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 Irreducibility and Unpredictability in Nature

By Harry Fu

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Research Paper


1  Introduction

Modern day science is unable to describe an entire universe that scientists might want to study.  While the most sophisticated mathematic formula and physics law only explain some of the concepts that models part of this ever expanding universe, scientist are constantly looking for  simple, yet comprehensive model to describe what human perceive. Finding a universal model that explains the entire universe can be challenging. The concept of emergence, for instance, consists of irreducibility and unpredictability properties that prevent scientists from concluding this finding. In A New Kind of Science, Wolfram stated in Principle of Computational Equivalence that asserts essentially, any process that is not obviously simple would equivalent in its computational sophistication [7]. It implies complicated processes that are not obviously understandable will be computationally irreducible. In spite of many processes in this universe with behavior that can be described with mathematic formulas or physics laws, there are unpredictable behaviors emerging, as one perceive in everyday experience. How to predict behavior using concise mathematic formulas or physics laws for a system becomes a consensus for science arenas. Nevertheless, how to explain a complex system behavior with a simple model becomes wonder for scientists. A cellular automaton model has been used to simulate life and artificial life. Some behaviors that emerged in cellular automata systems seemed random and highly unpredictable. For example, how do we determine if one cellular automaton rule indeed belongs to a certain class? Perhaps there is not sufficient time lapse for such an observation. One might argue that if we keep on observing every step of evolution for a particular cellular automaton experiment, ultimately, it must show a result. In fact, watching every single step of evolution and hoping to obtain an outcome turns out to be impractical.  There have been numerous mathematic formulas and physics laws modeled as a foundation to explain matters we experience and perceive in this universe. Many of these formulas or laws appear complicated, and others may seem incomprehensible. This led to the notion that no such simple model can ever describe everything in the universe. Through using cellular automata as a model for science exploration, we can analyze how these concepts emerge in emergent computing where property such as irreducibility and unpredictability exist. Consequently, these properties prevented us from perceiving and analyzing models that describe nature.

2  Using CA as Model for Nature

Why is Cellular Automata used as a model to elaborate these emergence properties in nature? Cellular automaton has been studied and modeled as one of the prime experiments for emergent computing [1]. It involved using a large number of cells that interconnected to form arrays. These arrays of cells update synchronously in parallel that resembles some properties of the physical world. Knowing its parallelism and evolutionary progression, cellular automaton can be viewed as a progress of events moving through space and time. Since cellular automata rules are given as initial condition for further evolution, it leads to notion that irreducibility and unpredictability are produced intrinsically. Yet, these properties still exist in nature even if the universe is likely to be running with simple rules.

According to Wolfram, the universe is likely to be running with certain simple rules. These simple rules may in fact produce some behaviors that are computationally irreducible, which lead to unpredictability. This phenomenon emerged in many parts of nature. Cellular automata model presents some of these phenomena in respect. If we model the universe with a computation of running a simple rule, will the simple rule illustrate every aspect of our nature? Alternatively, nature is operating at the level of phenomena that cannot be described by human perception. Despite these unknown controversies, irreducibility and unpredictability inevitably emerge in every scale of our nature.

3  Irreducibility

3.1  Cohesive Relation

Collier uses concept of cohesion to explain why there is a property of irreducibility in nature. According to Collier, cohesion can account for irreducibility. The basic criterion for natural physical objects depended on continuity of space and time. This also resembles the intuition that natural physical objects move together through space and time. Cohesion represents those factors that causally connect the components of its matters through space and time [2]. It manifests this stability for objects in its underlying property. It also makes the object resistant to change or fluctuate on its own [2].

Similar to cells in cellular automaton, the patterns and behaviors are created through each step of evolution through space and time. These cells are updated based on the state of neighboring cells  that progressively update in space and time. In class IV of cellular automata, information for change of behavior is carried over a long range [7]. Any change with its behavior is emerged without any way to reduce this information. Because of this long range of information communicated between structures or patterns being produced, cohesion has established a unity property for overall structure that inevitability makes the information irreducible.

The irreducibility of cohesive properties has implication for other areas of science such as biology. In biology, the organisms are cohesive with their structural and functional connections that make them unaffected to the continuing change in their composition and even their form [2]. For example, the “broken wing” behaviors of some birds in the presence of predators might seem to be a coherent deception to protect offspring [2]. Then after the distractions succeed, the bird will take an alternative route back to its baby birds.

a) Killdeer                                                          b) Thick Knee

Figure 1.  Killdeer and Thick Knee Distracting Predator

a) Killdeer and b) Thick Knee are displaying broken-wing behavior when they sense a predator to protect their offspring. They act pitifully to distract the predator when it approaches baby birds. The broken-wing makes the predator follow them while keeping themselves one-step away from predator. Once when the predator is off range from baby birds, the broken-wing behavior suddenly heals and they fly away. [8] [9]

This cohesive behavior appeared in many systems in our universe whether it is a biological process, mathematics, physics or even a general matter in our every day life. In many respect, scientists favor mathematics formulas or physics laws to explain some of these phenomena. Conceivably, irreducibility can also exist in basic mathematics or physics.

