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Neural automata, combining neural networks and symbolic dynamics via Gödel encoding, have dynamics that depend on arbitrary assignments. This study defines patterns of equality to identify intrinsic dynamics, finding only step functions are invariant under recoding.

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Area of Science:

  • Computational neuroscience
  • Artificial intelligence
  • Mathematical modeling

Background:

  • Computational modeling of neurodynamical systems integrates neural networks and symbolic dynamics.
  • Vector symbolic architectures enable neural automata through Gödel encoding, mapping symbols to numbers.
  • This encoding represents symbolic computation as state vector trajectories for data analysis.

Purpose of the Study:

  • To develop a rigorous mathematical framework for analyzing symmetries and invariants in neural automata under varying encodings.
  • To distinguish intrinsic dynamics from representation-dependent artifacts in Gödel-encoded systems.
  • To investigate the invariance properties of macroscopic observables in neural automata.

Main Methods:

  • Formal mathematical framework development.
  • Definition of "patterns of equality" as a central concept.
  • Analysis of macroscopic observables like mean activation level for invariance properties.

Main Results:

  • Identified "patterns of equality" as key to understanding neural automata dynamics.
  • Demonstrated that only step functions defined over these patterns are invariant under symbolic recoding.
  • Showed that common observables like mean activation are not invariant to encoding changes.

Conclusions:

  • The choice of Gödel encoding significantly impacts observed dynamics in neural automata.
  • Invariance is achieved only through step functions over patterns of equality, not simple averages.
  • Findings are crucial for neurosymbolic regression studies to avoid encoding-dependent confounds.