How it works

One string folds into an operator that acts on another. A binary string of length N is read as a sqrt(N) x sqrt(N) matrix, applied in chunks to a second string with a threshold, producing a third; both reactants survive, so every string is potentially an operator on every other. Self-replicators and small closed ecologies appear unprompted. Because the species set is finite yet built by the rule rather than listed, its closure provably terminates - which makes it the reference test for constructive machinery.

S

binary strings of length N, N a perfect square, named by the integer they encode (s(5) = (1,0,1,0)); |S| = 2^N - 1 (the all-zero 'destructor' s(0) is excluded from reacting) · implicit

R

s1 + s2 -> s1 + s2 + s3, where s1 folds into a sqrt(N) x sqrt(N) operator P and acts on s2 in sqrt(N)-sized chunks: s3[i + k*sqrt(N)] = 1 iff sum_j P[i][j] * s2[j + k*sqrt(N)] > Theta. Both reactants survive (catalytic). A product equal to s(0) is an elastic collision. · arity 2

A

well-stirred-multiset, ode · each reaction removes a randomly drawn string s4, keeping M constant; in the ODE a non-selective flow Phi = sum_i x_i' keeps sum x = 1

Parameters

What to look for

• emergence of self-replicators (N=4: s(1), s(8), s(9), s(15); table 3.3)
• closure of the reachable species set (N=9 from s1..s15 closes at s1..s27 minus s20..s23, book 3.3)
• focusing of all mass into one self-replicator (fig. 3.3: s(1) or s(15) at N=4; 13.2.1: s(2)..s(7) all react to produce s(1))
• ecosystem-like attractors of several strings (fig. 3.4: {s1, s2, s4, s8} and {s1, s2, s3, s4, s5, s8, s10, s12, s15})
• organisations (N=4: 54, as the prose of 12.5.2 says; table 12.3 lists 52 of them, IDs 0..51)

Implementation decisions

• Threshold is strict: an output bit is 1 iff sum_j P_ij s_j > Theta. Book eq. 3.5 says >=, which with Theta=0 maps every zero sum to 1 and contradicts table 3.3 (operator s(1) on s(2) yields s(0)). The strict form reproduces tables 3.3 and 3.5, the N=4 counts, table 3.4 and the 3.3 closure exactly. The original paper [63] (eq. 6) uses >= with Theta fixed at 1, which for integer sums is the same as > 0, so the book's Theta = 0 is a misprint.
• Foldings for N > 4 (book defers to [65]): 2 = transpose, 3 = row snake (odd rows reversed), 4 = column snake (odd columns reversed). These reproduce table 3.4 exactly for N=9: 14/12028, 122/21310, 18/11822, 94/16830 self-replications/replications. [64] and [65] (Biological Cybernetics) are not openly available; the author's page offers them only by email.
• Organisations of the 4-bit system: table 12.3 lists 52 (IDs 0..51, including the empty and full sets); the prose of 12.5.2 says 54. The table's 52 entries are all correct, but the table is incomplete: since no reaction consumes anything, a set is an organisation under the dilution flow iff it is closed and every member is produced inside it, and that gives exactly 54, the 52 of the table plus {1,2,3,4,5,7,8,10,11,12,13,15} and {1,2,3,4,5,8,10,11,12,13,14,15} (tested). The same table has 122 closed sets including the empty one (the prose says '70+'). Table 12.2 is table 3.3 (first folding) with the destructor row/column added, although 12.5.2 calls it 'nontopological horizontal folding'.
• Bit order: s(5) = (1,0,1,0), so component s_1 is the least significant bit of the integer name. Self-replication counts i with s(i)+s(i) -> +s(i); replication counts ordered pairs i != j whose product is s(i) or s(j); the destructor s(0) is excluded and producing s(0) is an elastic collision.
• Reactions are multisets: operator s(a) on s(b) and operator s(b) on s(a) giving the same product are one reaction. Its mass-action constant is the number of ordered (operator, string) pairs that give it (1 or 2), and outflow is constant-total, so the mass-action ODE is exactly eq. 3.24 with W_ijk = 1 and Phi of eq. 3.25 (tested). The book gives no other rate constants.
• Default network: the closure from seed_species (the procedure of 3.3, via expand over ordered pairs), restricted to reactions whose product is in the found set. The initial state gives the seeds equal concentrations summing to 1, as in figs. 3.7-3.8 ('nearly equal initial distribution'). Defaults N = 9, seeds s(1)..s(15) give the book's 23-string closure.
• Soup mode follows the book's algorithm listing through chemart.soup with constant dilution: two distinct molecules are drawn as operator and string, and after a productive collision one random molecule is removed. The book draws s4 before inserting s3; here the removal happens after insertion, so the new string can itself be removed (an O(1/M) difference). The initial population is M split equally among the seeds.
• The decay of eqs. 3.11-3.12 (p = (I/N)^n) is dropped with its exponent n: the book's algorithm listing does not use it. In [63] it is STEP 7 (a string is replaced by a random string with probability p), destructors are replaced by random strings (STEP 6), and the ODE (eq. 16) has matching D_i and A(t) terms; the book's eq. 3.24 omits them. Replacement by a random string is not a reaction, so none of this is represented. The mutation variants of 3.4 are not modelled either.
• The prose of 3.3 lists the new strings made by s(10) on s(1)..s(15) as 's(16), s(17), s(17) s(19)' (garbled); table 3.7 row 10 gives s(16), s(17), s(18), s(19).