3.2  Computational Irreducibility

In retrospect, many scientific experiments are practiced to simulate promising model or framework for solutions that will solve other similar problems. However, these models usually solve for one particular problem but not on other similar problems. There are numbers of fundamental mathematics where irreducibility emerged in number theories. Scientists use formulas to explain complicated solutions in a simpler term, though some of these terms are merely a symbol to another extrapolation for what really ought to be known. For instance, comparing rational and irrational numbers such as a base 10 rational number of the form p/q always reveals a sequence of repetition in the decimals place. On the contrary, irrational numbers such as √3 generate a seemingly random sequence [7]. Consider the following:

1/13 = 0.076923076923076923… (Geometric Series)

√3 = 1.7320508075688772935… (Random Sequence of Decimal Digits)

π = 3.141592653589793238462643383279502884197169399375105820974944592



(The exact value remains mystery)

When dividing 1 by 13, it generates a sequence of ‘076923’ as the repetition for decimal portion known as geometric series. Since 13 is a divisor of 999999 with a quotient of 76923, it is safe to predict its repeating region without using a high precision calculator. For irrational numbers, √3 generates a seemingly random sequence of ‘7320508075688772935…’ with continually growing digits. This sequence is computationally irreducible with any form of mathematical shortcut. It will take endless pages to write down digits to represent the actual sequence of √3.  As a result, mathematicians prefer to present answer in the form of ‘√3.’ Similar to the representation of π first discovered back in 1600s, a complete value of π remains mystery. Despite the simple definition of π as the ratio of the circumference to the diameter of a circle, its sequence is considered sufficiently random.

Can a similar phenomena exhibit in cellular automata models? When experimenting with 1D cellular automaton rules 98111117, 91111177, 93111117, 91151117, different kinds of behavior in constant, repetition, seemingly random, and localized structures are produced respectively.


a) Code 98111117 (Class I)                                            b) Code 91111177 (Class II)


c) Code 93111117 (Class III)                                                      d) Code 91151117 (Class IV)

Figure 2.  3-State 1D CA with 4 Classes

CA examples that showed in a) constant pattern b) repetitive pattern c) seemingly random looking pattern and d) seemingly random yet with some localized structures patterns.

While the first two systems on the top seemed to be computationally reducible, the behavior of the third and fourth systems appeared computationally irreducible. Indeed, whenever there is computational irreducibility existing in a given system, there is no way to predict the overall system behavior without going through every step of computation for the system itself [7]. Consequently, to reduce the amount of computations will require building another system with an equally difficult computational process for such special rules to exist. After all, the system itself needs to track all possible variables emerged. In the end, it does not help to reduce much of the computation but instead going through every single step of evolution. In many biological processes seen in nature, the growth development of a butterfly starts from eggs, caterpillar, cocoon, and adulthood. It will never skip intermediate development process. It goes through various stages of evolution before a newly emerged adult butterfly. Is it possible to make a reduction of overall growth cycle from caterpillar to butterfly directly? This coherent biological process does not allow such reduction to take place in nature. This led to the notion that information or process for many systems in nature is always increasing and highly dependent onto their causal relationship.

3.3  Entropy Increase

Entropy has led to the notion of irreducibility to some systems in nature. Entropy seems to increase over time as more information tends to be generated in any given system. This concept is known as Second of Law of Thermodynamic. A specific measurement for any entropy will depend on the system itself and future processes it generates [7].  For example, a cellular automaton that is setup with simple rules might generate behaviors of plain pattern initially; as evolutions progresses, a more chaotic behavior seemed to emerge. The behavior of seemingly random patterns has carried information that communicates with future evolution over long range. The patterns do not seem to die out nor conform to any regularity. Instead, the behavior of such patterns emerges as evolutions continue. To reduce the amount of computations and discover the final behavior is implausible.


a) Simple Repetitive Pattern                                            b) Random Looking Pattern

Figure 3.  3-State 1D CA Random Pattern

CA example using Code 39833579 showed a) simple repetitive pattern in the first hundred of evolution, b) after about 200 steps when patterns are overlapping each other, a seemingly random pattern emerged.

