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Computing devices are composed of spatial arrangements of simple funda-mental logic gates. These gates may be combined to form more complex addingcircuits and, ultimately, complete computer systems. Implementing classicaladding circuits using unconventional, or even living substrates such as slimemould Physarum polycephalum, is made difficult and often impracti- cal by thechallenges of branching fan-out of inputs and regions where circuit lines mustcross without interference. In this report we explore whether it is possible toavoid spatial propagation, branching and crossing completely in the design ofadding circuits. We analyse the input and output patterns of a single-bit fulladder circuit. A simple quantitative transformation of the input patterns whichconsiders the total number of bits in the input string allows us to map therespective input combinations to the correct outputs patterns of the full addercircuit, reducing the circuit combinations from a 2:1 mapping to a 1:1 mapping.The mapping of inputs to outputs also shows an incremental linear progression,suggesting its implementation in a range of physical systems. We demonstrate anexample implementation, first in simulation, inspired by self-oscillatorydynamics of the acellular slime mould Physarum polycephalum. We then assess thepotential implementation using plasmodium of slime mould itself. This simpletransformation may enrich the potential for using unconventional computingsubstrates to implement digital circuits.

Quantitative Transformation for Implementation of Adder Circuits in Physical Systems