Sources

• Banzhaf, W. (1993). Self-replicating sequences of binary numbers. Computers and Mathematics with Applications 26:1-8 (book ref [63]); eq. 6 (threshold), the algorithm STEPs 1-8 and eqs. 16-19. http://www.cs.mun.ca/~banzhaf/papers/comp_appl93.ps.gz

Banzhaf & Yamamoto, Artificial Chemistries (2015), §chapter 3; revisited in 12.5.2, 13.2

Preview at default parameters

23 species · 375 reactions · status complete · seed 1

2 s1 -> 3 s1  [mass-action k=1.0]
s1 + s3 -> 2 s1 + s3  [mass-action k=2.0]
s1 + s5 -> 2 s1 + s5  [mass-action k=2.0]
s1 + s7 -> 2 s1 + s7  [mass-action k=2.0]
s1 + s8 -> s1 + 2 s8  [mass-action k=1.0]
s1 + s9 -> s1 + 2 s9  [mass-action k=1.0]
s1 + s10 -> s1 + s10 + s8  [mass-action k=1.0]
s1 + s11 -> s1 + s11 + s9  [mass-action k=1.0]
s1 + s12 -> s1 + s12 + s8  [mass-action k=1.0]
s1 + s13 -> s1 + s13 + s9  [mass-action k=1.0]
s1 + s14 -> s1 + s14 + s8  [mass-action k=1.0]
s1 + s15 -> s1 + s15 + s9  [mass-action k=1.0]
s1 + s17 -> 2 s1 + s17  [mass-action k=2.0]
s1 + s19 -> 2 s1 + s19  [mass-action k=2.0]
s1 + s24 -> s1 + s24 + s8  [mass-action k=1.0]
s1 + s25 -> s1 + s25 + s9  [mass-action k=1.0]
s1 + s26 -> s1 + s26 + s8  [mass-action k=1.0]
s1 + s27 -> s1 + s27 + s9  [mass-action k=1.0]
2 s2 -> 2 s2 + s1  [mass-action k=1.0]
s2 + s3 -> s2 + s3 + s1  [mass-action k=2.0]
s2 + s6 -> s2 + s6 + s1  [mass-action k=2.0]
s2 + s7 -> s2 + s7 + s1  [mass-action k=2.0]
s2 + s10 -> s2 + s10 + s1  [mass-action k=2.0]
s2 + s11 -> s2 + s11 + s1  [mass-action k=2.0]
s2 + s14 -> s2 + s14 + s1  [mass-action k=2.0]
s2 + s15 -> s2 + s15 + s1  [mass-action k=2.0]
s2 + s16 -> s2 + s16 + s8  [mass-action k=1.0]
s2 + s17 -> s2 + s17 + s8  [mass-action k=1.0]
s2 + s18 -> s2 + s18 + s9  [mass-action k=1.0]
s2 + s19 -> s2 + s19 + s9  [mass-action k=1.0]
s2 + s24 -> s2 + s24 + s8  [mass-action k=1.0]
s2 + s25 -> s2 + s25 + s8  [mass-action k=1.0]
s2 + s26 -> s2 + s26 + s9  [mass-action k=1.0]
s2 + s27 -> s2 + s27 + s9  [mass-action k=1.0]
2 s3 -> 2 s3 + s1  [mass-action k=1.0]
s3 + s5 -> s3 + s5 + s1  [mass-action k=2.0]
s3 + s6 -> s3 + s6 + s1  [mass-action k=2.0]
s3 + s7 -> s3 + s7 + s1  [mass-action k=2.0]
s3 + s8 -> s3 + 2 s8  [mass-action k=1.0]
s3 + s9 -> s3 + 2 s9  [mass-action k=1.0]
… 335 more

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