Essay Abstract

I will propose the notion that the universe is digital, not as a claim about what the universe is made of but rather about the way it unfolds. Central to the argument will be the concepts of symmetry breaking and algorithmic probability, which will be used as tools to compare the way patterns are distributed in our world to the way patterns are distributed in a simulated digital one. These concepts will provide a framework for a discussion of the informational nature of reality. I will argue that if the universe were analog, then the world would likely be random, making it largely incomprehensible. The digital model has, however, an inherent beauty in its imposition of an upper limit and in the convergence in computational power to a maximal level of sophistication. Even if deterministic, that it is digital doesn't mean that the world is trivial or predictable, but rather that it is built up from operations that at the lowest scale are very simple but that at a higher scale look complex and even random, though only in appearance.

Author Bio

Hector Zenil (BSc. Math, UNAM, 2005; MPhil. Logic, Paris 1 Sorbonne, 2006; PhD. Computer Science, Lille 1, 2011) has held visiting positions at the Massachusetts Institute of Technology and Carnegie Mellon University. He is a senior research associate at Wolfram Research, member of the Turing Centenary Advisory Committee, founding honorary associate of the Algorithmic Social Science Research Unit of the University of Trento and editor of Randomness Through Computation (published by World Scientific). His main research interests lie at the intersection of several disciplines in connection or application to the concept of randomness and algorithmic complexity motivated by foundational questions.

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Dear Hector,

Thank you for your very interesting article. It does seem the rules in nature will turn out to be simple as opposed to needing very complicated constructs. I regularly see this in engineering where difficult problems taking much analysis boil down to only a few lines of code at their cores. It is interesting that you also regularly see this in your work. Thank you for your fine essay.

Kind Regards,

Russell

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    • Post #3

    NP-completeness and the NP Hardness Assumption suggest that Reality isn't algorithmizable. Where, for example, is the algorithm for protein folding?

    Arguably the emergence of complexity from Omega or quantum randomness can never be described for essentially the same reason: it's not a compressible process.

    Or am I full of it? Missing the point?

      Dear Russell,

      Thank you very much for your message.

      Dear nikman,

      Thanks for your message. In my definition of an algorithmic world problems do not need to belong to a particular computational complexity class. My use of algorithmic is independent and compatible with the theory of computational complexity.

      As you know, the framework and investigation of problems in the theory of computational complexity is based in the concept of the universal Turing machine. Problems, even if NP-complete, are studied as being carried out by a digital computer (the Turing machine), so even if NP-complete, they are algorithmic under my worldview.

      The fact that some problems may take a long time to be solved in the size of the input doesn't mean, in my definition, that something is no longer algorithmic. Problems don't need to belong to any time complexity class to be algorithmic. On the contrary, if a problem can be described in algorithmic terms, then it is algorithmic, so NP-complete problems can coexist with my algorithmic universe.

      On the other hand, we shouldn't forget that often instances of a NP problem may be easy to solve within polynomial time by a deterministic Turing machine, and that it is unknown whether there are any faster algorithms to solve NP-complete problems for all instances of the problem.

      I think it is different to ignore what the algorithm of protein folding is than claiming (I'm not sure you did) that protein folding cannot be carried out by a (deterministic) Turing machine (either in polynomial time or not , something that we yet don't know).

      Sincerely.

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      • Post #6

      Thanks for that. And yes, I'm suggesting what you suggest I'm suggesting. I know that Turing himself first posited the possibility of non-recursive adjuncts ("oracles") to computation, but in retrospect he may not have realized the immense other world that might lead to.

      Time will tell.

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      • Post #7

      Hector,

      To make near-poetry of computer science, and have it be factually accurate and scientifically well grounded besides, is a tour de force. I look forward to seeing this piece published in a prestigious venue, as by any objective standard I know, it deserves to be.

      It's so gratifying to see information theory getting the strong treatment in this contest, that I hoped it would.

      All best,

      Tom

        T H Ray,

        Thanks for your kind words. I'm glad you found the essay interesting and also to be me who stands in favor of information theory to support the digital view of the world in this exciting contest.

        Sincerely.

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        • Post #9

        I hope you've made plans to be in Boston for ICCS this summer. If my own plans go as expected I'd love to share some conversation over a cold Sam Adams.

        Tom

        Many thanks Tom,

        I will likely be in Boston during the Summer (not sure if I will attend ICCS this time though). Drop me a line if your plans go as you hope. My email is on the first page of the essay.

        Best.

        Hi Hector, I was very impressed with your essay. Very easy to read yet dealing with complex study matter. I have a question which relates to you talking about DNA incidentally:

        Q: Why can't an Archimedes screw be used as a particle/wave model of gravity? Why is no-one experimenting with this simple idea of a screw being the analogy needed to visualise a force-inducing particle of attraction? If it then travelled around a wraparound universe it would emerge on the other side as force of repulsion i.e. dark energy. I don't understand why no-one has grasped this simple idea yet.

        Many thanks.

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          • Post #12

          Hi Hector Zenil,

          Congratuations, that permits to understand better the computing and its randomness.

