B/S Notation: The Language of Life-Like Rules

Conway’s Life is written B3/S23.

That string of characters — five characters, a slash, two clusters of numbers — is a complete specification of one of the most famous mathematical systems ever created. It tells you everything you need to know to run the universe of Life: what makes a dead cell spring to life, and what keeps a living cell alive.

More than that, it is the first element in a language — a notation system that describes not just Conway’s rules but an entire family of universes. Every universe in this family is defined by a B/S string. The full family contains 262,144 members. Conway’s Life is one of them.

This page explains how to read and write that language, and what it reveals about the structure of rule space.

The Two Questions Every Rule Must Answer

Every cellular automaton of the Life-like family governs cells that are in one of two states — alive or dead — on a square grid where each cell has eight neighbors (the Moore neighborhood). At each step, every cell simultaneously updates its state according to the rule.

To define such a rule completely, you need to answer exactly two questions:

1. Birth: For each possible number of live neighbors (0 through 8), does a dead cell come to life?
2. Survival: For each possible number of live neighbors (0 through 8), does a live cell survive?

That’s all. The complete behavior of any life-like universe is determined by these two lists.

Birth (B)

A dead cell with exactly n live neighbors is born (becomes alive in the next generation) if n appears in the birth list.

Survival (S)

A live cell with exactly n live neighbors survives (remains alive in the next generation) if n appears in the survival list.

If a value is not in the birth list, a dead cell with that many live neighbors stays dead. If a value is not in the survival list, a live cell with that many live neighbors dies.

Reading B/S Notation

The notation B{birth}/S{survival} — where {birth} and {survival} are lists of neighbor counts — encodes exactly this.

Conway’s Life: B3/S23

• B3: A dead cell is born if it has exactly 3 live neighbors.
• S23: A live cell survives if it has exactly 2 or 3 live neighbors.

That’s the entire Game of Life, encoded in five characters.

More examples:

Seeds (B2/S) has an empty survival list — no live cell ever survives. Every live cell dies immediately; the only action is birth. This produces explosive, never-stabilizing growth from almost any starting configuration.

Life Without Death (B3/S012345678) has the full set in its survival list — every live cell always survives, no matter how many neighbors it has. Once a cell is alive, it stays alive forever. The grid can only grow, never shrink.

The Numbers Are Not Arbitrary

The digits in a B/S string are not ordered by convention — they are a direct encoding of neighbor counts. The digit 0 means “zero live neighbors,” 1 means “one live neighbor,” and so on up to 8.

Conventions:

• Digits within each list are written in ascending order (B36, not B63)
• The B and S sections are separated by a forward slash
• An empty section is written with no digits: B2/S (Seeds) has an empty survival section
• Some writers use the Golly notation: B3/S23 is equivalent to B3 S23 or 3/23 in older texts

How Many Rules Are There?

Each of the two lists (birth and survival) is a subset of {0, 1, 2, 3, 4, 5, 6, 7, 8}. A set with 9 elements has 2⁹ = 512 possible subsets. So:

• Birth: 512 possible lists
• Survival: 512 possible lists
• Total: 512 × 512 = 262,144 possible life-like rules

This is the full life-like rule space — a discrete space with 262,144 points, each representing a distinct universe with its own dynamics.

Conway’s Life occupies one point in this space. Most points produce boring universes: rules where everything immediately dies, or everything immediately solidifies into a static mass. A small fraction produce interesting behavior. An even smaller fraction produce behavior as rich as Life.

Finding those interesting points is, in part, what the study of life-like CA is about.

What Makes a Rule Interesting?

Not all 262,144 rules are equally interesting. Researchers have identified several properties that tend to characterize “interesting” rules:

Balance between birth and survival. If birth conditions are much more aggressive than survival conditions, the rule tends to explode — patterns grow without bound, filling the grid. If survival conditions are much more permissive than birth conditions, the grid tends to freeze — everything alive stays alive, nothing new is born. Interesting rules tend to balance expansion and consolidation.

Non-trivial steady states. Interesting rules tend to produce stable configurations that are more complex than “all dead” or “all alive.” If the only stable state is the empty grid, the rule is technically interesting but practically monotonous.

Sensitive dependence on initial conditions. In the most interesting rules, small changes to the initial configuration produce dramatically different long-term behavior. This is what makes Conway’s Life — and rules near it in rule space — essentially unpredictable without simulation.