When entropy is exhibited in a system, reducing its computation is nearly impossible. The Second Law of thermodynamics inferred if one repeats the same measurements at different times, then the entropy deduced from the system would tend to increase over time [7]. Just as how it appears in nature, behaviors emerged can be highly complex without a systematic way to reduce its overall pattern of behavior. Do patterns of these CA examples truly show complexity that cannot be perceived by human intuition, or is it merely a perception that cannot be accepted by human intellectual capacity. For the most part, patterns generated with this level of complexity undoubtedly cannot be captured with normal human perception. Even with a high-speed super computing device, it computes this information exhaustively without knowing what patterns or behaviors come next. With the amount of irreducible information increasing over time, the probability of getting accurate prediction over future behavior is diminishing.

4  Unpredictability

4.1  Defining Randomness 

Contradicting to what many believed that Second Law of Thermodynamics explains universality in the physical world, Wolfram has used the example of reversible cellular automaton with rule 37R to demonstrate a seemingly random behavior that cannot be predicted with this law. The Second Law of Thermodynamics dictates randomness and entropy will always increase after its initial condition. Conversely, patterns generated by rule 37R appear to fluctuate between order and disorder states.  As in consequence, rule 37R does not follow prediction of Second Law of Thermodynamics. Moreover, an unpredictable behavior such as rule 37R indeed cannot be predicted with any form of mathematic formula or physics laws. To explain universality, Wolfram argued that Second Law of Thermodynamics is not universally valid even if it is an important and quite general principle.

Figure 4.  2-State 1D CA rule 37R 

An example of CA that does not follow Second Law of Thermodynamics. [5]

Defining what a true randomness is could vary depending on how humans perceive the system. Most intuitively, randomness is described with system of behavior without apparent regularity. According to Wolfram, rule 30 in elementary 1D cellular automaton shows a seemingly intrinsic generation of randomness. Such randomness is believed to be generated without any form of insertion or external environmental effects. The intrinsic randomness is essentially what makes Wolfram believed that a simple underlying rule could still generate a behavior of great complexity. In essence, the steps of evolutions are determined, new states are created and updated through space and time. Assuming our universe behaves as cellular automata similar to rule 30, then each step of history event is created intrinsically without anyway to predict what future events will emerge. This history of events has exhibited causal relationship one above and one below its underlying system. However to skip interconnected neighboring relationships for the cells and leap to the very last stage of evolution is not possible. If history of events are in fact created and updated at each step through space and time, a notion of unpredictability will emerge in nature. Viewing this intricacy of overall history of causal connectivity exemplify the perception of complexity.

4.2  Perception of Complexity

Complexity and randomness are used interchangeably in the context of our everyday experience. Complexity can also be viewed as the level of perception from human intuition. For example, class III and IV of cellular automata can be perceived random, thus complex. However, it may well be human perception that does not accept its appearance of behavior at its base level. Given a cellular automaton behavior with pattern analogous to class III in picture a), it was merely a version of class II cellular automaton with its initial condition randomized.


a) Random Initial Condition                                            b) One Cell as Initial Condition

Figure 5.  3-State 1D CA Complex Pattern

Rule Code 91111177 showed a) seemingly complex and random behavior with randomized initial condition. b) Using the same rule with its initial condition set to one cell.

A notion of unpredictability can even be seen when describing the pattern from Code 9111177. In everyday experience, human tend to interpret things with complex behavior that must somewhat be created with rules of great complexity. Moreover, our intuition also suggests that if a pattern has no apparent regularity, it cannot possibly be generated from a pattern that looks so regular and repetitive. Likewise, in nature many behaviors and processes are quite unpredictable at its superficial level. Some behaviors may seem as complex as in picture a) , but without knowing its simpler case in picture b), human might be leading into different intuition of interpreting aspect of behavior in our nature. This intuition can be either developed or innate from human response; nevertheless, properties of nature reflect uncertainty that cannot be easily understood with human conception.

4.3  A Notion of Uncertainty

In the physical world, the most evocative model can be best described with laws of physics. These laws elaborate nature with use of examples from particle physics, theory of relativity and so on around our universe. One of the profound findings in the history of physics was Uncertainty Principle by Heisenberg. Heisenberg stated “uncertainty relation between the position and the momentum (mass times velocity) of a subatomic particle, such as an electron. This relation has reflective implications for such fundamental notions as causality and the determination of the future behavior of an atomic particle.” Why use particles to describe behavior of the universe? Physicists believed using the smallest defined and perceived unit could best mimic what is happening in nature. To sum up Heisenberg’s uncertainty principle, the more precisely the position of an object is determined, the less precisely the momentum is known in this instant, and vice versa. In another word, if we try to measure some moving object in the universe, we cannot both decide precisely what speed it is moving and what position it locates. If we try to measure one of them, we cannot measure the other [5]. 

