A369586 -id:A369586 - cp's OEIS frontend

A368946 Irregular triangle read by rows: row n lists the strings of the MIU formal system at the n-th level of the tree generated by recursively applying the system rules, starting from the MI string (see comments and example).

31, 310, 311, 31010, 3110, 31111, 310101010, 3110110, 311110, 311111111, 301, 310, 31010101010101010, 3110110110110, 31111011110, 3010, 3100, 3111111110, 31111111111111111, 3011111, 3101111, 3110111, 3111011, 3111101, 3111110, 3010, 30101, 31010

Offset: 0

Paolo Xausa, Jan 10 2024

In the MIU formal system, invented by Douglas R. Hofstadter, we start with the string (axiom) MI and build new strings (theorems) by applying the following rules.
Rule 1: if a string ends with I, we can append U at the end.
Rule 2: if a string is of the form Mx, we can build the string Mxx.
Rule 3: if a string contains III, we can replace it with U.
Rule 4: if a string contains UU, we can remove it.
Hofstadter asks whether the string MU can be obtained by repeatedly applying the above rules and then proves it cannot.
At each recursion level, strings are obtained by applying the rules in order (1 to 4) to each string of the previous level. When the same rule can be applied more than once, it is first applied to the leftmost substring that can be affected.
Following the mapping adopted by Hofstadter himself (see Hofstadter (1979), p. 261), we can encode the characters M, I, and U with 3, 1, and 0, respectively.
See A368953 for a variant where, within a row, duplicates (if present) are removed and encoded strings are lexicographically ordered.
A369173 gives all of the derivable strings within the MIU system.

			After recursively applying the rules three times, we get the following tree (cf. Hofstadter (1979), page 40, Figure 11).
.
                           MI
  0 ---------------------- 31
                         /    \
                        1      2 <--- Rule applied
                       /        \
                     MIU        MII
  1 ---------------- 310        311
                    /          /   \
                   2          1     2
                  /          /       \
              MIUIU       MIIU      MIIII
  2 --------- 31010       3110      31111         Here rule 3 can
               /          /       / |   | \       be applied twice:
              2          2       1  2   3  3 <--- it is first applied
             /          /       /   |   |   \     to the leftmost
        MIUIUIUIU   MIIUIIU  MIIIIU |  MUI  MIU   III substring
  3 --- 310101010   3110110  311110 |  301  310
                                MIIIIIIII
                                311111111
.
By reading the tree from top to bottom, left to right, the triangle begins:
  [0] 31;
  [1] 310 311;
  [2] 31010 3110 31111;
  [3] 310101010 3110110 311110 311111111 301 310;
  [4] 31010101010101010 3110110110110 ... 3010 ... 3010 30101 31010;
  ...
Please note that some strings may be duplicated in different rows and within the same row: within the first four rows, the string MIU (310) is present in rows 1 and 3. In row 4, the string MUIU (3010) is present twice: it can be generated both by applying rule 3 to MIIIIU (311110) and by applying rule 1 to MUI (301). The initial string MI (31) first reappears in row 5.

Douglas R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, 1979, pp. 33-41 and pp. 261-262.

Paolo Xausa, Table of n, a(n) for n = 0..3670 (rows 0..7 of the triangle, flattened).
Wikipedia, MU Puzzle.
Index entries for sequences from "Goedel, Escher, Bach"

Cf. A331536, A368953 (variant).
Cf. A368947 (row lengths), A369172 (length of strings), A369173 (all MIU strings), A369206 (number of zeros), A369207 (number of ones), A369409.
Cf. A369586 (shortest proofs), A369408 (length of shortest proofs), A369587 (number of symbols of shortest proofs).

