The paper establishes a novel connection between CA and many-valued (MV) logic, specifically Ćukasiewicz propositional logic. It is shown that the transition functions of general CA with arbitrary state sets can be expressed as formulae in MV logic.
The key insights are:
Binary CA essentially perform operations in Boolean logic, but no such relationship exists for general CA with arbitrary state sets. The paper demonstrates that MV logic constitutes a suitable language for characterizing the logical structure behind general CA.
The transition functions of CA are interpolated to continuous piecewise linear functions, which, by virtue of the McNaughton theorem, yield formulae in MV logic characterizing the CA.
Deep ReLU networks realize continuous piecewise linear functions and are therefore found to naturally extract the MV logic formulae from CA evolution traces.
The dynamical behavior of CA can be realized by recurrent neural networks, providing a complete neural network implementation of general CA.
The paper provides a corresponding algorithm and software implementation for extracting the logical rules governing CA from neural networks trained on CA evolution data.
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