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Given an initial condition {i, list, n} the initial condition for the 2-color Turing machine is With[{b = Ceiling[Log[2, k]]}, {i, Flatten[IntegerDigits[list, 2, b]], b n}]
But this is analogous to a digit sequence of a real number. And since any program for a universal system can be specified by an integer it follows that there must be many problems for which no such program can be given.
Such choices are particularly convenient on computers where machine integers are represented by 32 binary digits. … (Small values of a also lead to an excess of runs of identical digits, as mentioned on page 903 .) … In fact, the first known generator for digital computers was John von Neumann 's "middle square method" n  FromDigits[Take[IntegerDigits[n 2 , 10, 20], {5, 15}], 10] In practice this generator has too short a repetition period to be useful.
Viewed in terms of digit sequences, example (d) from page 83 was discussed by Georg Cantor in 1883 in connection with his investigations of the idea of continuity. General relations between digit sequences and sequences produced by neighbor-independent substitution systems were found in the 1960s. … As early as 1851, for example, Eugène Prouhet showed that if sequences of integers were partitioned according to sequence (b) on page 83 , then sums of powers of these integers would be equal: thus Apply[Plus, Flatten[Position[s, i]] k ] is equal for i = 0 and i = 1 if s is a sequence of the form (b) on page 83 with length 2 m , m > k .
Probably the simplest is a statement shown to be unprovable in Peano arithmetic by Laurence Kirby and Jeff Paris in 1982: that certain sequences g[n] defined by Reuben Goodstein in 1944 are of limited length for all n , where g[n_] := Map[First, NestWhileList[ {f[#] - 1, Last[#] + 1} &, {n, 3}, First[#] > 0 &]] f[{0, _}] = 0; f[{n_, k_}] := Apply[Plus, MapIndexed[#1 k^f[{#2 〚 1 〛 - 1, k}] &, Reverse[IntegerDigits[n, k - 1]]]] As in the pictures below, g[1] is {1, 0} , g[2] is {2, 2, 1, 0} and g[3] is {3, 3, 3, 2, 1, 0} . g[4] increases quadratically for a long time, with only element 3 × 2 402653211 - 2 finally being 0. … Every possible proof in Peano arithmetic can in principle be encoded as an ordinary integer. … In constructing g[n] the integer n is in effect treated like an ordinal number in Cantor normal form, and a sequence of numbers that should precede it are found.
An alternative formulation is to ask whether for all n FixedPoint[(3#/2^IntegerExponent[#, 2] + 1)/2 &, n]  2 With the rule n  If[EvenQ[n], 5n/2, (n + 1)/2] used in the main text, the sequence produced repeats if n ever reaches 2, 4 or 40 (and possibly higher numbers). … In case (b), the number of steps is equal to the number of base 2 digits in n , while in case (c) it is determined by the number of 1's in the base 2 digit sequence of n .
Nonlinear feedback shift registers Linear feedback shift registers of the kind discussed on page 974 can be generalized to allow any function f (note the slight analogy with cyclic tag systems): NLFSRStep[f_, taps_, list_] := Append[Rest[list], f[list 〚 taps 〛 ]] With the choice f=IntegerDigits[s, 2, 8] 〚 8 - # . {4, 2, 1} 〛 & and taps = {1, 2, 3} this is essentially a rule s elementary cellular automaton.
Fractal dimensions [of additive cellular automata] The total number of nonzero cells in the first t rows of the pattern generated by the evolution of an additive cellular automaton with k colors and weights w (see page 952 ) from a single initial 1 can be found using g[w_, k_, t_] := Apply[Plus, Sign[NestList[Mod[ ListCorrelate[w, #, {-1, 1}, 0], k] &, {1}, t - 1]], {0, 1}] The fractal dimension of this pattern is then given by the large m limit of Log[k,g[w, k,k m + 1 ]/g[w, k, k m ]] When k is prime it turns out that this can be computed as d[w_, k_:2] := Log[k,Max[Abs[Eigenvalues[With[ {s = Length[w] - 1}, Map[Function[u, Map[Count[u, #] &, #1]], Map[Flatten[Map[Partition[Take[#, k + s - 1], s, 1] &, NestList[Mod[ListConvolve[w, #], k] &, #, k - 1]], 1] &, Map[Flatten[Map[{Table[0, {k - 1}], #} &, Append[#, 0]]] &, #]]] &[Array[IntegerDigits[#, k, s] &, k s - 1]]]]]]] For rule 90 one gets d[{1, 0, 1}] = Log[2, 3] ≃ 1.58 .
In general, one condition for a solution to exist is that integer numbers of pairs can yield strings of the same length, so that given the length differences d = Map[StringLength, p, {2}] . {1, -1} there is a vector v of non-negative integers such that v . d  0 . … The undecidability of PCP can be seen to follow from the undecidability of the halting problem through the fact that the question of whether a tag system of the kind on page 93 with initial sequence s ever reaches a halting state (where none of its rules apply) is equivalent to the question of whether there is a way to satisfy the PCP constraint TSToPCP[{n_, rule_}, s_] := Map[Flatten[IntegerDigits[#, 2, 2]] &, Module[{f}, f[u_] := Flatten[Map[{1, #} &, 3u]]; Join[Map[{f[Last[#]], RotateLeft[f[First[#]]]} &, rule], {{f[s], {1}}}, Flatten[ Table[{{1, 2}, Append[RotateLeft[f[IntegerDigits[j, 2, i]]], 2]}, {i, 0, n - 1}, {j, 0, 2 i - 1}], 1]]], {2}] Any PCP constraint can also immediately be related to the evolution of a multiway tag system of the kind discussed in the note below.
The first n elements can be found efficiently using Module[{a = 1}, Table[First[IntegerDigits[ a, a = BitXor[a, BitOr[2a, 4a]]; 2, i]], {i, n}]] The sequence does not repeat in at least its first million steps, and I would amazed if it ever repeats, but as of now I know of no rigorous proof of this. ( Erica Jen showed in 1986 that no pair of columns can ever repeat, and the arguments on page 1087 suggest that neither can the center column together with occasional neighboring cells.)
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