On the necessity of complexity

Wolfram’s Principle of Computational Equivalence (PCE) implies that universal complexity abounds in nature. This paper comprises three sections. In the first section we consider the question why there are so many universal phenomena around. So, in a sense, we seek a driving force behind the PCE if any. We postulate a principle GNS that we call the Generalized Natural Selection principle that together with the Church-Turing thesis is seen to be equivalent in a sense to a weak version of PCE. In the second section we ask the question why we do not observe any phenomena that are complex but not-universal. We choose a cognitive setting to embark on this question and make some analogies with formal logic. In the third and final section we report on a case study where we see rich structures arise everywhere. 1 Why complexity abounds Throughout the literature one can find various different and sometimes contradicting definitions of what complexity is. The definition that we shall employ in the first section involves the notion of a universal computational process/device. For the second and third section of this paper we shall use slightly less formal and more relaxed notions of the word complexity. 1 ar X iv :1 21 1. 18 78 v1 [ cs .L O ] 6 N ov 2 01 2 1.1 What is complexity? Let us recall that a computational process Π is universal if it can simulate any other computational process Θ. In other words, Π is universal if for any other computational process Θ, we can find an easy coding protocol C and decoding protocol C−1 so that we can encode any input x for Θ as an input C(x) for Π so that after Π has performed its computation we can decode the answer Π(C(x)) to the answer that Θ would have given us. In symbols: C−1(Π(C(x))) = Θ(x). For the sake of this first section we can take as working definition that a system is complex if we can easily perceive it as a universal computational process. Note that we have used the word ‘easy’ a couple of times above. If we were to be more precise we should specify this and could for example choose for poly-time or some other technical notion that more or less covers the intuition of what is easy. We wish to be not too specific on these kind of details here. Thus, for the first section of this paper, a complex process is one that is computationally universal. However, great parts of the reasoning here will also hold for other definitions of complexity. For example, stating that a process is complex if comprehending or describing it exceeds or supersedes all available resources (time, space, description size). Note that our current definition of complexity need not necessarily manifest itself in a complex way. Remember that a universal process is one that can simulate any other process, thus also including the very simple and repetitive ones. One might thus equally well observe a universal process that temporarily exhibits very regular behavior. In this sense universal cannot be directly equated to our intuitive notion of complexity but rather to potentially complex. 1.2 The Principle of Computational Equivalence and the Church-Turing thesis In his NKS book [14], Wolfram postulates the Principle of Computational Equivalence (PCE): PCE: Almost all processes that are not obviously simple can be viewed as computations of equivalent and maximal sophistication.