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Collatz Automata and Compute Residue Class from Reduced Dynamics by Formula
- Citation Author(s):
- Wei Ren
- Submitted by:
- Wei Ren
- Last updated:
- Thu, 11/08/2018 - 10:34
- DOI:
- 10.21227/7z84-ms87
- License:
135 Views- Categories:
- Keywords:
Abstract
The data set provided source code in C on how to compute Collatz dynamics by automata in terms of residue classes. It also includes algorithms implemented by C codes that can output residue classes by inputting reduced dynamics. The formular for computing a residue class for a given reduced dynamics is as follows:
Function $Invrs(\cdot)$.
$Invrs: c \rightarrow rs$ takes as input \\
$c=O$ or \\
$c=I^{p_1}O^{q_1}I^{p_2}O^{q_2}...I^{p_n}O^{q_n} \in \{I,O\}^{\geq 2},$ $p_i,q_i\in \mathbb{N}^*, i=1,2,...,n, n \in \mathbb{N}^*$\\
$CntO(c)=\lceil \log_21.5*CntI(c)\rceil,$ \\
$CntO(s)<\lceil \log_21.5*CntI(s)\rceil, s=Substr(c,1,k), k=1,2,...,|c|-1,$ \\
and outputs \\
$rs=[0]_2$ when $c=O$, or\\
$rs=([-\sum_{i=1}^nA_iB_iC_{i-1}]_{2^{|c|}} \cap \{x|x>\Psi*\frac{2^{\sum_{i=1}^n(p_i+q_i)}}{2^{\sum_{i=1}^n(p_i+q_i)}-3^{\sum_{i=1}^np_i}}\})$
when $|c|\geq2,$ where $A_i=3^{p_i}-2^{p_i}$,
$B_i=(3^{\sum_{j=1}^ip_j})^{-1} \mod C_n$,
$C_i=2^{\sum_{j=1}^i(p_j+q_j)}, C_0=1,$\\
$\Psi=\prod_{i=2}^na_ib_1+\prod_{i=3}^na_ib_2+...+\prod_{i=n-1}^na_ib_{n-2}+\prod_{i=n}^na_ib_{n-1}+b_n,$\\
$a_i=\frac{3^{p_i}}{2^{p_i+q_i}},b_i=\frac{3^{p_i}-2^{p_i}}{2^{p_i+q_i}}.$
txpo20.c
//Function: Given U, compute D, and output x such that CODE(x)=I^UO^{D-U}
//Input: U
//Output: x, such that CODE(x)=I^UO^{D-U}
//Usage Example: tpxo20.exe U > txpo20-U
// txpo20.exe 1 > txpo20-1
// txpo20.exe 2 > txpo20-2
// txpo20.exe 3 > txpo20-3
// txpo20.exe 4 > txpo20-4
//Output Example:
//txpo20-1:
//D=2
//m=3^U=3,s=2^{D-U}=2,t=s mod m=2
//t_inverse=2,A=(m-1)*t_inverse mod m=1,v_min=(m-2^U)/(2^D-m)=1
//A=4>1,x=(s*A+1)*pow(2,U)/m-1=5
txpo21.c
//Function: Given n, p[1],q[1],p[2],q[2],...,p[n],q[n],
// output x such that CODE(x)=I^p[1]O^q[1]I^p[2]O^q[2]...I^p[n]O^q[n]
//Input: n, p[1],q[1],p[2],q[2],...,p[n],q[n],
//Output: x, such that CODE(x)=I^p[1]O^q[1]I^p[2]O^q[2]...I^p[n]O^q[n]
//Usage Example: tpxo21.exe 2 4 2 2 2 > txpo21-2-4-2-2-2
// txpo21.exe 3 4 1 1 1 1 2 > txpo21-3-4-1-1-1-1-2
// txpo21.exe 4 4 1 1 1 1 1 1 2 > txpo21-4-4-1-1-1-1-1-1-2
//Output Example:
//
//txpo21-2-4-2-2-2:
//p[1]=4,q[1]=2
//p[2]=2,q[2]=2
//D=10, L=1024, S[1]=81, m=16
//b=9, b_inverse=9, M=3, H=192
//In If(n==2), Z=65
//S1_inverse=177, x=975
txpo22.c
//Function: Given p,q, output x such that CODE(x)=I^pO^q
// This program has the same purpose as program 20, but the method is different
// in that U,D replaced by p,q. Indeed, here p equals U in Program 20, q equals D-U in Program 20.
//Input: p,q
//Output: x, such that CODE(x)=I^qO^q
//Usage Example: tpxo22.exe p q > txpo22-p-q
// txpo22.exe 1 1 > txpo22-1-1
// txpo22.exe 2 2 > txpo22-2-2
// txpo22.exe 3 2 > txpo22-3-2
// txpo22.exe 4 3 > txpo22-4-3
//Output Example:
//txpo22-1-1:
//m=2^{p+q}=4,s=3^p=3,t=s mod m=3
//t_inverse=3, x=2^p*t_inverse mod m=1,v_min=(3^p-2^p)/(2^{p+q}-3^p)=1
//x=5
//
txpo23.c
//Function: Output CODE(x) by Finite State Automata.
//Input: x
//Output: CODE(x)
//Usage Example: tpxo23.exe 10000003 > txpo23-10000003
// tpxo23.exe 1055 > txpo23-1055
// tpxo23.exe 2047 > txpo23-2047
// tpxo23.exe 103 > txpo23-103
// tpxo23.exe 27 > txpo23-27
// tpxo23.exe 155> txpo23-155
// tpxo23.exe 123 > txpo23-123
//Output Example:
//txpo23-1055
//t in [3]_4, x=5345
//x in [1]_4, x=4009
//x in [1]_4, x=3007
//t in [3]_4, x=15227
//t in [2]_4, x=25697
//x in [1]_4, x=19273
//x in [1]_4, x=14455
//t in [1]_4, x=24394
//x in [0]_2, x=12197
//x in [1]_4, x=9148
//x in [0]_2, x=4574
//x in [0]_2, x=2287
//t in [3]_4, x=11582
//x in [0]_2, x=5791
//t in [3]_4, x=29321
//x in [1]_4, x=21991
//t in [1]_4, x=37111
//t in [1]_4, x=62626
//x in [0]_2, x=31313
//x in [1]_4, x=23485
//x in [1]_4, x=17614
//x in [0]_2, x=8807
//t in [1]_4, x=14863
//t in [3]_4, x=75248
//x in [0]_2, x=37624
//x in [0]_2, x=18812
//x in [0]_2, x=9406
//x in [0]_2, x=4703
//t in [3]_4, x=23813
//x in [1]_4, x=17860
//x in [0]_2, x=8930
//x in [0]_2, x=4465
//x in [1]_4, x=3349
//x in [1]_4, x=2512
//x in [0]_2, x=1256
//x in [0]_2, x=628
//CODE(1055)=-----0-0------0--0-0---00-000----0-----0---0---00-0-00---0----0000-----000-0-000
//u=50, d=80, ratio=(d-u)/u=0.3750000
