Month: September 2017

The numbers k with k == 5 (mod 8), and s(k-2) = k with s(.) the hypothetical sequence generated by Turing Machine #4, are what I’ve called “b-numbers” for basic numbers. I haven’t succeeded in finding any rule that determines the whole sequence. Below, I copy the 26 b-numbers below 256. I’ve found this to… Continue reading The b-numbers below 256

I refer to an earlier post a few days ago as an introduction to the problem: The b-numbers again b-numbers are positive integers k with k == 5 (mod 8) such that  s(k-2) = k; here, s(1), s(2), s(3), … is the hypothetical integer sequence computed by TM #4 (chaotic) of Heiner Marxen and Buntrock… Continue reading b-numbers with up to 10 bits

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This is a continuation of the previous post on b-numbers. === For a b-number ‘x’ in the range 4096 to 8191 (13 bit length), it appears at first glance that, modulo 4096, x == y (mod 4096), where y is a b-number in the range 1-4095; Details:   k up to 4095:   5 //… Continue reading More on b-numbers, part B

If we write the (hypothetical, from now on understood) sequence from TM #4 (chaotic), 5-state, 2-symbol tape of Marxen and Buntrock in rows of 8, the 3rd column has a complex structure. Suppose we use “index 1”-notation, which just means that the first term of the sequence, 3, is written s(1) instead of s(0) as… Continue reading The b-numbers again

The code compiles and executes without problems using the GCC compiler on architecture x86_64 on Linux. It only needs the standard C library. It’s an accelerated simulator that combines many repated operations into one big operation. It was checked against a conventional un-accelerated TM simulator, and the outputs matched. (no name program, or sim12g84.c locally):… Continue reading C source code to current TM #4 (chaotic) simulator

Patterns in the b’-numbers related to TM #4 (chaotic) presumptive sequence   b’ numbers = ((( {1, 2, 3, 4, 5, 6, 7} Union the following disjoint sets, one per number of bits in the base 2 representations of these numbers, 4 bits for v8, 5 bits for v16, 6 bits for v32, 7 bits… Continue reading Patterns in the b’-numbers related to TM #4 (chaotic) presumptive sequence

? for(X=1,158, print(binary(yyy[X]) )) [1] [1, 0] [1, 1] [1, 0, 0] [1, 0, 1] [1, 1, 0] [1, 1, 1] [1, 0, 0, 1] [1, 0, 1, 0] [1, 0, 1, 1] [1, 1, 1, 0] [1, 0, 0, 0, 1] [1, 0, 0, 1, 0] [1, 0, 0, 1, 1] [1, 0, 1,… Continue reading b'(n) sequence in binary, n = 1 to 158:

If the n’th b-number is b(n), then b(n) == 3 (mod 8). Therefore, b(n)+5 is divisible by 8, and we can form: b'(n) := ( b(n) + 5)/8. For n = 1, 2, 3, … we have b'(n) as copied below, an irregular sequence: 1 2 3 4 5 6 7 9 10 11 14… Continue reading Irregular sequence derived from the b-numbers

For a b-number ‘b’ , empirically, s(2^m – b – 2) = 2^m – b, as long as b > 0 (always true) AND b < 2^(m-1). Being connected to Column 3 of 8, a “b-number” b is such that b == 3 (mod 8). A b-number ‘b’ predicts relationships of the type: s(y) =… Continue reading The sequence of “b-numbers” determines some values in Column 3 of the TM #4 presumptive sequence

For the TM #4 , hypothetically, an infinite sequence is generated a bit at a time. By arranging this sequence into rows of eight, some simplication occurs: (a) The first column is all 3s (b) The second column is all 1s. (c) And so on. Suppose we denote the sequence by s(1), s(2), s(3), s(4),… Continue reading Observations on column 3 of TM #4 sequence, and mystery remains

Below, the first 350 values of j for which 8j +5 is an allowable number:   0 1 2 3 4 5 6 7 9 10 11 12 13 14 15 18 21 22 23 25 26 27 28 29 30 31 38 42 45 46 47 50 53 54 55 57 58 59 60… Continue reading Allowable numbers: when is 8j + 5 allowable?

1000000110110101 l16 1000011111101101 l16 1000101101110101 l16 1000101111110101 l16 1000110011101101 l16 1000110110110101 l16 1000111011010101 l16 1001000110110101 l16 1001001011010101 l16 1001010111010101 l16 1001011011101101 l16 1001011110110101 l16 1001011111101101 l16 1001100111101101 l16 1001101011111101 l16 1001101101110101 l16 1001101111110101 l16 1001110011101101 l16 1001110110110101 l16 1001110111101101 l16 1001111011010101 l16 1001111011111101 l16 1001111101110101 l16 1001111111110101 l16 1010000110110101 l16 1010001011010101 l16 1010010111010101 l16 1010011011101101… Continue reading The 234 allowable numbers of bit-length 16 for TM #4 (chaotic)

