If the rules for a one-element-dependence tag system are given in the form {2, {{0, 1}, {0, 1, 1}}} (compare page 1114 ), the initial conditions for the Turing machine are TagToMTM[{2, rule_}, init_] := With[{b = FoldList[Plus, 1, Map[Length, rule] + 1]}, Drop[Flatten[{Reverse[Flatten[{1, Map[{Map[ {1, 0, Table[0, {b 〚 # + 1 〛 }]} &, #], 1} &, rule], 1}]], 0, 0, Map[{Table[2, {b 〚 # + 1 〛 }], 3} &, init]}], -1]] surrounded by 0 's, with the head on the leftmost 2 , in state 1 .

If the last 2 axioms are dropped any statement can readily be proved true or false essentially just by running rule 110 for a finite number of steps equal to the number of nested ↓ plus 〈 … 〉 in the statement.

Note that the rules are set up so that a string for which there are no applicable replacements at a given step is simply dropped.

The number of states with spatial period m is given by s[m_, k_]:= k m - Apply[Plus, Map[s[#, k] &, Drop[Divisors[m], -1]]] or equivalently s[m_, k_]:=Apply[Plus, (MoebiusMu[m/#] k # &)[Divisors[m]]] In a cellular automaton with a total of n cells, the maximum possible repetition period is thus s[n, k] .

Various schemes are used in practice, and almost all of them are based on the idea from traditional mathematics that by viewing data in terms of numbers it becomes possible to decompose the data into sums of fixed basic forms—some of which can be dropped in order to achieve compression.

{Take[#, k], Drop[ #, k](-1)^(Range[0, k - 1]/k)}] &, Partition[##]]] &, BitReverseOrder[data], 2^Range[Log[2, n]]]/ √ n ] (See also page 1080 .)

I have dropped all honorifics or titles, except when they significantly alter a name.

Select[PM[s], Count[#, 1] > 1 &], 2]] while blocks of length n (and at most one error) can be decoded with Drop[(If[#  0, data, MapAt[1 - # &, data, #]] &)[ FromDigits[Mod[data .

A number of variations of the basic procedure—using different orderings and with different schemes for dropping redundant rules—have been proposed for systems arising in different kinds of applications.

Sequential substitution systems [from cellular automata] Given a sequential substitution system with rules in the form used on page 893 , the rules for a cellular automaton which emulates it can be obtained from SSSToCA[rules_] := Flatten[{{v[_, _, _], u, _}  u, {_, v[rn_, x_, _], u}  r[rn + 1, x], {_, v[_, x_, _], _}  x, MapIndexed[ With[{r n = #2 〚 1 〛 , rs = #1 〚 1 〛 , rr = #1 〚 2 〛 }, {If[Length[rs]  1, {u, r[rn, First[rs]], _}  q[0, rr], {u, r[rn, First[rs]], _}  v[rn, First[rs], Take[rs, 1]]], {u, r[rn, x_], _}  v[rn, x, {}], {v[rn, _, Drop[rs, -1]], Last[rs], _}  q[Length[rs] - 1, rr], Table[{v[rn, _, Flatten[{___, Take[rs, i - 1]}]], rs 〚 i 〛 , _}  v[ rn, rs 〚 i 〛 , Take[rs, i]], {i, Length[rs] - 1, 1, -1}], {v[rn, _, _], y_, _}  v[rn, y, {}]}] & , rules /. s  List], {_, q[0, {x__, _}], _}  q[0, {x}], {_, q[0, {x_}], _}  r[1, x], {_, q[0, {}], x_}  r[1, x], {_, q[_, {___, x_}], _}  x, {_, q[_, {}], x_}  x, {_, x_, q[0, _]}  x, {_, _, q[n_, {}]}  q[n - 1, {}], {_, _, q[n_, {x___, _}]}  q[n - 1, {x}], {q[_, {}], _, _}  w, {q[0, {__, x_}], p[y_, _], _}  p[x, y], {q[0, {__, x_}], y_, _}  p[x, y], {p[_, x_], p[y_, _], _}  p[x, y], {p[_, x_], u, _}  x, {p[_, x_], y_, _}  p[x, y], {_, p[x_, _], _}  x, {w, u, _}  u, {w, x_, _}  w, {_, w, x_}  x, {_, r[rn_, x_], _}  x, {_, u, r[_, _]}  u, {_, x_, r[rn_, _]}  r[rn, x], {_, x_, _}  x}] The initial condition is obtained by applying the rule s[x_, y__]  {r[1, x], y} and then padding with u 's.