Non-uniform Cellular Automata

• Non-uniform Cellular Automata are extensions of classical cellular automata where local update rules vary by site, allowing diverse and flexible dynamical behaviors.
• They employ structured formalisms and algorithmic methods—including reachability trees and algebraic models—to analyze properties like reversibility, surjectivity, and injectivity.
• They have practical applications in computational emulation, reservoir computing, and modeling of heterogeneous systems, paving the way for innovative research directions.

Searching arXiv for the cited NUCA papers to ground the article in current sources. arxiv_search(query="(Rollier et al., 2024) Non-uniform Cellular Automata TensorFlow (Phung, 2022, Nichele et al., 2017, Adak et al., 2019, Phung, 30 Mar 2025, Provillard et al., 2011, Dennunzio et al., 2011, Kamilya et al., 21 May 2026, Hazari et al., 2016, Paturi et al., 9 Jul 2025, Phung, 2022, Phung, 2024, Phung, 2022, Phung, 2022)", max_results=10) arxiv_search(query="Non-uniform cellular automata", max_results=10) Non-uniform cellular automata (NUCA) are extensions of cellular automata in which the local update rule is allowed to depend on the site rather than being spatially uniform. In a widely used group-universe formulation, for a memory set $M\subset G$, an alphabet $A$, and a configuration of local rules $s\in S^G$ with $S=A^{A^M}$, the global map is

$\sigma_s(x)(g)=s(g)\big((g^{-1}x)|_M\big),$

so classical CA are exactly the special case where $s$ is constant. Other strands of the literature use finite one-dimensional rule vectors or broader taxonomies in which non-uniformity can also affect update pattern, lattice structure, or neighborhood dependency. Across these formalisms, NUCA provide the standard mathematical language for spatially heterogeneous local computation on symbolic dynamical systems (Paturi et al., 9 Jul 2025, Dennunzio et al., 2011, Bhattacharjee et al., 2016).

1. Formal models and terminological landscape

A persistent feature of the literature is that “NUCA” has both narrow and broad usages. In the narrow and historically dominant sense, the object is a one-dimensional system in which different cells use different local rules, often elementary Wolfram rules encoded in a rule vector

$$\mathcal{R}=\langle \mathcal{R}_0,\mathcal{R}_1,\ldots,\mathcal{R}_{n-1}\rangle,$$

with global update

$y_i=f_i(x_{i-1},x_i,x_{i+1}).$

This is the finite, binary, radius-$1$ setting used in work on reversibility, reachability, and number conservation (Adak et al., 2019, Hazari et al., 2016). In a broader symbolic-dynamical formulation on $A^{\mathbb Z}$, NUCA are structures $A$0 with

$A$1

and the full class coincides with the continuous self-maps of $A$2; this breadth is one reason later work concentrates on structured subclasses such as fixed-radius, default-rule, or eventually periodic systems (Dennunzio et al., 2011).

A complementary formalism treats non-uniformity as a distribution of rules $A$3 over a finite set of local maps. The induced global map is denoted $A$4, and when $A$5 is finite one may pad all rules to a common radius $A$6, thereby obtaining an $A$7-CA with site-dependent rule but fixed neighborhood size (Provillard et al., 2011). This rule-distribution viewpoint is central when the object of study is not one fixed automaton but the symbolic set of distributions producing surjective, injective, or number-conserving dynamics.

The survey literature also places NUCA inside a larger classification of non-classical CA. One may distinguish non-uniformity in update pattern, non-uniformity in lattice structure and neighborhood dependency, and non-uniformity in local rule. In that broader sense, asynchronous CA and automata networks are parallel non-uniform branches, while “hybrid” or “non-uniform” CA usually refer specifically to spatial variation of the local rule (Bhattacharjee et al., 2016). This suggests that the modern NUCA literature is unified less by one canonical definition than by a shared relaxation of spatial homogeneity.

2. Structured subclasses of non-uniformity

Much of the rigorous theory concerns classes that are non-uniform but still structured enough to admit finite-memory or algebraic analysis. One important class is that of local perturbations of CA: the local-rule configuration $A$8 is asymptotically constant, meaning there exists a constant configuration $A$9 and a finite set $s\in S^G$0 such that $s\in S^G$1. In this regime the automaton differs from a background CA only on finitely many sites, and this finite defect structure underlies several surjunctivity and invertibility results (Phung, 30 Mar 2025, Phung, 2024). A larger class consists of NUCA with uniformly bounded singularity, where for every finite $s\in S^G$2 with $s\in S^G$3, there exists a finite $s\in S^G$4 such that $s\in S^G$5 is constant. Here the non-uniformity may be global, but finite regions can still be surrounded by uniform shells (Phung, 30 Mar 2025, Phung, 2024).

