![]() Whereas a neural network looks nice and easy to understand on the first glance it is very complicated to understand its intended functionality inside our brain. They are just too complex for our technology. That is why we have no possibility to explain such effects with the help of maths or especially with the help of computers (super computers) today. To be more precise every component of the entire system increases its complexity in an exponential manner. You could say we need to hit the jackpot here. When it comes to emergence there is only one possibility to achieve a desired outcome. Although the starting conditions can be deterministic the interplay of its components and with it the outcome is very unpredictable. Changing initial conditions can cause the extinction of an emergent phenomenon. Very sensitive systems underlie emergent phenomena like the ones above. The point is that you often forget the importance of the corresponding details: the little components that are responsible for much mightier phenomena. Despite functioning in a different way from traditional, Turing machine- like devices, CA with suitable rules can emulate a universal Turing machine (see entry), and therefore compute, given Turing’s thesis (see entry on Church-Turing thesis), anything computable.There are a lot of things that seem to be very unspectacular in our daily life. Thirdly, CA are computational systems: they can compute functions and solve algorithmic problems. Secondly, CA are abstract: they can be specified in purely mathematical terms and physical structures can implement them. They evolve in parallel at discrete time steps, following state update functions or dynamical transition rules: the update of a cell state obtains by taking into account the states of cells in its local neighborhood (there are, therefore, no actions at a distance). At each time unit, the cells instantiate one of a finite set of states. Firstly, CA are (typically) spatially and temporally discrete: they are composed of a finite or denumerable set of homogenous, simple units, the atoms or cells. The considered adaptations of Godel’s proof distinguish between computational universality and undecidability, and show how the diagonalization argument exploits, on several levels, the self-referential basis of undecidability.Ĭellular automata (henceforth: CA) are discrete, abstract computational systems that have proved useful both as general models of complexity and as more specific representations of non-linear dynamics in a variety of scientific fields. ![]() The comparative analysis emphasizes three factors which underlie the capacity to generate undecidable dynamics within the examined computational frameworks: (i) the program-data duality (ii) the potential to access an infinite computational medium and (iii) the ability to implement negation. In particular, we elaborate on the diagonalization argument applied to distributed computation carried out by CAs, illustrating the key elements of G¨odel’s proof for CAs. Some of these interconnections are well-known, while some are clarified in this study as a result of a fine-grained comparison between recursive formal systems, Turing machines, and Cellular Automata (CAs). In this paper we explore several fundamental relations between formal systems, algorithms, and dynamical systems, focussing on the roles of undecidability, universality, diagonalization, and self-reference in each of these computational frameworks. ![]()
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