When Information Theory deals with the complexity and randomness of objects, usually data in computer science, it employs the concept of algorithms. A particular branch has evolved under the name Algorithmic Information Theory, and the basis for it is the Kolmogorov Complexity. With its interpretations through the principles of complexity, randomness, and Turing machine, this theory is not just an interesting mathematical concept but a significant tool for understanding the intricacies of God’s creation and the nature of our universe.
The computational framework of Kolmogorov Complexity, which measures the length of the shortest program that can describe an object or data set, is crucial for understanding simulated environments, as proposed in the theory of Simulation Creationism created by Nir Ziso, founder of The Global Architect Institute. This groundbreaking concept opens up a world of intriguing implications.
An Introduction to the Kolmogorov Complexity
The discovery of the Kolmogorov Complexity, a scientific concept of immense significance, was a testament to the intellectual prowess of three remarkable scientists in the mid-1960s. Ray Solomonoff, Andrei Kolmogorov, and Gregory Chaitin independently defined the same notion, now known as Kolmogorov Complexity. This simultaneous discovery, set against the backdrop of a rapidly advancing scientific community, not only underscores the importance of the concept but also highlights the scientific community’s readiness to embrace it, leading to significant advancements in Algorithmic Information Theory.
Each of the three scientists made a profound and unique contribution to the development of Algorithmic Information Theory. In 1964, Solomonoff, a machine learning expert, grappled with the induction problem. His groundbreaking proposal—that among the many hypotheses fitting the observed data, the simplest should be preferred—laid the foundation for Kolmogorov Complexity, highlighting his innovative thinking.
In 1965, Kolmogorov, a renowned mathematician and theoretical physicist, was sixty-nine years old. Thirty-two years prior, he laid down fundamental axioms to define probability theory, a significant achievement in his career. Continuing his work on the mathematical definition of the quantity of information, he searched for a correct definition of randomness in 1965. This pursuit, built on his extensive experience and previous contributions to the field, would eventually lead to the concept of Kolmogorov Complexity.
Chaitin, a mere nineteen years old in 1966, had a unique and forward-thinking perspective on information. He believed that it should be grounded in computation rather than probability. He aimed to demonstrate that the informational content of mathematical theories has inherent limits. This innovative perspective would significantly influence the development of Algorithmic Information Theory, shaping its future direction and applications. Although the concept was named after Kolmogorov, Chaitin is more interesting for Simulation Creationism since he developed an understanding of God as “a programmer.”
Despite their different paths, these three scientists converged on the same concept, named after the most renowned among them. The notion of Kolmogorov Complexity, a formula that could be involved in many aspects of God’s Simulation, unifies and connects seemingly disparate areas of study.
The concept can be formalized by the following equation:
In this equation:
- C(x) represents the Kolmogorov complexity of x, which is defined as the length of the shortest program p that can generate x when executed on a universal Turing machine U.
- minₚ indicates that we are searching for the minimum length across all possible programs p.
- length(p) refers to the size of the program p in bits.
- U(p) = x means that when program p is run on the universal Turing machine U, it outputs x.
- U is the universal Turing machine, an idealized computational model with unlimited memory and the ability to simulate any algorithmic process.
The Kolmogorov Complexity measures the informational content of x by finding the shortest possible program that can describe it. Since p is the minimal set of instructions necessary to produce x, it effectively captures all the information contained in x. Additionally, there is a related concept known as prefix-free complexity, denoted by K(x). This represents the minimal description length of any program encoded in a prefix-free code, where no code is a prefix of any other. The benefit of a prefix-free code is that it allows a machine to read words from the code in one direction. As soon as the last symbol of a word is read, the machine knows that the word has ended and does not need to backtrack, similar to how a Turing machine processes information.
Per its definition, the Kolmogorov Complexity of an object, such as a piece of text, is the length of the shortest computer program (in a predetermined programming language) that produces the object as output. It is the functional encoding of the message of a string or an object.
Consider these two strings:
- abababababababababababababababab
- 4c1j5b2p0cv4w1x8rx2y39umgw5q85s7
The first string is made of a repeating “ab” pattern repeated 16 times, and the second string is purely random. The first string has a lower Kolmogorov Complexity because it can be compressed easily (e.g., write ab 16 times), while the second one has a higher Kolmogorov Complexity because it appears random and cannot be compressed efficiently.