MIUStepO[s_] := Flatten[Map[{If[StringEndsQ[#, "1"], # <> "0", Nothing], # <> StringDrop[#, 1], StringReplaceList[#, "111" -> "0"], StringReplaceList[#, "00" -> ""]}&, s]];
With[{rowmax = 4}, Map[FromDigits, NestList[MIUStepO, {"31"}, rowmax], {2}]]

from itertools import islice
def occurrence_swaps(w, s, t):
    out, oi = [], w.find(s)
    while oi != -1:
        out.append(w[:oi] + t + w[oi+len(s):])
        oi = w.find(s, oi+1)
    return out
def moves(w): # moves for word w in MIU system, encoded as 310
    nxt = []
    if w[-1] == '1': nxt.append(w + '0')        # Rule 1
    if w[0] == '3': nxt.append(w + w[1:])       # Rule 2
    nxt.extend(occurrence_swaps(w, '111', '0')) # Rule 3
    nxt.extend(occurrence_swaps(w, '00', ''))   # Rule 4
    return nxt
def agen(): # generator of terms
    frontier = ['31']
    while len(frontier) > 0:
        yield from [int(t) for t in frontier]
        reach1 = [m for p in frontier for m in moves(p)]
        frontier, reach1 = reach1, []
print(list(islice(agen(), 28))) # Michael S. Branicky, Jan 14 2024

A369173 Irregular triangle read by rows: row n lists all of the distinct derivable strings in the MIU formal system that are n characters long.

Original entry on oeis.org

31, 301, 310, 311, 3001, 3010, 3011, 3100, 3101, 3110, 30001, 30010, 30011, 30100, 30101, 30110, 31000, 31001, 31010, 31100, 31111, 300001, 300010, 300011, 300100, 300101, 300110, 301000, 301001, 301010, 301100, 301111, 310000, 310001, 310010, 310100, 310111, 311000, 311011, 311101, 311110, 311111

Offset: 2

Paolo Xausa, Jan 15 2024

See A368946 for the description of the MIU formal system.
A string S can be derived in the MIU formal system if and only if S contains just one M (as its first character) and an arbitrary number of I and U characters, where the number of I characters is not divisible by 3 (see Wikipedia link).
Strings are encoded using the map M -> 3, I -> 1 and U -> 0, and then sorted.
Row n has length A024495(n).

			Triangle begins:
  [2] 31;
  [3] 301 310 311;
  [4] 3001 3010 3011 3100 3101 3110;
  [5] 30001 30010 30011 30100 30101 30110 31000 31001 31010 31100 31111;
  ...

Douglas R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, 1979, pp. 33-41 and pp. 261-262.

Paolo Xausa, Table of n, a(n) for n = 2..10922 (rows 2..14 of the triangle, flattened).
Wikipedia, MU Puzzle.
Index entries for sequences from "Goedel, Escher, Bach".

Cf. A368946, A024495 (row lengths), A369174 (number of zeros), A369179 (number of ones), A369409.

Cf. A369586 (shortest proofs), A369408 (length of shortest proofs), A369587 (number of symbols of shortest proofs).

A369173row[n_] := Map[FromDigits[Join[{3}, #]]&, Select[Tuples[{0, 1}, n - 1], !Divisible[Count[#, 1], 3]&]]; Array[A369173row, 5, 2]

A369409 Irregular triangle read by rows: row n lists the lines of a "normal" proof (see comments) for the MIU formal system string (theorem) given by A369173(n+1).

Original entry on oeis.org

31, 31, 311, 31111, 301, 31, 311, 31111, 310, 31, 311, 31, 311, 31111, 311111111, 3111111110, 31111100, 311111, 31111111111, 311111111110, 3111111100, 31111111, 311101, 3001, 31, 311, 31111, 311111111, 3111111110, 31111100, 311111, 31111111111, 311111111110, 3111111100, 31111111

Offset: 1

Paolo Xausa, Jan 23 2024

See A368946 for the description of the MIU formal system.
Matos and Antunes (1998) define a "normal" proof for a string (theorem) S the output of the following algorithm, which generates the lines (strings) of the proof.
Step 1. Replace in turn every occurrence of U in S by III.
Step 2. Until we get MI:
Step 2a. If the number of I characters is even, remove half of them.
Step 2b. Otherwise, apply in turn the following transformations: append UU, replace the first U by III, remove the remaining U, remove half of the I characters.
Step 3. Reverse the order of the lines.
This algorithm generates a short proof, but not necessarily the shortest one.
Strings are encoded by mapping the characters M, I, and U to 3, 1, and 0, respectively.