100000000000000101 l18 100000000000001101 l18 100000000000010101 l18 100000000000011101 l18 100000000000100101 l18 100000000000101101 l18 100000000000110101 l18 100000000000111101 l18 100000000001001101 l18 100000000001010101 l18 100000000001011101 l18 100000000001100101 l18 100000000001101101 l18 100000000001110101 l18 100000000001111101 l18 100000000010010101 l18 100000000010101101 l18 100000000010110101 l18 100000000010111101 l18 100000000011001101 l18 100000000011010101 l18 100000000011011101 l18 100000000011100101 l18 100000000011101101 l18 100000000011110101 l18 100000000011111101 l18 100000000100110101 l18 100000000101010101… Continue reading List of the 543 atomic forbidden numbers of bit-length 18

1000101 10000101 10001101 10011101 10100101 100000101 100001101 100010101 100011101 100100101 100101101 100111101 101001101 101011101 101100101 1000000101 1000001101 1000010101 1000011101 1000100101 1000101101 1000110101 1000111101 1001001101 1001010101 1001011101 1001100101 1001101101 1001111101 1010010101 1010101101 1010111101 1011001101 1011011101 1011100101 10000000101 10000001101 10000010101 10000011101 10000100101 10000101101 10000110101 10000111101 10001001101 10001010101 10001011101 10001100101 10001101101 10001110101 10001111101 10010010101 10010101101 10010111101 10011001101 10011011101… Continue reading List of the first 200 atomic forbidden numbers for TM #4 (chaotic)

Please refer to the earlier post about the subsequence expressed in base 8 (octal).   Here,  I changed base 8 to base 2, and secondly, I’ve had the program print the difference between the first and second number in third position: 101 101 000 1101 1101 0000 10101 10101 00000 11101 11101 00000 100101 100101… Continue reading Terms from subsequence of TM #4 chaotic in base 2

A pattern shows up if we put the output (the runs of 1s and runs of 0s) of TM #4 in rows of 8 numbers. So far, columns 1 and 2 appear to be “all 3” and “all 1”. We turn to column 3, the terms with index 8*J + 3, J = 0, 1,… Continue reading Terms from subsequence of TM #4 chaotic in octal

3 1 5 3 7 1 9 3 3 1 13 3 15 1 17 11 3 1 21 3 7 1 25 3 3 1 29 3 31 1 33 27 3 1 37 3 7 1 9 3 3 1 45 3 15 1 49 11 3 1 53 3 7 1 57… Continue reading Content of Tape of TM #4 (chaos machine) in rows of 8

These are exceptions to the general rule that n1 >= n0 , that the count of 1 bits is equal to or greater than the count of 0 bits in the base 2 representation of numbers in the hypothetical sequence computed by TM #4 chaotic, 5-state, Marxen and Buntrock, 1990. Please refer to previous post… Continue reading List of exceptions

The numbers of the hypothetical sequence are represented in base 2, for a fixed width of 18 bits. It’s observed that, with few exceptions, if n1 is the number of 1 bits in standard base 2 notation, and n0 is the number of 0 bits in standard base 2 notation, which always begins with a… Continue reading Content of Tape of TM #4 or “chaotic” in binary

The variation begins with: 23. Ra8 Rxb7 24. Rda1 , which sets up connected white rooks on the A-file on A1 and A8. ================================================================= [Event “World Cup 2017”] [Site “Tbilisi”] [Date “2017.09.06”] [Round “9.10”] [White “Anand Viswanathan (IND)”] [Black “Kovalyov Anton (CAN)”] [Result “0-1”] [ECO “B90”] [WhiteElo “2794”] [BlackElo “2649”] [Annotator “,david250DavidBernier”] [PlyCount “86”] [EventDate… Continue reading Variation on 23. Nc5 in Anand-Kovalyov round 2 game

3 1 5 3 7 1 9 3 3 1 13 3 15 1 17 11 3 1 21 3 7 1 25 3 3 1 29 3 31 1 33 27 3 1 37 3 7 1 9 3 3 1 45 3 15 1 49 11 3 1 53 3 7 1 57… Continue reading Content of Tape of TM #4 (chaos machine)

From the series of 200 lines beginning alternately with “3” then “1” from the previous post, I removed all lines that began with a “1”. That left 100 lines beginning with a “3” followed by one or more “4”s. I removed from each line the initial “3”. What remains is a series of 100 lines… Continue reading On TM #4 of Marxen & Buntrock, Part 2

Turing Machine number 4 (5 states, 2 symbols) of Heiner Marxen and Buntrock ( “chaotic”): Transition table = on A read 0: B1L on A read 1: B1R on B read 0: C1R on B read 1: E0L on C read 0: D0R on C read 1: A0L on D read 0: A1L on D… Continue reading Observations on TM #4 of Marxen & Buntrock (chaotic 5-state TM)