This principle has become the basis for data compression algorithms used in everyday technology like zip files. If we use the Python computer script, we can easily say, “write ab 16 times,” but the second string would necessarily be written as it is because it cannot be shortened or generated consistently. Python was the preferred language in the initial research stage, while other languages are now used as well. Still, the effect of changing languages is bounded (a result called the invariance theorem).
The issue of programming language choice is quite significant when discussing the Kolmogorov Complexity. The invariance theorem states that the Kolmogorov Complexity C(x) of an object remains invariant up to an additive constant, regardless of the programming language used. In other words, whether you describe an object in Python, Java, or any other universal programming language, the length of the shortest program that generates that object will differ only by a constant amount, reflecting the fixed overhead of translating between languages.
This property makes Kolmogorov Complexity a robust and reliable measure of information content, independent of the specific computational framework chosen. The invariance theorem states that no matter which programming language you choose, the Kolmogorov Complexity of an object remains invariant up to a constant factor. For instance, whether you describe an object in Python or Java, the length of the shortest program that describes it will only differ by a constant amount.
Of note, the invariance theory is very significant for the Christian story of creation. God’s language is the Word or the Logos, the ultimate source of all creation: “In the beginning was the Word, and the Word was with God, and the Word was God. He was with God in the beginning. Through Him all things were made; without Him nothing was made that has been made” (John 1:1-3). Therefore, the fundamental truths of Christianity are invariant regardless of the framework used to understand them. God’s Logos is the universal language that underpins all reality.
The Kolmogorov Complexity is the foundation for Algorithmic Information Theory. Since the early 1970s, many scientists have researched its implications, particularly in the domain of Artificial Intelligence. However, recent developments suggest that much remains to be discovered. One such development is Simulation Creationism as mentioned above. Measuring the complexity of a power that simulates the world means finding the shortest program’s length that can generate observable phenomena of our perceived world. A lower Kolmogorov Complexity might be used to measure the current simulated world, our Earth, and our universe. At the same time, a higher Kolmogorov Complexity would be necessary to explain God’s Simulation in its totality as the complex product of relatively simple God’s Singularity: “By faith, we understand that the universe was formed at God’s command, so that what is seen was not made out of what was visible” (Hebrews 11:3).
Unfortunately, we cannot still measure just how complex our universe is. It would be valuable information since a low Kolmogorov Complexity would mean that The Simulation contains only a few clear laws and rules. In fact, it would be a simple program. A high Kolmogorov Complexity necessitates a more sophisticated system. Christian theology tells us that there are four laws through which we can understand how God creates and sustains His creation: eternal law, natural law, divine law, and human law. Apart from human law, these are not computable, and most of them cannot be known in their totality: “By wisdom the Lord laid the earth’s foundations, by understanding He set the heavens in place” (Proverbs 3:19).
Through the Kolmogorov Complexity, we perceive patterns and regularities. Effective data compression about our simulated world means a low Kolmogorov Complexity prevails. Such patterns indeed exist, as evidenced by the Mandelbrot Set and fractal geometry, for instance: “Where were you when I laid earth’s foundation? Tell me, if you understand. Who marked off its dimensions? Surely you know! Who stretched a measuring line across it? On what were its footings set, or who laid its cornerstone – while the morning stars sang together and all the angels shouted for joy?” (Job 38: 4-7). It would also mean that The Simulation is governed by simple and consistent rules of simulating reality, wherein the products only seem complex and random.
Complexity – discovering God’s wisdom
We often use the term “complexity”, but we rarely define it. Complexity occurs somewhere between strict order and maximum disorder, i.e., between very low and very high entropy. According to recent astronomy, the universe started as very orderly, with energy and matter smoothly distributed in space. It was not perfectly smooth, as some tiny quantum fluctuations occurred. Then, the universe began to expand (or at least the theory suggests so, with huge question marks according to recent research), and gravity changed it. Quantum fluctuations in the early universe created regions of slightly higher density, which, under the influence of gravity, collapsed to form stars, galaxies, and galaxy clusters.
This transition from a nearly uniform state to complex structures is a key example of how simplicity can evolve into complexity. Some of these stars have planets, and life evolved on at least one of them. What began as simple became astonishingly complex. So, how does complexity come out from simplicity?
