查看“︁PPS”︁的源代码

'''Parented Predecessor Sequence(PPS)'''是由3184创造的一个序列记号，其父项定位方式是[[项定位方式#2.标记父项位置|标记父项位置]]。

PPS有着较为简单的定义，但分析它却极为复杂和困难。

2025年8月，PPS2被发现[[无穷降链]]。

2025年12月10日，PPS1被发现无穷降链。所有PPS衍生物亦未能幸免。

截止到2026年2月23日，目前PPS的已知良序极限(实际可能更低)为0,1,<span style="color:red;">0</span>,<span style="color:red;">2</span>,<span style="color:red;">2</span>,<span style="color:red;">0</span>,<span style="color:red;">5</span>,<span style="color:red;">2</span>,<span style="color:red;">0</span>,<span style="color:red;">8</span>,<span style="color:red;">8</span>,<span style="color:red;">6</span>,<span style="color:red;">0</span>,<span style="color:red;">12</span>,<span style="color:red;">8</span>,<span style="color:red;">6</span>,<span style="color:red;">0</span>,<span style="color:red;">16</span>,<span style="color:red;">16</span>,<span style="color:red;">15</span>,<span style="color:red;">0</span>,<span style="color:red;">16</span>,<span style="color:red;">16</span>,<span style="color:red;">15</span>,<span style="color:red;">0</span>,<span style="color:red;">15</span>,<span style="color:red;">0</span>,<span style="color:red;">26</span>,<span style="color:red;">26</span>,<span style="color:red;">15</span>,<span style="color:red;">0</span>,<span style="color:red;">26</span>,<span style="color:red;">26</span>,<span style="color:red;">13</span>,<span style="color:red;">6</span>,<span style="color:red;">2</span>,<span style="color:red;">0</span>,<span style="color:red;">15</span>,<span style="color:red;">0</span>,<span style="color:red;">38</span>,<span style="color:red;">38</span>,<span style="color:red;">15</span>,<span style="color:red;">0</span>,<span style="color:red;">38</span>,<span style="color:red;">15</span>,<span style="color:red;">0</span>,<span style="color:red;">45</span>,<span style="color:red;">45</span>,<span style="color:red;">13</span>,<span style="color:red;">8</span>,<span style="color:red;">6</span>,<span style="color:red;">0</span>,<span style="color:red;">51</span>,<span style="color:red;">43</span>,<span style="color:red;">15</span>,<span style="color:red;">0</span>,<span style="color:red;">0</span>,<span style="color:red;">45</span>,<span style="color:red;">45</span>,<span style="color:red;">13</span>,<span style="color:blue;">0</span>,<span style="color:blue;">60</span>,<span style="color:blue;">60</span>,<span style="color:blue;">0</span>,<span style="color:blue;">63</span>,<span style="color:blue;">60</span>,<span style="color:blue;">0</span>,<span style="color:blue;">66</span>,<span style="color:blue;">66</span>,<span style="color:blue;">64</span>,<span style="color:blue;">0</span>,<span style="color:blue;">70</span>,<span style="color:blue;">66</span>,<span style="color:blue;">64</span>,<span style="color:blue;">0</span>,<span style="color:blue;">74</span>,<span style="color:blue;">74</span>,<span style="color:blue;">73</span>,<span style="color:blue;">0</span>,<span style="color:blue;">74</span>,<span style="color:blue;">74</span>,<span style="color:blue;">73</span>,<span style="color:blue;">0</span>,<span style="color:blue;">73</span>,<span style="color:blue;">0</span>,<span style="color:blue;">84</span>,<span style="color:blue;">84</span>,<span style="color:blue;">73</span>,<span style="color:blue;">0</span>,<span style="color:blue;">84</span>,<span style="color:blue;">84</span>,<span style="color:blue;">71</span>,<span style="color:blue;">64</span>,<span style="color:blue;">60</span>,<span style="color:blue;">0</span>,<span style="color:blue;">73</span>,<span style="color:blue;">0</span>,<span style="color:blue;">96</span>,<span style="color:blue;">96</span>,<span style="color:blue;">73</span>,<span style="color:blue;">0</span>,<span style="color:blue;">96</span>,<span style="color:blue;">73</span>,<span style="color:blue;">0</span>,<span style="color:blue;">103</span>,<span style="color:blue;">103</span>,<span style="color:blue;">71</span>,<span style="color:blue;">66</span>,<span style="color:blue;">64</span>,<span style="color:blue;">0</span>,<span style="color:blue;">109</span>,<span style="color:blue;">101</span>,<span style="color:blue;">73</span>,<span style="color:blue;">0</span>,<span style="color:blue;">0</span>,<span style="color:blue;">103</span>,<span style="color:blue;">103</span>,<span style="color:blue;">71</span>,<span style="color:green;">0</span>,<span style="color:green;">118</span>,<span style="color:green;">118</span>,<span style="color:green;">0</span>,<span style="color:green;">121</span>,<span style="color:green;">118</span>,<span style="color:green;">0</span>,<span style="color:green;">124</span>,<span style="color:green;">124</span>,<span style="color:green;">122</span>,<span style="color:green;">0</span>,<span