The “glider criterion.” Conway set a specific target when designing his rules: he wanted a rule that could support patterns that moved, patterns that grew, and patterns that stabilized. The existence of spaceships (especially gliders) is a strong indicator of richness. Rules that support spaceships tend to support other complex behavior as well.

Life in Rule Space: Neighborhoods

The rule space has a natural geometry. Two rules are “close” if their B/S strings differ in only a few digits. Conway’s Life (B3/S23) and HighLife (B36/S23) differ in exactly one position: HighLife adds 6 to the birth list. They produce qualitatively similar universes with one important difference: HighLife supports a replicator, Life does not.

This suggests that the interesting region of rule space is not isolated — it has neighborhoods. Rules near Conway’s Life tend to produce Life-like dynamics (sparse, complex, with gliders and oscillators). Rules far from it tend to produce qualitatively different dynamics.

Mapping these neighborhoods — understanding which regions of rule space produce which types of behavior — is an ongoing research project. Some regions have been systematically surveyed. Most of the space remains unexplored.

Outer Totalistic vs. Inner Totalistic

The B/S notation describes what are technically called outer totalistic rules: the fate of a cell depends only on its current state (alive or dead) and the total count of its live neighbors, not on the arrangement of those neighbors.

This is a significant constraint. A cell with three neighbors in a diagonal line is treated identically to a cell with three neighbors clustered in an L-shape — in both cases, the rule sees “3 neighbors” and applies the same birth or survival probability.

Inner totalistic rules relax this constraint. They distinguish between different configurations of the same number of neighbors — for example, treating “3 neighbors arranged in a row” differently from “3 neighbors arranged in an L.” Inner totalistic rules define a much larger space (the space grows exponentially with the number of distinguishable configurations), and most of that space is unexplored.

The distinction matters: the richness of outer totalistic Life-like rules suggests that inner totalistic rules — which can distinguish finer-grained patterns — might be even richer. The tradeoff is complexity: inner totalistic rules are harder to specify, harder to analyze, and harder to implement efficiently.

Beyond Binary: More States, More Neighborhoods

B/S notation as described here applies specifically to:

• Two-state cells (alive/dead)
• Square grid
• Moore neighborhood (8 neighbors)

Extensions exist:

• Generations rules add a “dying” state — live cells don’t die immediately but pass through an intermediate state, leaving “traces” on the grid. This produces beautifully different dynamics.
• Von Neumann neighborhood rules use only the 4 orthogonal neighbors, producing more angular, cross-shaped dynamics.
• Larger neighborhoods (r=2, r=3…) examine more distant cells, allowing for larger-scale coordination and the possibility of high-speed spaceships.
• Multiple states allow for more complex local interactions, approaching reaction-diffusion chemistry.

Explore the variants →

Writing Your Own Rule

To define a new rule:

1. Choose which neighbor counts (0–8) trigger birth in a dead cell
2. Choose which neighbor counts (0–8) allow survival of a live cell
3. Write them as B{birth}/S{survival}

For example, suppose you want a rule where:

• Dead cells are born with 2, 3, or 4 neighbors
• Live cells survive with 3, 4, or 5 neighbors

That rule is: B234/S345

You can run this rule immediately in the sandbox → and observe what it does to a random initial configuration.

Most rules you invent will produce boring results — instant death or instant solidification. But occasionally, by accident or by careful analysis of the neighbor-count arithmetic, you will find something surprising: a universe with its own patterns, its own creatures, its own dynamics. This is the pleasure of exploring rule space.

A Note on Probabilistic Rules

The B/S notation describes deterministic rules: for a given neighbor count, birth or survival either definitely happens or definitely does not. This site also explores probabilistic rules, in which each neighbor count is associated with a probability rather than a boolean. Conway’s Life is equivalent to a probabilistic rule with all active probabilities set to 1.0.

Probabilistic rules introduce a richer vocabulary: a rule can specify not just whether birth happens at 3 neighbors but how likely it is — say, 70% probability. The deterministic rules are a special case. The probabilistic rules fill out a much larger, continuous space of behavior. Explore the sandbox →

Further Reading

• The Mathematics of Life →
• Beyond Conway: The Universe of Life-Like Rules →
• Universal Computation in a Two-State Grid →
• Try It: Live Sandbox →