Humankind has made scientific progress and found definitive rules in nature around us. The issue is whether we understand these rules correctly. What reasonable rules must nature use for what it does? These might differ from the rules we set up as humans in our science. Computer programs help us find the answer. The issue is that we can get an incredible complexity from a simple start. It is not difficult for God to make all these complexities. It is the “natural” thing to happen.
A large number of divine and natural laws immediately create complexities. It seems peculiar to us because, as humans, we have restricted ourselves to the rules where we can immediately see the consequences of the rules. God’s creative process involves setting up the initial conditions (simplicity) from which the universe (complexity) would emerge as a simulated world. The initial conditions of the simulated world may be seen as a simple set of fundamental rules or parameters, designed to unfold into the rich complexity we observe. God’s creation is purposeful and directed. Therefore, the concept of emergence, where complex systems and structures arise from the interaction of simpler components, is a reflection of God’s wisdom and foreknowledge: “I made known the end from the beginning, from ancient times, what is still to come. I say, My purpose will stand, and I will do all that I please” (Isaiah 46:10). God, knowing the end from the beginning, simulates our world that is not static but dynamic, capable of developing and evolving in complexity.
Many scientists see information as the fundamental of reality. A computational universe necessarily generates complexity. Of course, this is a mysterious notion, as we do not know why a world that starts from simplicity grows into complexity. Such an idea goes deep. We could not have human DNA at the beginning of our world because there was insufficient information about it. But, the key to it is randomness, which is the second core thing about the Kolmogorov Complexity, is that a tiny fluctuation in the initial stage was a random cause for our world to become as it is. Or is it?
A funny example explains it. Suppose monkeys type random messages on a typewriter. The consequence would be indeed random. However, the program might interpret the instructions if they type them on a computer. It is different. The Algorithmic Information Theory explains what happens when a monkey types random instructions to a computer. The outcome is complexity. Thus, if tiny quantum fluctuations program our world, it necessarily produces interesting and complex outcomes. These things necessarily happen because the universe is computing. According to such a scientific view, the entire universe is, in fact, a giant computer.
Here comes the Kolmogorov Complexity, which is a significant help. Indeed, we need a more difficult algorithm to calculate what a human body does than a stone does. Still, both the human body and the stone are made of atoms, themselves made of elementary particles. The behavior of all those particles is always described by the same mathematics within the standard model of particle physics, coupled with Einstein’s general relativity. In principle, everything that we know exists can be described by that method.
The Kolmogorov Complexity provides a way to measure the complexity of an object based on the length of the shortest program that describes it. This does not mean that all objects are equally complex. The complexity varies depending upon the object and its description. For example, the arrangement of particles in different objects, such as a stone versus a human body, leads to varying levels of complexity. While the Kolmogorov Complexity focuses on the minimal description length, the concept of structural complexity is more appropriate for describing the inherent complexity that arises from the specific arrangement of particles in physical systems.
A complex system should be surprising as we celebrate the diversity surrounding us: “How many are your works, Lord! In wisdom, you made them all; the earth is full of your creatures” (Psalm 104:24). It does things that are difficult to predict and often hard to understand. It has many emergent properties and behaviors. While everything is made of elementary particles and obeys the same laws, combining those particles with more significant things often makes sense to describe them differently. These things have properties and natural laws of their own right. This is what we mean by emergent.
For instance, a metal’s conductivity tells us how easily the material transports electrons and how well it conducts electricity. This is something we find outside the standard model of particle physics. It does not make any sense to speak of a particle’s conductivity. Still, it is helpful to talk about a metal’s conductivity. These are emergent properties.
The number of emergent properties and their behaviors increase with complexity. We also expect their behavior to be complex but still manageable. The most complex systems live on the “edge of chaos.” Complexity increases until chaos sets in, and a transformation from chaos to order happens as the pivotal moment of creation: “Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters” (Genesis 1:2). Within God’s Singularity lies the potential for all levels of complexity, from the simplest elements to the most intricate systems.
