Rule 184¶
A synchronous binary cellular-automaton rule whose local update moves each unblocked occupied site one step right while conserving occupancy on a ring.
Core Idea¶
Rule 184 is the exact synchronous binary radius-one cellular-automaton update f(a,b,c)=(b AND c) OR (a AND NOT b). For neighborhoods 111 through 000 its outputs are 10111000, or decimal 184. When 1 means an occupied site and 0 a gap, each unblocked 1 moves one site right in a parallel step; on a periodic ring the 1-count is conserved. Traffic and surface growth are readings of this specified rule, not its definition.[ref-c3bc52858a35][ref-a4d902ac34f2]
Scope of Application¶
The rule concerns binary states on a one-dimensional lattice, a fixed local table, and simultaneous iteration. It is used as a deterministic vehicle-gap model and can be mapped, within stated image limits, to ballistic-annihilation and surface-growth shape dynamics. It is not an empirical law for every road or a probabilistic hopping process.[ref-c3bc52858a35][ref-a4d902ac34f2]
Clarity¶
A verbal instruction to “move cars” can hide update order. Rule 184 uses the old time slice for every site's new state: an occupied site facing a gap moves right, while one facing another occupied site remains. Replacing even one output bit or updating sequentially changes the automaton.[^ref-c3bc52858a35]
Manages Complexity¶
The fixed local table makes global motion analyzable without individual driver rules. Count conservation on a ring is distinct from motion and flux. In Belitsky and Ferrari's recoding, ζ(i)=1−η(i)−η(i+1) gives signed defect dynamics, then successive ζ values define surface slopes. The transfer is limited to image states and shape up to vertical offset; deposited grains and annihilating particles are not the original conserved 1-cells.[ref-c3bc52858a35][ref-a4d902ac34f2]
Abstract Reasoning¶
To identify Rule 184, verify all eight outputs and simultaneous application. For a traffic reading, state the boundary conditions before claiming count conservation. For a recoded reading, check the map's domain before transferring a result: the annihilation map is not onto all annihilation configurations, and the two alternating bi-infinite binary states both map to ζ≡0, so it is not globally one-to-one on that domain even though the cited paper's Remark 1.1 uses that word.[ref-c3bc52858a35][ref-a4d902ac34f2]
Knowledge Transfer¶
The same update can be read as rightward vehicle exclusion or, after an explicit signed-defect and slope mapping, as deterministic surface-shape change. The shared content is the mapped update, not a literal identity between a vehicle, a defect, and a deposited grain. The model flux J(ρ)=min(ρ,1−ρ) is supported for extremal translation-invariant invariant measures; it is not a formula for arbitrary transient states, stationary mixtures, or measured road traffic.[^ref-a4d902ac34f2]
Example¶
Traffic: on a periodic binary road lattice, 1 marks a vehicle and 0 a gap. Every vehicle facing 0 moves one site right at the same step; a blocked vehicle stays. The vehicle count remains fixed, though positions and flow can change.[^ref-c3bc52858a35]
Surface shape: starting from a Rule 184 state η, form ζ(i)=1−η(i)−η(i+1) and integrate ζ as neighboring slopes. Belitsky and Ferrari's intertwined dynamics change local minima of the corresponding piecewise-linear surface. The correspondence ignores absolute vertical height and covers only the recoding's image, so surface deposition is not conservation of the original 1-cells.[^ref-a4d902ac34f2]
Relationships to Other Abstractions¶
Current abstraction Rule 184 Domain-specific
Parents (1) — more general patterns this builds on
-
Rule 184 is a kind of Automaton Domain-specific
Rule 184 is a specified cellular automaton with binary lattice states, local transitions, and synchronous runs.
Hierarchy paths (2) — routes to 2 parentless roots
- Rule 184 → Automaton → Abstract Machine → Formal System → Formalization → Representation → Abstraction
Neighborhood in Abstraction Space¶
Rule 184 sits in a sparse region of the domain-specific corpus (66th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Condensed Matter & Many-Body Physics (16 abstractions)
Nearest neighbors
- Particle in a one-dimensional lattice — 0.84
- Dual lattice — 0.84
- Haefliger structure — 0.84
- Lattice Model (Physics) — 0.84
- Ewald summation — 0.84
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
Rule 184 is a strict kind of the live Automaton identity: it specifies formal states, a local transition, and an iterative run, while adding this particular binary table. A microscopic traffic-flow model is one use rather than an all-instance parent; a quantum cellular automaton has different carrier and evolution conditions. Fukś's perfect density-classification construction uses Rule 184 and Rule 232, not Rule 184 alone. Conservation, a traffic image, or a surface-growth metaphor cannot replace the exact truth table.[ref-c3bc52858a35][ref-a4d902ac34f2]
References¶
[^ref-c3bc52858a35]: Henryk Fukś, Solution of the Density Classification Problem with Two Cellular Automata Rules (1997), section II, Rule 184 table and traffic interpretation; abstract and section I for the Rule 184 plus Rule 232 classifier. arXiv:comp-gas/9703001. https://arxiv.org/pdf/comp-gas/9703001
[^ref-a4d902ac34f2]: Vladimir Belitsky and Pablo A. Ferrari, Invariant Measures and Convergence for Cellular Automaton 184 and Related Processes (1998), introduction, Eq. (1.1), Assertions 1.1–1.2, Eqs. (1.3)–(1.4), Remarks 1.1–1.2, Theorem 2.2 proof (printed p. 7) for the alternating-state exception, and §2 extremal invariant-measure flux discussion (printed pp. 10–11). arXiv:math/9811103. https://arxiv.org/pdf/math/9811103