Emergent Model - EM43 - Number Doubler

How it Works

EM-43 is a one-dimensional cellular automaton with a neighborhood of 3 and 4 possible cell states:

• 0 (blank), 1 (P – program), 2 (R – marker), and 3 (B – separator/halt).

Tape Structure: Program & Separator

The initial state of the automaton (the tape) is constructed as follows:

[program] BB 0^(n+1) R 0

The program is a sequence of fixed length (in this model: 32 cells) placed at the beginning of the tape.
During training, this program is searched/optimized. Together with the rules, it defines the behavious of the model.

The separator BB acts as a clear boundary between program and input.

The encoded input is placed after the separator: it consists of n+1 zeroes, followed by a red marker R, and a trailing 0.

This structure is critical: the automaton's computation unfolds starting from this initialized state, processing the interaction between the program, the input beacon, and the evolving cell dynamics.

Encoding the Input

To provide an input number n, the tape is initialized as:
[program] BB 0^(n+1) R 0
This creates a beacon, where the number of 0s before the R encodes the value n.

Decoding the Output

The automaton halts when blue (B) cells occupy ≥50% of the non-blank tape.
The rightmost R is located, and the output is decoded by:

output = position(R) − position(last B) − 2

This mirrors the encoding procedure.

Training

The system was trained using genetic algorithms to solve the task:

output = 2 × input

Only the rule table and initial state program were evolved.
Through this process, the automaton learned to perform emergent computation — solving the task without explicit programming.

Generalization

The model was trained exclusively on inputs from 1 to 30, but it discovered a general algorithm for doubling — not just memorizing examples.
It generalizes perfectly to all natural numbers (excluded n = 0), despite never having seen them during training.