God’s Singularity contains a divine blueprint that results in the structured and purposeful world. The simplest level includes the fundamental laws of nature that provide the groundwork upon which more complex forms can emerge. Each system is more complex than the last but still governed by the underlying principles established by God. For example, the emergence of life from a primordial soup of chemicals can be seen as a transition from simplicity to complexity, balancing between stable and chaotic states. Similarly, in computer science, cellular automata like Conway’s Game of Life demonstrate how complex patterns can emerge from simple rules operating at the edge of chaos. At the highest level there is life itself, humans with souls, capable of self-awareness, creativity, and morality. Each level reflects a different aspect of God’s nature and glory, but also a progressive revelation of His plan.
The defining feature of chaos is that minimal changes in the configuration of a system can have enormous consequences later on. And since we never know the configuration exactly, chaos makes it impossible to make predictions beyond a specific time. The edge of chaos is also the edge of unpredictability. Complex systems can also evolve. They learn, adapt, self-organize, and improve over time. It is not a property of the structure per se but how that structure changes. The complexity of creation reflects divine wisdom: “The heavens declare the glory of God; the skies proclaim the work of his hands” (Psalm 19:1). This Biblical verse celebrates the intricate complexity and beauty of the universe that arises from simple conceptional rules, similar to the complexity stemming from simple computation.
Omega Number – Bits of God’s Singularity
Chaitin sees the complexity and order of the visible world as evidence of God’s creative power and wisdom. If Christian theology sees simple principles of creation and translates them into an ordered and intricate world full of complexities, we can match Kolmogorov Complexity and God’s creation. Chaitin proposed the Omega number, which he describes as “the DNA of mathematical truth”. This number encapsulates the essence of mathematical truth in a form that appears random due to its maximal complexity. Unlike typical file compression, where the result may seem random, Omega represents a specific form of randomness tied to the halting problem in Turing machines. It is also the worst nightmare of any rational being because something is only valid for a sensible person if it can be understood or proven.
Gödel’s incompleteness theorem asserts that any finite system of mathematical axioms—any mathematical theory—is incomplete. More precisely, he demonstrated that there will always be arithmetic assertions (assertions about the positive integers, as well as addition and multiplication, known as number-theoretic assertions) that are true but unprovable! For instance, the “liar paradox” (the statement “this sentence is false”) is a simple illustration of how a system can contain statements that cannot be consistently proven true or false. In philosophy and logic, the classical liar paradox or liar’s paradox or antinomy of the liar is the statement that they are lying: for instance, declaring that “I am lying”. If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied.
The Omega number requires an infinite number of digits for complete accuracy, with each bit representing an irreducible mathematical truth. These bits appear to have no discernible structure, yet they can be understood in the context of the Halting probability for a general-purpose Turing machine, which encapsulates the inherent randomness of Omega. It is easily defined numerically as a number between 0 and 1. But it is maximally unknowable if we want to know Omega digit by digit. It is an inherent limitation of human knowledge and asks for faith. With the eyes of faith, we can learn more about the deeper truths of existence and God’s nature, as rationality stops at some point: “And we know that in all things God works for the good of those who love Him, who have been called according to His purpose” (Romans 8:28). Indeed, the Omega number serves as a powerful symbol of the inherent limitations in human understanding and the finite nature of human reason.
By contrast, God’s Singularity represents the infinite wisdom and knowledge that transcends all human limitations. It bridges the gap between human reason and divine wisdom by providing a source of ultimate truth and understanding. Through faith and divine revelation in the Bible, humans can access aspects of God’s wisdom and participate in God’s plan. It calls for humility, trust, and awe: “Trust in the Lord with all your heart and lean not on your own understanding: in all your ways submit to Him, and He will make your paths straight” (Proverbs 3:5-6).
Each of these bits is an independent mathematical fact. It is one bit of divine inspiration, as we cannot grasp the delicate balance between zero and one. Chaitin’s Omega is a radical opposition to traditional mathematics, which looks at all mathematical truths as coming from a small set of axioms that we can all agree on just by using logic. Acting against it, Chaitin shows we do not understand everything. Omega is a remarkable number because it embodies the essence of algorithmic randomness and incompleteness with a single value, where every digit is a complete and unexpected surprise. These are mathematical truths that cannot be known or comprehended. Therefore, Omega is connected to the mystery of God’s Singularity and His creation. Many aspects of God’s essence are entirely unknown or incomprehensible to humans.
