Collatz Automata and Compute Residue Class from Reduced Dynamics by Formula

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)\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}}.$

所提供的数据集包含用 C 语言编写的源代码，旨在通过自动机在余数类的前提下计算柯勒茨动态。此外，数据集还包含了通过 C 语言实现的算法，能够根据输入的简化动态输出余数类。计算给定简化动态的余数类的公式如下：函数 $Invrs(cdot)$。函数 $Invrs: c
ightarrow rs$ 以 $c=O$ 或 $c=I^{p_1}O^{q_1}I^{p_2}O^{q_2}...I^{p_n}O^{q_n} in {I,O}^{geq 2},$ 其中 $p_i,q_iin mathbb{N}^*, i=1,2,...,n, n in mathbb{N}^*$ 为输入，$CntO(c)=lceil log_21.5*CntI(c)
ceil,$ $CntO(s)Psi*frac{2^{sum_{i=1}^n(p_i+q_i)}}{2^{sum_{i=1}^n(p_i+q_i)}-3^{sum_{i=1}^np_i}}})$ 当 $|c|geq2,$ 其中 $A_i=3^{p_i}-2^{p_i}$,$B_i=(3^{sum_{j=1}^ip_j})^{-1} mod C_n$,$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}}.$