			Triangle begins:
  [1] 31;
  [2] 31 311 31111 301;
  [3] 31 311 31111 310;
  [4] 31 311;
  [5] 31 311 31111 ... 31111100 ... 31111111111 ... 3111111100 31111111 311101 3001;
  ...
For the theorem MUI (301), which is given by A369173(3,1) or (after flattening) A369173(3), steps 1 and 2 of the algorithm generate the lines 301 -> 31111 -> 311 -> 31 which, when reversed (step 3), give row 2 of the present sequence.
For the theorem MIUU (3100), which is given by A369173(4,4) or (after flattening) A369173(9), steps 1 and 2 of the algorithm generate the lines 3100 -> 311110 -> 31111111 -> 3111111100 -> 311111111110 -> 31111111111 -> 311111 -> 31111100 -> 3111111110 -> 311111111 -> 31111 -> 311 -> 31 which, when reversed (step 3), give row 8 of the present sequence.

Douglas R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, 1979, pp. 33-41 and pp. 261-262.

Paolo Xausa, Table of n, a(n) for n = 1..10823 (rows 1..682 of the triangle, flattened).
Armando B. Matos and Luis Filipe Antunes, Short Proofs for MIU theorems, Technical Report Series DCC-98-01, University of Porto, 1998.
Wikipedia, MU Puzzle.
Index entries for sequences from "Goedel, Escher, Bach".

Cf. A368946, A369173, A369408, A369410 (row lengths), A369411, A369414.

Cf. A369586 (analog for shortest proofs).

MIUStrings[n_] := Map["3" <> FromCharacterCode[# + 48]&, Select[Tuples[{0, 1}, n - 1], !Divisible[Count[#, 1], 3]&]];
MIUProofLines[t_] := Module[{s = t}, Reverse[Flatten[Reap[Sow[s]; While[StringCount[s, "0"] > 0, Sow[s = StringReplace[s, "0" -> "111", 1]]]; While[s != "31", If[EvenQ[StringCount[s, "1"]], Sow[s = StringDrop[s, -(StringLength[s]-1)/2]], Sow[{s <> "00", s <> "1110", s <> "111", s = StringDrop[s, -(StringLength[s]-4)/2]}]]]][[2,1]]]]];
Map[FromDigits, Map[MIUProofLines, Flatten[Array[MIUStrings, 3, 2]]], {2}]

A369408 Irregular triangle read by rows: T(n,k) is the length of the shortest proof for the MIU formal system string (theorem) given by A369173(n,k).

Original entry on oeis.org

1, 4, 2, 2, 11, 5, 8, 5, 8, 3, 9, 9, 6, 9, 5, 6, 9, 6, 3, 6, 3

Offset: 2

Paolo Xausa, Jan 22 2024

See A368946 for the description of the MIU formal system and A369173 for the triangle of the corresponding derivable strings.
The length of the shortest proof for a string (theorem) S is the number of lines of the shortest possible derivation of S.
A369173(n,k) first appears in row T(n,k) - 1 in triangle A368946.

			Triangle begins:
  [2]  1;
  [3]  4  2  2;
  [4] 11  5  8  5  8  3;
  [5]  9  9  6  9  5  6  9  6  3  6  3;
  ...
For the theorem MUI (301), which is given by A369173(3,1), the shortest derivation from the axiom MI is MI (31) -> MII (311) -> MIIII (31111) -> MIU (301) (4 lines), so T(3,1) = 4.

Douglas R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, 1979, pp. 33-41 and pp. 261-262.

Wikipedia, MU Puzzle.
Index entries for sequences from "Goedel, Escher, Bach".

Cf. A024495 (row lengths), A331536, A368946, A369173, A369410.

Cf. A369586 (proofs), A369587 (number of symbols).