Randomness – to understand, we need to compress
Our computers should generate numbers that appear to be random. But what does it mean for a sequence to be truly random? We could ask how random if we look at any sequence of 0 and 1. All sequences of the same length are equally probable; however, the feeling of randomness comes from the lack of discernible pattern. If we have a sequence of 16 bits and they are all one, it does not feel random at all. For believers, randomness is a chance to strengthen one’s faith when challenges occur. In Simulation Creationism, challenges are seen as specific trials within God’s Simulation through which a soul can learn more about divine reality: “Consider it pure joy, my brothers and sisters, whenever you face trials of many kinds, because you know that the testing of your faith produces perseverance. Let perseverance finish its work so that you may be mature and complete, not lacking anything” (James 1:2-4).
A random sequence should be patternless, i.e., we should not identify regularities, like all zeros or all ones, alternation, growing runs of consecutive zeros and ones, or even less obvious patterns that are harder to find. Here, we need the Kolmogorov Complexity. Still, the defining feature of finding the shortest explanation depends upon the language we use. Since human languages vary, there is no single way to describe a pattern or a random sequence unless we use a programming language. The Kolmogorov Complexity of a sequence ???? measures the shortest description or program that can generate ????. If no shorter description exists than the sequence itself, the sequence is considered to be Kolmogorov random.
We use this concept in data compression, converting more extended data representations into shorter ones. Data compression algorithms, like ZIP or JPEG, work by identifying and removing redundancies in the data, effectively reducing its Kolmogorov Complexity. For example, a repeated pattern of pixels in an image can be encoded more efficiently, leading to a smaller file size. However, the Kolmogorov Complexity is inherently non-computable, meaning that not only is no existing function that can take any sequence and determine its Kolmogorov Complexity, but it has also been proven impossible for such a function to exist at all.
Calculating the Kolmogorov Complexity for any given input is impossible since some contain infinite loops and encounter a halting problem. These are strings of arbitrary large Kolmogorov complexity: for each natural number n, there is a string s with K(s) ≥ n. Thus, the Kolmogorov Complexity is often considered an estimate with an upper bound, as no program can take any object as input and produce the exact integer value of its Kolmogorov Complexity.
Instead, we rely on the following relation:
It illustrates the limitations of human understanding and the necessity of faith. Through such faith, we can approach the profound nature of God’s creation through His transcendent wisdom.
We need energy for a simple thing to turn into a complex one. The power of light is paramount for almost all organisms on this planet, coupled with a long period of time. Complex organisms have more ability to adapt to their surroundings and to reproduce. It is very consistent with a prearranged purpose: God uses this mechanism to create complex creatures with intelligence and the ability to understand the words you are reading right now. It would be hard to imagine the same thing happening with a murky and random soup of particles. The laws of physics do not have to be violated for the divine purpose to be fulfilled. Some events appear random to humans, such as casting lots: “The lot is cast into the lap, but its every decision is from the Lord” (Proverbs 16:33).
Randomness is a challenging concept for theology. Some arguments posit that God created the world but then left it outside of His intervention, which cannot be easily reconciled with Christian theology. Instead, we can focus on human incapability to see the full picture of reality. We might interpret something as random without seeing it as an element of God’s divine plan: “For my thoughts are not your thoughts, neither are your ways my ways, declares the Lord. As the heavens are higher than the earth, so are my ways higher than yours and my thoughts than your thoughts” (Isaiah 55:8-9). In The Simulation, what seems like randomness or chance may be a result of complex interactions governed by underlying rules set by God. While we may perceive these events as random due to our limited understanding, God, in His omniscience, simulates these events to fulfill His purposes: “The lot is cast into the lap, but its every decision is from the Lord” (Proverbs 16:33). Apparent randomness can be part of God’s way of testing, refining, or growing His creation. God allows seemingly random events in our lives to strengthen faith, build character, and fulfill His greater plan: “Consider it pure joy, my brothers and sisters, whenever you face trials of many kinds, because you know that the testing of your faith produces perseverance. Let perseverance finish its work so that you may be mature and complete, not lacking anything” (James 1:2-4).
