User blog:Syst3ms/Using cellular automata to generate a potentially uncomputable function

Sup.

Formal definition of a cellular automaton
We define a cellular automaton \(A\) as a 6-tuple \((d,\Sigma,N,\delta,C,F)\) :
 * \(d\) is the dimension of the automaton
 * \(\Sigma\) is its alphabet (which we assume here to be an element of \(\mathcal{P}(\mathbb{Z})\))
 * \(N \subset \mathbb{Z}^d\) is its neighbourhood
 * \(\delta : Q^a \mapsto Q\) is the automaton's local rule, where \(a = |N|\), its arity
 * \(C: \mathbb{Z}^d \mapsto Q\) is the configuration function of the automaton
 * \(F: Q^{\mathbb{Z}^d} \mapsto Q^{\mathbb{Z}^d}\) is the global rule. It is defined as \(F(c) = z \mapsto \delta(c(z+v_1),\ldots,c(z+v_a))\) where \(V = \{v_1,\ldots,v_a\}\)

Related functions
We define \(\mathcal{A} = \mathbb{Z} \times \mathcal{P}(\mathbb{Z}) \times \mathbb{Z}^d \times Q^{Q^a} \times Q^{mathbb{Z}^d} \times {Q^{\mathbb{Z}^d}}^{Q^{\mathbb{Z}^d}}\) as the set of all possible CAs and \(\mathcal{C} = Q^{mathbb{Z}^d}\) as the set of all possible configurations.

For any given \(A \in \mathcal{A}\) and its initial configuration function \(C_I\), we define the generation function \(G_n : \mathcal{A} \mapsto \mathcal{A}\) recursively as such :

\(G_0(A) = A \\ G_{n+1}(A) = (A_d,A_Q,A_N,A_\delta,{A_F}({G_n(A)}_C),A_F)\)

Where \(A_d,A_Q,A_N,A_\delta,A_C,A_F\) refer to \(A\)'s respective elements. We refer to \(G^n(A)\) as "A at generation n"

We define the population \(|C|\) of a configuration \(C\) as \(|C| = |\{a \in \mathbb{Z}^d : C(a) > 0\}|\).

We then define the redundancy as the following predicate over \(\mathcal{A}\) : there exists a configuration \(C_M\) such that \(|A_C| > |C_M|\) and A_C and C_M are the same at two (possibly equal) generations.. We say A is redundant iff \(R(A)\).

We define the stability \(S(A)\) of a CA \(A\) as the smallest i such that A be the same at generations \(i\) and \(i+jk\) for some \(j\) and any \(k\).

We finally define the swift snake function \(\Lambda_A(n)\) of a CA \(A\) as the biggest stability reached with a non-redundant intial condition with a population of \(n\).

TODO : fix the redundancy to accept a configuration, write using set theory

Defining Conway's Game of Life and its lower bounds of \(\Lambda(n)\)
TODO