(Note from the author: This story documents a train of thought about linking the expressiveness of computation with the origin of life. After writing this story I studied more about mathematics and entropy, and came to realize this story to be a mere speck in the grand scheme of all mathematics. I hope this article inspires you to explore those fascinating subjects.)

This article explores an unproven theory about computing machines being naturally occurring phenomena in large dynamic systems, possibly explaining the origin of life. The unproven theory states as follows:

Samshuijzen’s theory:

“Computing machines are mathematical attractors in large dynamic systems.”

The first image taken by humans of the whole Earth. AS8–16–2593. William Anders, Apollo 8.

The Turing machine

What is a Turing machine? If you know what a Turing machine is then you can skip this section.

A Turing machine is a hypothetical machine thought up by Alan Turing. Conceptually, the Turing machine consists of an infinite tape and a tape head. The tape consists of infinitely many squares along its length, where each square can be empty or store a symbol. The tape head is an active component that: reads the symbol on the square, follows a set of rules to determine whether to change the symbol on the square, determines whether to move left or right, and repeats this procedure until the outcome of the rules says it should halt.

The Turing machine is not a real machine. It is a model of computation, a model of a (simplest conceivable) machine that is Turing Complete. A machine that can simulate the Turing machine is Turing complete, thereby capable of simulating all other models of computation.

We can simulate Turing machines (with finite tape) on our computers, but they are notoriously cumbersome to program. Simplicity at the cost of expression. Our computers are based on the more expressive and easier-to-program von Neumann architecture.

The thesis

Alan Turing showed us that you can build a universal computing machine using relatively simple parts. Hypothetically speaking, if we would have to design a general purpose computing machine, using only very simple components, starting with a small set of such components, adding new components one by one, then we can expect the components of the first successful design to resemble the components of a Turing machine.

The process of evolution is similar to this design process, because it follows a similar breadth-first search. A bottom-up design process for “machines” that survive best on planet Earth, so to speak. If there exist simple algorithms for survival that will run on simple energy-efficient machines, then surely evolution will have discovered these machines and algorithms. If so, then evolution is the process of selecting the winning survival algorithms and machines, and life is the result of this process.

I find it conceivable that Turing-like machines naturally arise in the expressive language of molecules and chemistry, given enough time. After all, these machines are so simple to build, yet give access to the expressive world of computation. If the world of computation is indeed so easily and readily accessible (on evolutionary timescales), then surely properties of computation will naturally emerge in dynamic systems. There is evidence of this in molecular biology, where the structure and behavior of molecules show a striking resemblance with lambda calculus. Lambda calculus is an equivalent to Turing machines, as lambda calculus is all you need to simulate a Turing machine.

Turing-like machines

The machines discussed in this article are not the same as the hypothetical Turing machine as Alan Turing defined it. The machines discussed here are machines that show a resemblance with the design of the Turing machine, i.e. a “Turing-like machine”. They are similar in the sense that they are the “simplest conceivable computing machines given a finite set of symbols and rules”. I admit to misusing the term “Turing machine” beyond its intended mathematical definition, such as in the title. By doing so, my intention is to honor Alan Turing, because the machines discussed here closely resemble Turing’s invention. Furthermore, the title “We are all Turing-like machines” is not so catchy.

So what is a “Turing-like machine”? In this article we assume the following definition of the simplest Turing-like machine:

A Turing-like machine consists of at least these components:

1. A finite tape with symbols.

- The tape is of any type of medium that can hold symbols in locations.

2. A tape head.

- The tape head reads or reacts to symbols on the tape according to a set of rules.

An artistic representation of a Turing Machine. (commons.wikimedia.org)

Variations of Turing-like machines

We can expand the space of Turing-like machines by applying modifications to its construction. Here are some examples:

Different set of symbols (binary, number, letter, words, etc.).

Different tape head read/write rules.

Different initial states.

Multiple tape heads.

Stacked tape heads.

Tape head connected to an outside source.

Tape that moves continuously, independently from the tape head.

Tapes that are communicated between Turing machines.

And so on, whatever modification you can think of. Hypothetically there exist infinitely many variations of Turing-like machines, and infinitely many codes that will run on it.

Modifying a machine may make it’s construction more complex, but some modifications will help to increase the expressiveness of the machine’s language, i.e. the rules of the tape head. With the right choice of modifications, a language can be designed that allows for the most efficient description of the machine’s environment. Modifications allow for a large complex program to be reduced to a smaller, simpler and more efficient program.

