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

Author

Paolo Xausa, Jan 10 2024

Keywords

• nonn
• tabf

Comments

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.

Examples

			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.
		

References

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

Links

• 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"

Crossrefs

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).

Programs

Mathematica

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}]]

Python

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