A second major subclass is linear NUCA. With alphabet $s\in S^G$6 a finite-dimensional vector space over a field $s\in S^G$7, a NUCA is linear when all local maps are linear. The paper on noisy linear automata isolates

$s\in S^G$8

the class of linear NUCA with finite memory and asymptotically constant local defining maps, i.e. local perturbations of linear CA. This class admits a precise ring-theoretic model via a twisted group ring $s\in S^G$9, and finite-memory linear NUCA with finite non-uniformity become the dynamical realization of matrix rings over $S=A^{A^M}$0 (Phung, 2022). Related work also studies sparse global perturbations, in which infinitely many defects are allowed provided they occur in well-separated clusters (Phung, 2022).

The literature also contains operationally motivated subclasses. In reservoir computing, “quasi-uniform” CA denotes a piecewise-uniform one-dimensional binary reservoir split into two spatial regions using different rules, with interaction at the shared boundaries and under wrap-around topology (Nichele et al., 2017). In temporally non-uniform cellular automata, the non-uniformity is not spatial but temporal: a global rule sequence $S=A^{A^M}$1 alternates between rules $S=A^{A^M}$2 and $S=A^{A^M}$3, yielding a temporal analogue of NUCA that is formally distinct from the spatial models (Paul et al., 2024). At a more geometric extreme, higher-order NUCA on translationally invariant Euclidean and hyperbolic lattices use position-dependent update rules induced by lattice deformation; here non-uniformity is geometry-controlled rather than defect-like (Huang et al., 13 May 2026).

3. Dynamical properties, reversibility, and surjunctivity

The dynamical vocabulary of NUCA extends the CA notions of injectivity, surjectivity, pre-injectivity, and post-surjectivity, but stable versions become indispensable. In the group-universe formulation, two configurations are asymptotic if they agree outside a finite set; pre-injectivity forbids collisions between distinct asymptotic configurations, while post-surjectivity requires asymptotic liftability of asymptotic perturbations of an image. Because the local-rule configuration itself has a shift orbit closure $S=A^{A^M}$4, one defines stable injectivity and stable post-surjectivity by requiring the corresponding property for every $S=A^{A^M}$5 (Phung, 30 Mar 2025, Phung, 2022). This orbitwise strengthening has no analogue in uniform CA because constant rule fields have trivial orbit closure.

Several classical implications fail for general NUCA. Injectivity need not imply surjectivity, and even reversibility and invertibility separate: for finite-memory NUCA, a map may be injective and even reversible without being surjective, while in other examples bijectivity fails to imply reversibility in the finite-memory sense (Phung, 30 Mar 2025, Phung, 2022). In topological dynamics, the uniform-CA implication “transitive $S=A^{A^M}$6 sensitive” also fails: a two-dimensional NUCA is constructed that is minimal, hence transitive, but not sensitive, via an odometer-based nearly uniform rule distribution in which only the first cell uses a different local rule (Kamilya et al., 21 May 2026). These failures are not peripheral; they mark the loss of the rigid equivariant structure of classical CA.

Controlled non-uniformity restores part of the classical picture. For local perturbations and, more generally, finite-memory NUCA with uniformly bounded singularity over countable surjunctive or finitely generated sofic groups, stable injectivity implies stable invertibility; dual results hold for stable post-surjectivity (Phung, 30 Mar 2025, Phung, 2024). In a different direction, the Garden of Eden theorem is recovered when the rule distribution is recurrent enough: if the distribution of local rules is uniformly recurrent, or recurrent in the one-dimensional case, then

$S=A^{A^M}$7

whereas every non-recurrent one-dimensional template admits a substitution of local rules producing a NUCA that violates either Moore or Myhill direction (Paturi et al., 9 Jul 2025). This sharply localizes the obstruction: unique or non-repeating rule patterns can destroy Garden-of-Eden behavior, while repetitive heterogeneity preserves it.

For asynchronous non-uniform cellular automata, stable notions become even more central. Over a countable group with finite alphabet and finite memory, reversibility, stable reversibility, and stable injectivity are equivalent. Thus stable injectivity, not ordinary injectivity, is the correct finite-memory reversibility criterion in that setting (Phung, 2022). For linear NUCA, duality adds another layer: pre-injectivity, injectivity, stable injectivity, and invertibility of a linear NUCA correspond respectively to surjectivity, post-surjectivity, stable post-surjectivity, and invertibility of the dual linear NUCA, while bijectivity is no longer self-dual as it is for linear CA (Phung, 2022).

