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Cause of rotational/mirror non-invariance in PyPhi IIT outputs for CGOL

Investigate why configurations that are rotations or mirrorings of each other on a 2×2 toroidal grid under Conway’s Game of Life yield different integrated information (Φ) values when computed with PyPhi, and determine whether the discrepancy arises from transition probability matrix construction, connectivity matrix specification, grid orientation handling, or algorithmic aspects of the IIT calculation.

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Background

The authors apply Integrated Information Theory (IIT) using PyPhi to a minimal 2×2 toroidal grid for each ruleset due to computational constraints. In principle, rotational or mirror symmetries of cellular automaton configurations might be expected to produce identical Φ values if the system and analysis respect those symmetries.

However, for CGOL, the authors observe that rotated or mirrored configurations sometimes produce different Φ outputs, suggesting an unexplained asymmetry in the analysis pipeline or underlying model representation.

References

A strange result of Test 3 was that configurations that were simply rotations or mirrorings of each other would sometimes give different outputs from the IIT algorithm (Figure 1). It's unclear why this is.

— Conways game of life as an analogue to a habitable world Livingness beyond the biological (2410.22389 - McCrum et al., 29 Oct 2024) in Section Results, Figure 6