Evaluating One-Dimensional Continuous & Discrete Heterotic Systems
Jack Dewhirst, Matt Dale, Angelika Sebald, Susan Stepney
York Cross-disciplinary Centre for Systems Analysis, University of York, UK
In order to characterise computational systems, for example different proposed substrates forReservoir Computing (RC), measures such as memory capacity, kernel rank, and generalisation
capacity have been created. The broad applicability of the reservoir computing paradigm to manydifferent dynamical systems has recently lead to the development of a framework for theCharacterisation of Reservoir Computers (ChaRC), for comparing between substrates using
these measures [Dale et al 2018, 2019]. ChaRC exploits several computational measures toassess the ability of a substrate to realise different reservoirs, and therefore its capacity as ageneric RC substrate. However it is not yet clear how ChaRC may be applied to discrete statesystems such as Cellular Automata or Random Boolean Networks.
To investigate discrete state space reservoirs, we take as examples two analogous systems:
Threshold Coupled Maps (TCMs) [Sinha & Ditto 1998] (coupled logistic maps) and onedimensional cellular automata (CA) [Wolfram 1983]. These two systems are comparable in their
1D topologies, and their discrete time behaviour. However, TCMs have continuous-valued statevariables, whereas CAs have discrete valued variables.
TCMs and their ease of simulation provide a benchmark for many measures of spatiotemporal chaos such as Lyapunov spectra [Isola et al 1990], Kolmogorov-Sinai entropy density[Batista & Viana 2002], and Mutual Information and correlation in space-time [Schrieber 1990].
These are potential behaviour measures for use by ChaRC. Given TCMs similarities to someaspects of CAs, we determine how to transfer ChaRC-related measures to the discrete domain.
We also use these systems to look for insights into characterising heterotic computing
systems [Kendon et al 2015]. We compare quasi-uniform cellular automata [Gundersen 2017] inwhich different regions of cells contain different rules, and TCMs in which each map possesses a
different parameter value. By investigating the analogy between these two systems we determine
a method of applying ChaRC to heterotic continuous and discrete state systems.
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