But while it is known that many systems in nature are made up of discrete elements, it is still almost universally believed that there are some things that are fundamentally continuous—notably positions in space and values of quantum mechanical probability amplitudes.

And so, for example, in the rule 30 pattern below one can tell whether a cell at a given position has any chance of not being white just by doing a An example of a pattern where it is difficult to compute directly the color of a particular cell.

very short computation that tests whether that position lies outside the center triangular region of the pattern.

where we explicitly set up the color of each cell to be determined by some short computation from the numbers that represent its position.

The translation from the Turing machine problem is achieved by representing the behavior of the Turing machine by saying which of a sequence of elementary statements are true about it at each step: whether the head is in one state or another, whether the cell under the head is black or white, and whether the head is at each of the possible positions it can be in.

Implementation [of network systems] The nodes in a network system can conveniently be labelled by numbers 1 , 2 , … n , and the network obtained at a particular step can be represented by a list of pairs, where the pair at position i gives the numbers corresponding to the nodes reached by following the above and below connections from node i . … 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.

The inverse rule, corresponding to multiplication by 1/m , can be obtained by applying the rule for multiplication by the integer k q /m , then shifting right by q positions.

Implementation [of register machines] The state of a register machine at a particular step can be represented by the pair {n, list} , where n gives the position in the program of current instruction being executed (the "program counter") and list gives the values of the registers.

Instead, as discussed on page 940 , a PDE is essentially just a constraint on the values of a function at different times or different positions.

The rules for the multiway system can then be given for example as {"AAB"  "BB", "BA"  "ABB"} The evolution of the system is given by the functions MWStep[rule_List, slist_List] := Union[Flatten[ Map[Function[s, Map[MWStep1[#, s] &, rule]], slist]]] MWStep1[p_String  q_String, s_String] := Map[StringReplacePart[s, q, #] &, StringPosition[s, p]] MWEvolveList[rule_, init_List, t_Integer] := NestList[MWStep[rule, #] &, init, t] An alternative approach uses lists instead of strings, and in effect works by tracing the internal steps that Mathematica goes through in trying out possible matchings.