On a Constraint on Invariant Measures of Certain Cellular Automata
Pith reviewed 2026-05-10 16:29 UTC · model grok-4.3
The pith
Fixed positive indices determine uniform conditional probabilities on a coset at zero for invariant measures of bi-permutative cellular automata.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
In bi-permutative cellular automata, fixed values at the positive indices determine almost-surely a uniform conditional probability on the subset of values of positive conditional probability at the zero index; in the finite group multiplication case this set is a coset of some subgroup. The constraint induces a factor with respect to the shift, and zero-entropy invariant measures on that factor correspond to positive-entropy measures on the original system. The results extend to RLP subshifts, a class conjectured to be much larger than bi-permutative cellular automata but with only one further example proved to belong to it.
What carries the argument
The constraint on invariant measures that forces uniform conditional probability at the origin given the positive half-line, which defines an induced factor under the shift map.
Load-bearing premise
The prior observation for bi-permutative cellular automata continues to hold and the RLP subshifts class contains at least one additional example beyond the bi-permutative cases, without needing a fresh derivation of the base constraint.
What would settle it
An explicit invariant measure on a bi-permutative cellular automaton for which the conditional distribution at position zero given the positive half-line is non-uniform on the set of symbols with positive conditional probability.
read the original abstract
In [6], a constraint on invariant measures of bi-permutative cellular automata has been observed: fixed values at the positive indices determine almost-surely a uniform conditional probability on the subset of values of positive conditional probability at the zero index. When the alphabet is a finite group and the automaton is multiplication of two neighbors, that set is in fact a coset of some subgroup. In the present paper, we strengthen the formulations in [6] and investigate further the implications of this constraint. In the finite group case mentioned above, relations between some attributes of the group structure and the invariant measures are examined. We also inspect a factor, with respect to the shift, that this constraint induces, and analyze the special case in which it has zero measure-theoretical entropy, thus observing an interplay between existence of zero entropy invariant measures on that factor and existence of positive entropy measures corresponding to them on the original system. Then, we leave the setting of bi-permutative cellular automata and generalize our results to a wider class which we named RLP subshifts. The peculiar situation is that although this class may be much larger than the class of bi-permutative cellular automata, we were able to prove only for essentially one other example - the symbolic coding of the times 2 times 3 system on the circle (and its generalizations) - that it belongs to it.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper strengthens the constraint on invariant measures from [6] for bi-permutative cellular automata: fixed positive-index values determine almost-surely a uniform conditional probability on the positive-probability symbols at the zero index. In the finite-group multiplication case, this set is a coset of a subgroup. The authors examine relations between group attributes and measures, construct a shift factor induced by the constraint, and analyze the zero-entropy case on the factor, which corresponds to positive-entropy measures on the original system. They then generalize the results to RLP subshifts (defined by the same conditional-probability property), proving membership for bi-permutative CA and one additional example while conjecturing that the class is substantially larger.
Significance. If the derivations hold, the work provides a useful restriction on invariant measures for these systems and highlights an entropy interplay via the induced factor. The strengthening of [6] and explicit group-structure analysis are concrete contributions. The RLP generalization offers a potential broader framework, though its impact is currently constrained by the small number of verified examples. The paper gives credit to prior work, derives new implications without free parameters or circularity, and states its conjecture transparently.
major comments (1)
- [RLP subshifts generalization] In the section introducing and generalizing to RLP subshifts: the factor construction and entropy correspondence are derived from the RLP property, yet membership is verified only for bi-permutative cellular automata and essentially one other example (as stated in the abstract). Without a general criterion for membership or additional explicit examples, the claimed widening of the setting to a substantially larger class remains conjectural, limiting the demonstrated scope of the generalized results.
minor comments (1)
- [Abstract] The abstract phrasing 'we were able to prove only for essentially one other example that it belongs to it' is slightly unclear; rephrasing to indicate what 'essentially unique' means would improve readability.
Simulated Author's Rebuttal
We are grateful to the referee for the thorough review and insightful comments. Below we provide a point-by-point response to the major comment.
read point-by-point responses
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Referee: In the section introducing and generalizing to RLP subshifts: the factor construction and entropy correspondence are derived from the RLP property, yet membership is verified only for bi-permutative cellular automata and essentially one other example (as stated in the abstract). Without a general criterion for membership or additional explicit examples, the claimed widening of the setting to a substantially larger class remains conjectural, limiting the demonstrated scope of the generalized results.
Authors: We thank the referee for this observation. It is correct that membership in the RLP class has been verified only for bi-permutative cellular automata and one additional example. The manuscript defines RLP subshifts directly via the conditional-probability property, from which the factor construction, the entropy correspondence, and all subsequent results are derived without reference to the origin of the subshift. Consequently the theorems apply to every member of the class. The two explicit examples demonstrate that the class is nonempty and strictly larger than the bi-permutative cellular automata, while the conjecture that it is substantially larger is stated explicitly as a conjecture. We will revise the text to separate the general statements (which hold for any RLP subshift) from the concrete verifications and to make the conjectural status of the broader class even more prominent. revision: partial
Circularity Check
No circularity: results follow from definitional RLP property with explicit verifications
full rationale
The manuscript defines RLP subshifts precisely by the uniform conditional probability property first observed for bi-permutative CA in [6], then derives the factor, entropy-interplay statements, and group-structure relations directly from that property. It strengthens the [6] formulation, verifies one additional explicit example belongs to the class, and states the conjecture that the class is larger without claiming a general membership criterion. No equation or claim reduces by construction to a fitted parameter, self-referential definition, or unverified self-citation chain; the self-citation supplies the initial observation while the new implications and generalization rest on independent logical steps from the stated property.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard definitions and properties of cellular automata, invariant measures, and factors in symbolic dynamics
invented entities (1)
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RLP subshifts
no independent evidence
discussion (0)
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