By means of 1D cellular automaton according to Wolfram, the edge of the pattern produced by cellular automata rule has a maximum slope equal to one cell per step. It is also considered the absolute upper limit on the rate of information transfer, similar to the speed of light in physics [7]. Imagine again that cellular automaton represents universe with its history created through space and time at each step of evolution without wrapping. Theoretically, there should not be a precise way to predict the next evolution if only one measurement of speed or position can be captured but not both! With this assumption, recall rule 30 in elementary 1D cellular automaton. It tends to increase its randomness and complexity if it were running at an edgeless space in the universe. To a certain extent, it is incomprehensible even with human perception and analysis. Moreover, if precise measurement of either speed or position of any matters in the universe is not possible, the entire emergences we see in nature inevitably exist without our knowledge. This led to the notion of free will when any process and behavior emerge in nature.

5  Determinism and Free Will

What defines free will in cellular automata? Free will means there must be at least two or more possibilities when facing a given choice. Free will also means no coercion and choice is not forced. Many systems in our nature generate behaviors that are random and complex. The entropy and information tend to increase as time elapsed with the amount of information that is disordered.  Computational irreducibility is the origin of the apparent freedom of human [7]. If the evolution of a system corresponds to an irreducible computation then this means that the only way to work out how a system will behave is essentially to perform this computation. It is our perception that dictates how a complex system eventually will lead to a computation that seemed irreducible. It also suggests there can fundamentally be no laws that allow one to work out the behavior more directly [7]. A cellular automaton whose behavior displays characteristic of computational irreducibility could show an analog of free will. Even though its underlying laws are definite and simple, the behavior is complicated enough that in many respects follow no definite laws [7]. This phenomenon appeared in many disciplines of our universe. For instance, behaviors and processes are evolving and changing through space and time. Even so, human are evolving and changing with their intellectual capacity over time. When choices are present, evolutionary paths evolve forever.

Figure 6.  2-State 1D CA

Rule Code 1599 showed the patterns follow no definite laws. Most patterns and structures seem emerged without any way to predict. The only possible way to analyze this rule is to run through the whole computation.

Insofar, many of these existing behaviors were based on human assumption, perception, and analysis as an insider point of view since we are ourselves part of this universe. Instead of running our universe as what has been described as cellular automata, we are merely exploring a part of this universe where the complete history exists. What would it be like? From an observer’s perspective, if the universe is modeled as a Multiway System given a set of simple rules, multiple histories at any given step of evolution seem to emerge [7]. Each history created could escort in different paths from each other. Taking one slice of the Multiway System, another history perspective is being perceived by an observer. What happened if our universe is operating in the way where multiple histories are created and updated synchronously or even asynchronously? The degree of this indeterminism might trigger why these properties such as irreducibility and unpredictability exist in nature.

a) Universe with unique history                     b) Universe with multiple histories

 Figure 7.  Universe illustrated with Multiway System

As an observer perspective, there might be multiple histories created in our universe. Using a set of simple rules, a) a universe with more than one choice to update at each evolution. b) If slicing what has been created in the universe, multiple histories seem to emerge.


6  Conclusions

The concept of emergence consists of irreducibility and unpredictability properties that prevent scientists from concluding certain findings. These properties appeared in many areas of our everyday experience. For instance, in mathematics, physics, or even in biological behaviors. Scientists in various fields are constantly looking for better models to describe this ever-changing universe. During this endless research, new methods or findings were discovered, in many instances with the notion of the emergence emerged in nature. If scientists do not perceive this emergence existed in systems that create randomness and complexity, subsequently it does not raise much of an attention or interest to their discovery. Correspondingly, Heisenberg stated, “we only observe what we can observe, if anything that we cannot be observer, it is equally not observable.” Cellular automaton is certainly one of the most descriptive emergent computing models that help us to begin our science exploration. We can analyze some of these concepts emerged in emergent computing where irreducibility and unpredictability exist. Consequently, irreducibility and unpredictability prevented us from perceiving and analyzing models that conclude our nature.


[1] Klaus A. Brunner, What's Emergent in Emergent Computing? 2002.


[2] John D. Collier and Scott J. Muller The Dynamical Basis of Emergence in Natural Hierarchies, George Farre and Tarko Oksala (eds) Emergence, Complexity, Hierarchy and Organization, and Selected and Edited Papers from ECHOS III Conference, 1998.

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

[4] Claus Emmeche et al. Levels, Emergence, and Three Versions of Downward Causation. In: Peter Bogh Andersen et al.(eds.), Downward Causation: Minds, Bodies and Matter. Aarhus University Press, 2000.

[5] Werner, Heisenberg History Museum, 1976


[6] Timothy O’Conner, Emergent Property. 2002


[7] Stephen Wolfram, A New Kind of Science, Wolfram Media, Champaign, IL 2002, p 138, 140,   p 518, p 301, p 737-750, p750, 752, 967, 1132, 1135

[8] Outdoor Photographing. Killdeer Photo Source:




[9] TrekEarth. Thick Knee Photo Source:



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