Reversible mobile automata can for instance be constructed using Table[(IntegerDigits[i, 2, 3]  If[First[#]  0, {#, -1}, {Reverse[#], 1}]&)[IntegerDigits[perm 〚 i 〛 , 2, 3]], {i, 8}] where perm is an element of Permutations[Range[8]] .

However, the straightforward method for converting a t -digit number x to base k takes about t divisions, though this can be reduced to around Log[t] by using a recursive method such as FixedPoint[Flatten[Map[If[# < k, #, With[ {e = Ceiling[Log[k, #]/2]}, {Quotient[#, k e ], With[ {s = Mod[#, k e ]}, If[s  0, Table[0, {e}], {Table[0, {e - Floor[Log[k, s]] - 1}], s}]]}]] &, #]] &, {x}] The pictures below show stages in the computation of 3 20 (a) by a power tree in base 2 and (b) by conversion from base 3.

Concatenation sequences One can consider forming sequences by concatenating digits of successive integers in base k , as in Flatten[Table[IntegerDigits[i, k], {i, n}]] . … This is similar to picture (c) on page 131 , and is a digit-by-digit version of FoldList[Plus, 0, Table[Apply[Plus, 2 Rest[IntegerDigits[i, 2]] - 1], {i, n}]] Note that although the picture above has a nested structure, the original concatenation sequences are not nested, and so cannot be generated by substitution systems.

The table below gives the digit sequences for several rational numbers obtained by dividing pairs of whole numbers.

But as the table at the top of the next page shows, other square roots have much more complicated digit sequences.

But the point is that because of certain mathematical features of powers it turns out to be possible—as indicated in the table below—to find m t with many fewer than t operations; indeed, one or two operations for every base 2 digit in t is always for example sufficient.

With a list s of possible symbols, c[s, n] gives all possible expressions with LeafCount[expr]  n : c[s_, 1] = s; c[s_, n_] := Flatten[ Table[Outer[#1[#2] &, c[s, n - m], c[s, m]], {m, n - 1}]] There are a total of Binomial[2n - 2, n - 1] Length[s] n /n such expressions.

Finite groups and semigroups Any finite group or semigroup can be thought of as defined by having a multiplication table which satisfies the constraints given on page 887 .

A way to find candidates for q is to compute NullSpace[Table[With[{u = Table[Random[Integer, {0, k - 1}], {m}]}, BC[CAStep[u]] - BC[u]], {s}]] for progressively larger m and s , and to see what lists continue to appear.

Implementation [of operators from axioms] Given an axiom system in the form {f[a, f[a, a]]  a, f[a, b]  f[b, a]} one can find rule numbers for the operators f[x, y] with k values for each variable that are consistent with the axiom system by using Module[{c, v}, c = Apply[Function, {v = Union[Level[axioms, {-1}]], Apply[And, axioms]}]; Select[Range[0, k k 2 - 1], With[{u = IntegerDigits[#, k, k 2 ]}, Block[{f}, f[x_, y_] := u 〚 -1 - k x - y 〛 ; Array[c, Table[k, {Length[v]}], 0, And]]] &]] For k = 4 this involves checking nearly 16 4 or 4 billion cases, though many of these can often be avoided, for example by using analogs of the so-called Davis–Putnam rules.