To get all the familiar properties of additivity one needs an addition operation that is associative ( Flat ) and commutative ( Orderless ), and has an identity element (white or 0 in the cases above)—so that it defines a commutative monoid. (Usually it is also convenient to be able to get all possible elements by combining a small number of basic generator elements.) … But in all cases the general results for associative rules on page 956 show that the patterns obtained must be at most nested.

Then in 1970 Roger Banks managed to show that the 2-state 5-neighbor symmetric 2D rule 4005091440 was able to reproduce all the same logical elements. … From the discoveries I have made, I have no doubt at all that the Game of Life is in the end universal, and indeed I believe that the kind of elaborate behavior needed to support various components is in fact good evidence for this.

(Even if one can do operations on all digits in parallel it still takes of order n steps in a system like a cellular automaton for the effects of different digits to mix together—though see also page 1149 .) … It is known how to evaluate π (see page 912 ) and all standard elementary functions to n -digit precision using about Log[n] m[n] operations.

For example, rule 90 on page 25 corresponds to f[1, _, 1] = 0; f[0, _, 1] = 1; f[1, _, 0] =1; f[0, _, 0] = 0 One can specify initial conditions for example by a[0, 0] = 1; a[0, _] = 0 (the cell on step 0 at position 0 has value 1, but all other cells on that step have value 0). … (For efficiency, the main definition should in practice be given as a[t_, i_] := a[t, i] = f[a[t - 1, i - 1], a[t - 1, i], a[t - 1, i + 1]] so that all intermediate values which are computed are automatically stored.)

With this setup, a network consisting of just one node is {{1, 1}} and a 1D array of n nodes can be obtained with CyclicNet[n_] := RotateRight[ Table[Mod[{i - 1, i + 1}, n] + 1, {i, n}]] With above connections represented as 1 and the below connections as 2 , the node reached by following a succession s of connections from node i is given by Follow[list_, i_, s_List] := Fold[list 〚 #1 〛 〚 #2 〛 &, i, s] The total number of distinct nodes reached by following all possible succession of connections up to length d is given by NeighborNumbers[list_, i_Integer, d_Integer] := Map[Length, NestList[Union[Flatten[list 〚 # 〛 ]] &, Union[list 〚 i 〛 ], d - 1]] For each such list the rules for the network system then specify how the connections from node i should be rerouted. … With rules set up in this way, each step in the evolution of a network system is given by NetEvolveStep[{depth_Integer, rule_List}, list_List] := Block[ {new = {}}, Join[Table[Map[NetEvolveStep1[#, list, i] &, Replace[NeighborNumbers[list, i, depth], rule]], {i, Length[list]}], new]] NetEvolveStep1[s : {___Integer}, list_, i_] := Follow[list, i, s] NetEvolveStep1[{s1 : {___Integer}, s2 : {___Integer}}, list_, i_] := Length[list] + Length[ AppendTo[new, {Follow[list, i, s1], Follow[list, i, s2]}]] The set of nodes that can be reached from node i is given by ConnectedNodes[list_, i_] := FixedPoint[Union[Flatten[{#, list 〚 # 〛 }]] &, {i}] and disconnected nodes can be removed using RenumberNodes[list_, seq_] := Map[Position[seq, #] 〚 1, 1 〛 &, list 〚 seq 〛 , {2}] The sequence of networks obtained on successive steps by applying the rules and then removing all nodes not connected to node number 1 is given by NetEvolveList[rule_, init_, t_Integer] := NestList[(RenumberNodes[#, ConnectedNodes[#, 1]] &)[ NetEvolveStep[rule, #]] &, init, t] Note that the nodes in each network are not necessarily numbered in the order that they appear on successive lines in the pictures in the main text.

Given p = Array[Prime, Length[list], PrimePi[Max[list]] + 1] or any list of integers that are all relatively prime and above Max[list] (the integers in list are assumed positive) CRT[list_, p_] := With[{m = Apply[Times, p]}, Mod[Apply[Plus, MapThread[#1 (m/#2)^EulerPhi[#2] &, {list, p}]], m]] yields a number x such that Mod[x, p]  list .

One approach is to use {1, 1} to indicate the boundary of each block, and then within each block to use all possible digit sequences which do not contain {1, 1} , as in the Fibonacci number system discussed on page 892 .

The axioms given can prove most of the results in an elementary geometry textbook—indeed all results that are about geometrical figures such as triangles and circles specified by a fixed finite number of points, but which do not involve concepts like area.

The basic axioms for this allow forms of operators corresponding to multiplication tables for all possible commutative groups (see note above ).

Essentially all computer languages support And , Or and Not as ways to combine logical statements; many support And , Or and Xor as bitwise operations.