4. Algorithmic, combinatorial, and algebraic methods

One major line of work studies which rule distributions produce a desired global property. For fixed-radius one-dimensional NUCA, partial transition functions

$S=A^{A^M}$8

encode finite windows of the global map. A variant of the De Bruijn graph recognizes the language

$S=A^{A^M}$9

leading to the result that the set $\sigma_s(x)(g)=s(g)\big((g^{-1}x)|_M\big),$0 of surjective rule distributions is a sofic subshift. Injectivity is handled with a product graph $\sigma_s(x)(g)=s(g)\big((g^{-1}x)|_M\big),$1, and the corresponding set $\sigma_s(x)(g)=s(g)\big((g^{-1}x)|_M\big),$2 is a $\sigma_s(x)(g)=s(g)\big((g^{-1}x)|_M\big),$3-rational language (Provillard et al., 2011). For $\sigma_s(x)(g)=s(g)\big((g^{-1}x)|_M\big),$4-NUCA, generalized De Bruijn and reduced product graphs yield decidability of surjectivity and injectivity (Dennunzio et al., 2011).

Finite one-dimensional non-uniform elementary CA support especially detailed combinatorial tools. The reachability tree organizes valid Rule Min Terms (RMTs) level by level and turns the configuration reachability problem $\sigma_s(x)(g)=s(g)\big((g^{-1}x)|_M\big),$5 into a path-existence problem between destination and source edges. The resulting CREP algorithm is exponential in worst-case time and space, but the paper reports average explored-tree size behaving like $\sigma_s(x)(g)=s(g)\big((g^{-1}x)|_M\big),$6 (Adak et al., 2019). For number conservation under periodic boundary conditions, reachability-tree weights on RMTs characterize when the number of $\sigma_s(x)(g)=s(g)\big((g^{-1}x)|_M\big),$7s is preserved by every global update. This yields an $\sigma_s(x)(g)=s(g)\big((g^{-1}x)|_M\big),$8-time decision algorithm and an $\sigma_s(x)(g)=s(g)\big((g^{-1}x)|_M\big),$9-time synthesis procedure for $s$0-cell non-uniform number-conserving CA; for $s$1, only nine Wolfram rules can participate: $s$2 Even then, admissible local rules must be arranged compatibly in the rule vector (Hazari et al., 2016).

The algebraic theory is strongest for linear NUCA. Finite-memory linear NUCA with asymptotically constant local rules form a class $s$3 in which stable injectivity is equivalent to left-invertibility, and a twisted ring

$s$4

with twisted multiplication is constructed so that $s$5 for infinite $s$6. This identifies surjunctivity-type questions for noisy linear automata with stable finiteness of $s$7, and yields equivalences between $s$8-surjunctivity, dual $s$9-surjunctivity, and stable finiteness of the twisted group ring (Phung, 2022). A separate line proves that for finite-memory NUCA with multiple local transition rules, pointwise nilpotency, pointwise periodicity, and pointwise eventual periodicity are equivalent to their global counterparts; for linear NUCA, even a pointwise polynomial identity $$\mathcal{R}=\langle \mathcal{R}_0,\mathcal{R}_1,\ldots,\mathcal{R}_{n-1}\rangle,$$0 globalizes, and in the finite-vector-space case it forces eventual periodicity (Phung, 2022). This suggests that non-uniformity does not automatically destroy operator-level rigidity when finite memory and linear structure are present.

5. Computational realizations and application domains

A recent computational development shows that spatially non-uniform binary nearest-neighbor automata can be embedded exactly into mainstream CNN software. For a finite one-dimensional NUCA with rule allocation $$\mathcal{R}=\langle \mathcal{R}_0,\mathcal{R}_1,\ldots,\mathcal{R}_{n-1}\rangle,$$1, one global step is implemented in TensorFlow by: a fixed width-$$\mathcal{R}=\langle \mathcal{R}_0,\mathcal{R}_1,\ldots,\mathcal{R}_{n-1}\rangle,$$2 convolution with kernel $$\mathcal{R}=\langle \mathcal{R}_0,\mathcal{R}_1,\ldots,\mathcal{R}_{n-1}\rangle,$$3 to encode each neighborhood as an integer in $$\mathcal{R}=\langle \mathcal{R}_0,\mathcal{R}_1,\ldots,\mathcal{R}_{n-1}\rangle,$$4; one-hot expansion into eight channels; rulewise $$\mathcal{R}=\langle \mathcal{R}_0,\mathcal{R}_1,\ldots,\mathcal{R}_{n-1}\rangle,$$5 convolutions using the 8-bit truth tables of the candidate elementary rules; and a position-dependent selection stage realized either as a locally connected layer or a sparse dense layer. The method is an exact emulator rather than a learned predictor, and the reported practical benefit is that TensorFlow’s batched and parallel local computation becomes advantageous when many samples and many cells are simulated (Rollier et al., 2024).