The premise

We will explore the theory by asking these questions:

If a large dynamic system gradually increases in complexity, as dynamic systems do, and there exist infinitely many designs for simple machines and codes, given enough time, can we expect simple Turing-like machines to emerge and interact? If so, could life on Earth be the result of this?

The following chapters are an imaginary exploration of the conditions that may have given rise to the first machines on Earth.

The first Turing-like machines on Earth

Earth is a very large dynamic system. Our Sun consistently bombards our rotating Earth with light and heat. Its energy disperses throughout our atmosphere, causing winds, rains, rivers, currents and cycles.

At the smallest of scales, currents of molecules flow and interact according to the rules of the atomic and the sub-atomic. When molecules interact according to rules, then we could say that these rules describe a language. In this view, molecules are the symbols of a language, albeit a very complex and unstable language. Most combinations of molecules will not define any interesting language, but some combinations will. If an expressive language were by chance to occur somewhere in the interactions of molecular structures, then it has the opportunity of being read and reacted upon.

Suppose a small cycle of molecules were to interact with a slightly larger cycle of molecules. At this point, according to our definition, we already have something that is starting to resemble a Turing machine, a “proto-Turing machine”.

The larger cycle resembles a simple Turing-like machine with circular tape. The smaller cycle serves as a tape head with a cyclic code of molecules. The rules of chemical interaction define the language and instruction set. The molecules at the boundary will interact according to the rules of chemistry, analogous to the execution of code.

Simplified model of an early proto-Turing machine.

Of course this flimsy arrangement is far from being close to our definition of a computation machine. But there will be billions upon billions of variations of such interactions across the planet, in many forms and sizes. Some may flow through channels, some might be encapsulated in membranes, and so on. We can expect there to be snapshots in time when there exists an arrangement of molecular interactions for which there exists a hypothetical (Turing-like) computing machine that efficiently describes its arrangement and interactions. At such a moment, then by mathematics we can expect the arrangement to perform computation like that hypothetical machine, even if it were for a short moment.

Most will fade out instantly as they get broken apart. Even if it were stable for a while, the machine needs an expressive language to have any chance of being a computation machine. A good candidate for such a language is of course the very expressive lambda calculus. This language is so small that it can be modeled with molecules.

Even so, if the molecules were to describe an expressive language, most random sequences of codes lead to nothing interesting. It will take eons to find the simplest code that does anything interesting. Given enough time, some of these short-living machines will acquire codes that combat the elements for survival, so to speak. These machines will hang around for a little while longer than average.

Evolution of machines

At one point in history, a particular machine emerged which had the “right” code, the code for self-replication. Any machine can be described in code, and evolution has found a way to encode a self-replicating survival machine in DNA. (Surely it is no coincidence that the structure of DNA resembles lambda calculus’ Y-combinator. If the most basic machines are Turing-like machines, then we can expect the most basic codes and languages to be lambda calculus-like.)

This is when life began, the evolution of self-replicating computing machines, searching for codes and modifications that contribute to survival. The theory in this article predicts this emergence, because it states that computing machines are naturally occurring phenomena.

Some progressions took many millions of years, such as the time it took between prokaryotes and eukaryotes. Some advancements require an additional component to the machine that is hard to come by, or a piece of code that is very particular.

Our lowest common ancestor was a Turing-like computing machine. The tree of life is a map of variations of DNA code for machines and algorithms that survive well on planet Earth.

Ribosome mRNA translation. (commons.wikimedia.org)

Turing machines all the way down

If you have not yet seen those wonderful 3D animations of the molecular machinery in living cells, then please do. You will not be disappointed. You can find some impressive ones on YouTube.

Do you agree that the ribosome looks like the tape head of a molecular Turing machine? And that the strand of mRNA is the code of this machine? I find it quite convincing. Our blood vessels serve as a shared tape, a communication highway between the trillions of computing machines in our cells. A city of machines that communicate molecular messages written in the language of amino acids, codons and proteins.

Summary

This article claims that life is a mathematical phenomena, naturally emerging in large dynamic systems. From a dynamic system that has a large set of elements of different types that interact according to fixed rules, expressive languages and computing machines will tend to emerge, given enough time. We can expect the first forms of computation to be the simplest forms of computation, such a lambda calculus and Turing machines. To summarize the progression:

dynamic system → language → computation → replication

As this is a mathematical theory, I expect there exists a mathematical proof. I found very little written on this subject, which I find surprising, because merging the work of Turing, Church and Darwin just seems to make sense. If anyone knows about work done in this direction, please drop me a message.

Tim Samshuijzen

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