          I ask me how is the basis of these systems and laguages? The simulations can be optimized !

          Good luck.

          Best Regards

          Steve

            Hello,

            Very interesting essay and at the same time hard to understand if someone does not have formal exposure to complexity theory. I still don't see two things: why an analog universe can't have an algorithmic representation, which is obviously what relativity has offered with very high accuracy, and how can one decide the fundamental question from all these, namely whether there is a smallest interval of space(time) or not. Another question: do you take for granted that algorithmic representations supervene on laws of nature?

            Regards.

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              • Post #14

              Dear Dr. Hector Zenil,

              I am just now downloading your essay. I will read it to the best of my ability. I will be looking for whether or not your view of the universe is one of an unfolding 'program' in somewhat of a computer sense. Your statement:

              "On the contrary, if a problem can be described in algorithmic terms, then it is algorithmic, ..."

              Seems to me to suggest that you believe that the universe can be properly described and defined by the means of establishing 'steps'. I will be looking for your support for that view. If I am wrongly anticipating your position, then, I will learn this by reading your essay. Thank you for participating.

              James Putnam

              Thank you very much Alan.

              Concerning your question, I don't know why nobody has used the idea of an Archimedes screw. It sounds to me that something similar has been explored in the form of some topological spaces that behave as you describe, although I'm not sure whether they have been connected to the dark energy phenomenon.

              Best regards.

              Thanks Steve. I think it is the first time I read a post from you with nothing about the Spheres model =)

              Thanks again.

              Hello Efthimios,

              Good question. What I claim, and the reason I believe my model is stronger than claiming directly that the world is digital, is because in my view one doesn't have to presume discreteness as a basic assumption of the world. One starts asking how the universe looks like in terms of the distribution of patterns in the world. Then one can conclude either that the world is digital because it looks like so (or does not), or that it is algorithmic (in the digital sense) even if it is not digital, case in which I argue we have no reasons to think it is not digital. We provide some evidence in favor of the resemblance between empirical and digital datasets and means to continue the investigation (investigation that has already provided some applications by the way, such as the calculation of the complexity of short strings where compression algorithms use to fail).

              You may also mean that the world could be algorithmic in other different sense, in an analog fashion (in the sense of being carried out by an analog computer) and still remain algorithmic. It could be, but so far we have had a hard time trying to define analog computation, at least in feasible terms, and our best model to understanding the world has turned out to be digital (Turing) computation. This is why I focus on discussing the way the world seems to unfold and whether it may do so in one or another way. Our claims are supported by statistical results (statistics are not proofs though), and we found that patterns in the real world and the digital ones, that we simulated, seem to be distributed alike.

              Among the things I argue in my essay is that the world would have greater chances to look random (or more random if you prefer) if it were analog. If one throws digits into the air of an analog (infinitely divisible) world and if this hypothetical world allows 'true' (indeterministic) randomness unlike in a digital one, then one would expect every digit to be like the digits of a Chaitin Omega (see definition in the Appendix of the essay), this is a number that is random by definition under our standard model of computation. You can perform the same thought experiment with a real number line and see that chances of picking a random number among all numbers, in a finite interval, is 1, that is complete certainty that you will pick a random number. Yet we don't experience that in our everyday life, but quite the opposite. Chaitin has proven that one cannot calculate most digits of an Omega number (for some Omega numbers not even a single digit), so in a world where random numbers persist, things might just have greater chances to look like Chaitin Omegas. The fact that we can do science in this world seems to be an indication favoring that this is not the case.

              You are right, it is very interesting how physical models based on mathematical theories assume continuous variables, yet when one solves the equations of, let's say general relativity to take the example you mention, the model becomes algorithmic in the strict digital sense, either by the mechanistic way in which equations are solved by hand, or literally when solved by a digital computers. The algorithmic view might turn out to help as a shortcut to understanding the digital nature of the world without having to assume it at first.

              Sincerely.

              Dear Hector,

              Thank you for the detailed response. I now understand better your work (I hope), which I think is very interesting and original.

              However, relativity theory tells us that the world is analog and fully deterministic. You say: " I will argue that if the universe were analog, then the world would likely be random, making it largely incomprehensible."

              The above statement is contrary to the best scientific theory we have available that in based on continuity of spacetime and it is fully deterministic at the same time, contrary to your claim. In addition, this type of analog mode of a universe is comprehensible and falsifiable by experimentation, but hasn't been falsified to this date.

              Regardless analog computational machines, If the universe is analog, it is the best analog computer, we should not need to find another one.

              I would like to know more about how your quoted statement above reconciles with relativity theory.

              Thanks and regards.

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              • Post #19

              :)..Peter says I am sphericentrist,probably a problem of vanity due to my young age(35):)

              Regards

              Thanks Hector! You're the third person to appreciate the connection. If Newton had hit on this idea we would never have had Einstein's spacetime continuum imo. It leads on to the idea explaining the 100,000 year ice age problems which are encountered with Milankovitch cycles. Nevermind..

              Alan