NUCA also appear as computational substrates. In reservoir computing, quasi-uniform binary CA reservoirs with two loosely coupled rule regions were evaluated on the 5-bit memory task. Strong combinations such as $$\mathcal{R}=\langle \mathcal{R}_0,\mathcal{R}_1,\ldots,\mathcal{R}_{n-1}\rangle,$$6, $$\mathcal{R}=\langle \mathcal{R}_0,\mathcal{R}_1,\ldots,\mathcal{R}_{n-1}\rangle,$$7, and $$\mathcal{R}=\langle \mathcal{R}_0,\mathcal{R}_1,\ldots,\mathcal{R}_{n-1}\rangle,$$8 achieve perfect success in all tested settings, while pairs such as $$\mathcal{R}=\langle \mathcal{R}_0,\mathcal{R}_1,\ldots,\mathcal{R}_{n-1}\rangle,$$9, $y_i=f_i(x_{i-1},x_i,x_{i+1}).$0, and $y_i=f_i(x_{i-1},x_i,x_{i+1}).$1 fail almost completely. The experiments suggest that rule compatibility, not merely single-rule quality, governs performance, and that boundary interactions can either support or suppress reservoir capacity (Nichele et al., 2017). Temporally non-uniform CA extend this design axis further by varying the global rule sequence over time; this setting motivates notions such as restricted surjectivity, restricted reversibility, and weak reversibility, because reachability can depend on the temporal schedule even when the constituent rules are individually irreversible (Paul et al., 2024).

Application-oriented survey work reports extensive use of non-uniform elementary CA in VLSI testing, pseudo-random pattern generation, classifier design, compression, cryptography, image processing, and biological sequence problems. In linear/additive finite settings, matrix algebra with

$y_i=f_i(x_{i-1},x_i,x_{i+1}).$2

is the dominant tool; in non-linear finite settings, reachability trees and related graph constructions are the main characterization devices (Bhattacharjee et al., 2016). More recently, geometry-induced higher-order NUCA on hyperbolic lattices have been used to generate subsystem symmetry-protected topological states, spontaneous subsystem symmetry-breaking states, non-uniform Clifford QCA structures, and probabilistic directed-percolation processes on the hyperbolic $y_i=f_i(x_{i-1},x_i,x_{i+1}).$3 lattice. There the non-uniform rule field is not an arbitrary perturbation but a computational encoding of lattice deformation and non-Abelian translation symmetry (Huang et al., 13 May 2026).

6. Limitations, misconceptions, and open directions

A recurring misconception is that NUCA are merely CA with “a few different rules.” In fact, the literature spans objects as broad as all continuous self-maps of $y_i=f_i(x_{i-1},x_i,x_{i+1}).$4 in one formalization, group-universe finite-memory maps $y_i=f_i(x_{i-1},x_i,x_{i+1}).$5, local perturbations of CA, globally distributed but bounded-singularity rule fields, quasi-uniform piecewise reservoirs, temporal rule sequences, and geometry-induced higher-order constructions (Dennunzio et al., 2011, Phung, 30 Mar 2025, Huang et al., 13 May 2026). Results almost always apply to a specific subclass, not to arbitrary NUCA. The sharpest theorems in surjunctivity, direct finiteness, or decidability are correspondingly conditional on linearity, recurrence, asymptotic constancy, bounded singularity, or finite-dimensional alphabets.

Another misconception is that CA theorems routinely transfer to NUCA. They do not. Injectivity need not imply surjectivity; bijectivity need not imply finite-memory reversibility; transitivity need not imply sensitivity; and rulewise admissibility need not ensure global compatibility. Where classical behavior reappears, it does so for identifiable structural reasons such as recurrent rule distributions, stable injectivity, or finite perturbation of a surjunctive background (Paturi et al., 9 Jul 2025, Phung, 2022, Kamilya et al., 21 May 2026). This suggests that “non-uniformity” is not a single perturbative parameter but a collection of mechanisms that change the ambient dynamical category.

Open directions identified in the literature are correspondingly diverse. Reservoir-computing work proposes more-than-two-rule systems, dynamical descriptors such as Langton’s $y_i=f_i(x_{i-1},x_i,x_{i+1}).$6 and Lyapunov exponent, and evolutionary search over the combinatorially large rule space (Nichele et al., 2017). Temporally non-uniform CA motivate algorithmic classification of restricted reversibility and weak reversibility, as well as extensions from two rules to larger temporal alphabets (Paul et al., 2024). Hyperbolic-lattice NUCA raise the problem of classifying which subsystem symmetries arise from linear NUCA, while current translation-invariance criteria are only sufficient and the CQCA treatment remains constructive rather than classificatory (Huang et al., 13 May 2026). More generally, the accumulated record suggests that the central challenge of NUCA theory is to determine which forms of heterogeneity preserve the rigid algebraic and symbolic properties of classical CA, and which create genuinely new dynamical regimes.