MIUStringsW3[n_] := Map[FromCharacterCode[# + 48]&, Select[Tuples[{0, 1}, n - 1], ! Divisible[Count[#, 1], 3] &]];
MIUStepDW3[s_] := DeleteDuplicates[Flatten[Map[{If[StringEndsQ[#, "1"], # <> "0", Nothing], # <> #, StringReplaceList[#, {"111" -> "0", "00" -> ""}]} &, s]]];
Module[{rowmax = 5, treedepth = 10, tree}, tree = NestList[MIUStepDW3, {"1"}, treedepth]; Map[Quiet[Check[Position[tree, #, {2}][[1,1]], "Not found"]]&, Array[MIUStringsW3, rowmax - 1, 2], {2}]]

T(n,k) <= A369410(n,k).

A369587 Irregular triangle read by rows: row n lists the number of symbols of the shortest proof (see comments) for each of the distinct derivable strings (theorems) in the MIU formal system that are n characters long.

Original entry on oeis.org

2, 13, 5, 5, 68, 17, 44, 20, 44, 9, 52, 52, 31, 52, 18, 31, 52, 31, 10, 31, 10

Offset: 2

Paolo Xausa, Jan 26 2024

See A368946 for the description of the MIU formal system and A369173 for the triangle of corresponding strings.
Strings are encoded by mapping the characters M, I, and U to 3, 1, and 0, respectively.
In case more than one shortest proof exists, the lexicographically earliest one (after encoding) is chosen.

			Triangle begins:
  [2]  2;
  [3] 13  5  5;
  [4] 68 17 44 20 44  9;
  [5] 52 52 31 52 18 31 52 31 10 31 10;
  ...
For the theorem MIUU (3100), which is given by A369173(4,4), the shortest proof is MI (31) -> MII (311) -> MIIII (31111) -> MIIIIU (311110) -> MIUU (3100), which consists of a total of 20 symbols (counting only M, I and U characters): T(4,4) is therefore 20.

Douglas R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, 1979, pp. 33-41 and pp. 261-262.

Wikipedia, MU Puzzle.
Index entries for sequences from "Goedel, Escher, Bach".

Cf. A368946, A369173, A369411 (analog for "normal" proofs).

Cf. A024495 (row lengths), A369408 (number of lines), A369586 (proofs).

MIUStrings[n_] := Map["3" <> FromCharacterCode[# + 48] &, Select[Tuples[{0, 1}, n - 1], ! Divisible[Count[#, 1], 3] &]];
MIUNext[s_] := Flatten[{If[StringEndsQ[s, "1"], s <> "0", Nothing], s <> StringDrop[s, 1], StringReplaceList[s, {"111" -> "0", "00" -> ""}]}];
Module[{uptolen = 5, searchdepth = 10, mi = "31", g}, g = NestGraph[MIUNext, mi, searchdepth]; Map[Quiet[Check[Map[FromDigits, StringLength[StringJoin[If[# == mi, #, First[Sort[FindPath[g, mi, #, {GraphDistance[g, mi, #]}, All]]]]]]], "Not found"]] &, Array[MIUStrings, uptolen - 1, 2], {2}]] (* Considers theorems up to 5 characters long, looking up to 10 steps away from the MI axiom -- please note it takes a while *)

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A368946 Irregular triangle read by rows: row n lists the strings of the MIU formal system at the n-th level of the tree generated by recursively applying the system rules, starting from the MI string (see comments and example).

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A369173 Irregular triangle read by rows: row n lists all of the distinct derivable strings in the MIU formal system that are n characters long.

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A369409 Irregular triangle read by rows: row n lists the lines of a "normal" proof (see comments) for the MIU formal system string (theorem) given by A369173(n+1).

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A369408 Irregular triangle read by rows: T(n,k) is the length of the shortest proof for the MIU formal system string (theorem) given by A369173(n,k).

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A369587 Irregular triangle read by rows: row n lists the number of symbols of the shortest proof (see comments) for each of the distinct derivable strings (theorems) in the MIU formal system that are n characters long.

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