There are limits to what can be known or predicted: randomness implies incompleteness. Thus, both faith and reason are paths to understanding God. The famous mathematician Leibniz summarizes his view of what precisely it is that makes the scientific enterprise possible: “God has chosen that which is the most perfect, that is to say, in which at the same time the hypotheses are as simple as possible, and the phenomena are as rich as possible.” The scientist’s job, of course, is to figure out these most straightforward possible hypotheses. Leibniz argues that the ideas are simple, but the universe is very complicated if science applies. Through His mind, God minimizes the ideas, maximizes the universe, and is always involved in His creation to the smallest detail: “Are not two sparrows sold for a penny? Yet none of them will fall to the ground outside your Father’s care. And even the very hairs of your head are all numbered. So don’t be afraid; you are worth more than many sparrows” (Matthew 10:29-31). The Kolmogorov Complexity teaches that we must compress vast data into the theory. To comprehend is to compress, and we need a powerful computer to do that!
Turing Machine – God’s Machine?
At the start of the 20th century, mathematicians were fascinated by just how much math we could do, not by using our intuitions but by following rules very carefully—by following an algorithm. To make that notion precise, we need to answer what we mean by a program or algorithm. Alan Turing answered the question. The computer is a powerful new mathematical concept illuminating many questions in mathematics and metamathematics. Where formal axiomatic systems fail, formalisms for computing are a brilliant success.
Turing described these machines as writing down information in a coded form. He envisioned a tape as long as it needs to be, divided into squares, each holding a symbol from a finite alphabet (often 0 and 1, but not limited to these). The machine reads these symbols one square at a time, processes them according to a set of rules, and then updates the tape. At any moment in time, the Turing machine is in a particular state, and it looks at one square on the tape. It has a logbook or a program book. For instance, a simple Turing machine that performs binary addition might start in an initial state where it reads a “1”, changes it to a “0”, moves one square to the right, and transitions to a state representing the carry-over operation. This step-by-step process continues until the machine reaches a halting state, with the result of the addition left on the tape.
The machine starts with a specific pattern of ones and zeros. It follows these rules one square at a time, transforming that string of ones and zeros into a different string of ones and zeros. Eventually, the machine moves into a halting state, although no one can predict when this halting occurs. It continues to run based on its initial code, which is a necessary trait of a “divine computer” that is the ongoing sustenance of creation: “The Son is the radiance of God’s glory, and the exact representation of his being, sustaining all things by his powerful word” (Hebrews 1:3). What is left on the tape is the answer to a problem. If we consider God’s Singularity in Simulation Creationism similar to the Turing machine, we can envision it as the emergence of complex order from simple rules. For instance, God commands: “Let there be light,” and there was light as a complex and diverse creation output.
It is a simple process, indeed. However, it is the essence of computation. Whatever any computer can do, it can be done by that system looking at ones and zeros, leading to the limitless potential of computational processes. If God uses such computation, it points to the depth of His understanding of continuous creation: “Do you not know? Have you not heard? The Lord is the everlasting God, the Creator of the ends of the earth. He will not grow tired or weary, and His understanding no one can fathom” (Isaiah 40:28). Turing came up with his idea of Turing machines with a blueprint for modern digital computers. This straightforward idea of Turing machines is so powerful that all modern computers do what Turing machines do. We use this idea to evaluate the strength of contemporary computer programs.
When a computer program can do what the Turing machine does, we call it Turing complete. It is at the top of the hierarchy of program strength. As humans, we have never devised a way of computing things that could do more than a Turing machine. These machines would follow a predetermined set of rules to achieve the maximum mathematical function. From a theological point of view, the Turing machine describes God’s providence as the events and challenges in this world following a divine plan. Algorithmic processes may seem random, but they are logical in God’s mind and have a specific purpose people cannot comprehend in advance: “Oh the depths of the riches of the wisdom and knowledge of God! How unsearchable His judgments, and His paths beyond tracing out” (Romans 11:33).
Chaitin considered the Turing machine to be a technical way of seeing God as “a grand programmer”: “For in Him all things were created: things in heaven and on earth, visible and invisible, whether thrones or powers or rulers or authorities; all things have been created through Him and for Him. He is before all things, and in Him all things hold together” (Colossians 1:16-17). For Chaitin, there are many reasons to suspect that we might live in a digital universe! Just as a Turing machine can simulate any algorithmic process, we might see the universe as a vast computational system, with God as the ultimate “programmer”. In modern artificial intelligence, for instance, the algorithms that govern machine learning reflect this complexity—simple rules giving rise to highly sophisticated behaviors, much like the divine process of creation. In the end, the creative power of God is analogous to the Turing machine!
