uncertaintynotion of Uncertainty At micro levelAt micro level

– Uncertainty Principle states Uncertainty Principle states uncertainty uncertainty relation between the position and the relation between the position and the momentum (mass times velocity) of a momentum (mass times velocity) of a subatomic particle, such as an electron. subatomic particle, such as an electron. [Heisenberg][Heisenberg]

This relation has reflective implications This relation has reflective implications for such fundamental notions as for such fundamental notions as causality and the determination of the causality and the determination of the future behavior of an atomic particle. future behavior of an atomic particle.

UnpredictabilityUnpredictability – A – A notion of Uncertainty notion of Uncertainty cont.cont. The more precisely the position of The more precisely the position of

an object is determined, the less an object is determined, the less precisely the momentum is known precisely the momentum is known in this instant, and vice versa. in this instant, and vice versa.

In another word, if we try to In another word, if we try to measure some moving object in measure some moving object in the universe, we cannot both the universe, we cannot both decide precisely what speed it is decide precisely what speed it is moving and what position it moving and what position it locates.locates.

UnpredictabilityUnpredictability – A – A notion of Uncertainty notion of Uncertainty cont.cont. If precise measurement of either If precise measurement of either

speed or position of any matters in speed or position of any matters in the universe is not possible, then the universe is not possible, then the entire emergence we see in the entire emergence we see in nature inevitably exist without our nature inevitably exist without our knowledge. This led to the notion of knowledge. This led to the notion of free will when any process and free will when any process and behavior emerge in nature.behavior emerge in nature.

History created or History created or already existedalready existed?? Are we merely exploring part of the Are we merely exploring part of the

system at certain time and position system at certain time and position of universe where the complete of universe where the complete space-time history is always exist? space-time history is always exist?

Or are we more like Cellular Or are we more like Cellular Automata where the new states of Automata where the new states of universe get updated and created universe get updated and created and old one is lost?and old one is lost?

Multiway SystemMultiway System

Multiway SystemMultiway System– One kind of substitution systemOne kind of substitution system– Use to explain the history evolution created in Use to explain the history evolution created in

a systema system Using Multiway System as example, we Using Multiway System as example, we

can apply simple rule that generate can apply simple rule that generate multiple histories with different choice of multiple histories with different choice of evolution. evolution.

As an observer, we are able to visualize As an observer, we are able to visualize possibility of multiple histories instead of possibility of multiple histories instead of unique history.unique history.

Perspective of an Perspective of an ObserverObserver We are part ofWe are part of

– Universe Universe – Complexity Complexity – RandomnessRandomness

Our intuition told us there is one Our intuition told us there is one unique history.unique history.

If we can act as an observer of the If we can act as an observer of the system, would that make any system, would that make any difference?difference?

Imagine for a moment, we are the Imagine for a moment, we are the observer of the system. observer of the system.

Model our Universe as Model our Universe as Multiway SystemMultiway System

1 2Our Universe

Another History

Determinism and Free Determinism and Free WillWill Free will means Free will means

– There must be at least two or more possibilities There must be at least two or more possibilities when facing a given choice when facing a given choice

– No coercion and choice is not forced. No coercion and choice is not forced. Many systems in our nature seem to Many systems in our nature seem to

generate behaviors that are random and generate behaviors that are random and complex.complex.

Computational irreducibility is the origin of Computational irreducibility is the origin of the apparent freedom of human. the apparent freedom of human. [Wolfram][Wolfram]

It is also our perception that dictates It is also our perception that dictates complex system that lead to computationally complex system that lead to computationally irreducible hence unpredictable. irreducible hence unpredictable.

CA rule that doesn’t CA rule that doesn’t obey definite lawsobey definite laws

Code 15993 State Totalistic

ConclusionsConclusions

The concept of emergence consist irreducibility The concept of emergence consist irreducibility and unpredictability that prevents scientist from and unpredictability that prevents scientist from concluding certain finding.concluding certain finding.

These properties appeared in many disciplines These properties appeared in many disciplines of our science, for example mathematics, of our science, for example mathematics, physics, or even biological process. physics, or even biological process.

Cellular automaton is one of emergent Cellular automaton is one of emergent computing model that help us to explore computing model that help us to explore phenomena in science.phenomena in science.

We can analyze this emergence where We can analyze this emergence where irreducibility and unpredictability exit. irreducibility and unpredictability exit.

ReferencesReferences

1. Klaus A. Brunner, 1. Klaus A. Brunner, What's Emergent in Emergent Computing?What's Emergent in Emergent Computing? 2002 2002 http://winf.at/~klaus/emcsr2002.pdfhttp://winf.at/~klaus/emcsr2002.pdf 2. John D. Collier and Scott J. Muller2. John D. Collier and Scott J. Muller The Dynamical Basis of The Dynamical Basis of

Emergence in Natural HierarchiesEmergence in Natural Hierarchies, George Farre and Tarko Oksala , George Farre and Tarko Oksala (eds) Emergence, Complexity, Hierarchy and Organization, and (eds) Emergence, Complexity, Hierarchy and Organization, and Selected and Edited Papers from ECHOS III Conference, 1998.Selected and Edited Papers from ECHOS III Conference, 1998.

3. John D. Collier, 3. John D. Collier, Causation is the transfer of informationCausation is the transfer of information; Causation ; Causation of Law and Nature, (ed, Howard Sanky) Kluwer, 1998.of Law and Nature, (ed, Howard Sanky) Kluwer, 1998.

4. Werner, Heisenberg History Museum, 1976 4. Werner, Heisenberg History Museum, 1976 – http://www.aip.org/history/heisenberg/p08a.htmhttp://www.aip.org/history/heisenberg/p08a.htm

5. Stephen Wolfram, 5. Stephen Wolfram, A New Kind of ScienceA New Kind of Science, Wolfram Media, , Wolfram Media, Champaign, IL 2002, p 138, 140, p 518, p 301, p 737-750, p750, Champaign, IL 2002, p 138, 140, p 518, p 301, p 737-750, p750, 752, 967, 1132, 1135752, 967, 1132, 1135

6. Outdoor Photographing. Killdeer Photo Source:6. Outdoor Photographing. Killdeer Photo Source:– http://www.outdoorphoto.com/birdtips.htmhttp://www.outdoorphoto.com/birdtips.htm– http://home.eol.ca/~birder/plovers/kl.htmlhttp://home.eol.ca/~birder/plovers/kl.html

7. TrekEarth. Thick Knee Photo Source:7. TrekEarth. Thick Knee Photo Source:– http://www.trekearth.com/gallery/South_America/photo1197.htmhttp://www.trekearth.com/gallery/South_America/photo1197.htm