style="color:green;">128</span>,<span style="color:green;">124</span>,<span style="color:green;">122</span>,<span style="color:green;">0</span>,<span style="color:green;">132</span>,<span style="color:green;">132</span>,<span style="color:green;">131</span>,<span style="color:green;">0</span>,<span style="color:green;">132</span>,<span style="color:green;">132</span>,<span style="color:green;">131</span>,<span style="color:green;">0</span>,<span style="color:green;">131</span>,<span style="color:green;">0</span>,<span style="color:green;">142</span>,<span style="color:green;">142</span>,<span style="color:green;">131</span>,<span style="color:green;">0</span>,<span style="color:green;">142</span>,<span style="color:green;">142</span>,<span style="color:green;">129</span>,<span style="color:green;">122</span>,<span style="color:green;">118</span>,<span style="color:green;">0</span>,<span style="color:green;">131</span>,<span style="color:green;">0</span>,<span style="color:green;">154</span>,<span style="color:green;">154</span>,<span style="color:green;">131</span>,<span style="color:green;">0</span>,<span style="color:green;">154</span>,<span style="color:green;">131</span>,<span style="color:green;">0</span>,<span style="color:green;">161</span>,<span style="color:green;">161</span>,<span style="color:green;">129</span>,<span style="color:green;">124</span>,<span style="color:green;">122</span>,<span style="color:green;">0</span>,<span style="color:green;">167</span>,<span style="color:green;">159</span>,<span style="color:green;">131</span>,<span style="color:green;">0</span>,<span style="color:green;">0</span>,<span style="color:green;">161</span>,<span style="color:green;">161</span>,<span style="color:green;">129</span>,<span style="color:purple;">0</span>,<span style="color:purple;">176</span>,<span style="color:purple;">176</span>,<span style="color:purple;">0</span>,<span style="color:purple;">179</span>,<span style="color:purple;">176</span>,<span style="color:purple;">0</span>,<span style="color:purple;">182</span>,<span style="color:purple;">182</span>,<span style="color:purple;">180</span>,<span style="color:purple;">0</span>,<span style="color:purple;">186</span>,<span style="color:purple;">182</span>,<span style="color:purple;">180</span>,<span style="color:purple;">0</span>,<span style="color:purple;">190</span>,<span style="color:purple;">190</span>,<span style="color:purple;">189</span>,<span style="color:purple;">0</span>,<span style="color:purple;">190</span>,<span style="color:purple;">190</span>,<span style="color:purple;">189</span>,<span style="color:purple;">0</span>,<span style="color:purple;">189</span>,<span style="color:purple;">0</span>,<span style="color:purple;">200</span>,<span style="color:purple;">200</span>,<span style="color:purple;">189</span>,<span style="color:purple;">0</span>,<span style="color:purple;">200</span>,<span style="color:purple;">200</span>,<span style="color:purple;">187</span>,<span style="color:purple;">180</span>,<span style="color:purple;">176</span>,<span style="color:purple;">0</span>,<span style="color:purple;">189</span>,<span style="color:purple;">0</span>,<span style="color:purple;">212</span>,<span style="color:purple;">212</span>,<span style="color:purple;">189</span>,<span style="color:purple;">0</span>,<span style="color:purple;">212</span>,<span style="color:purple;">189</span>,<span style="color:purple;">0</span>,<span style="color:purple;">219</span>,<span style="color:purple;">219</span>,<span style="color:purple;">187</span>,<span style="color:purple;">182</span>,<span style="color:purple;">180</span>,<span style="color:purple;">0</span>,<span style="color:purple;">225</span>,<span style="color:purple;">217</span>,<span style="color:purple;">189</span>,<span style="color:purple;">0</span>,<span style="color:purple;">0</span>,<span style="color:purple;">219</span>,<span style="color:purple;">219</span>,<span style="color:purple;">187</span>,<span style="color:orange;">0</span>,<span style="color:orange;">234</span>,<span style="color:orange;">234</span>,<span style="color:orange;">0</span>,<span style="color:orange;">237</span>,<span style="color:orange;">234</span>,<span style="color:orange;">0</span>,<span style="color:orange;">240</span>,<span style="color:orange;">240</span>,<span style="color:orange;">238</span>,<span style="color:orange;">0</span>,<span style="color:orange;">244</span>,<span style="color:orange;">240</span>,<span style="color:orange;">238</span>,<span style="color:orange;">0</span>,<span style="color:orange;">248</span>,<span style="color:orange;">248</span>,<span style="color:orange;">247</span>,<span style="color:orange;">0</span>,<span style="color:orange;">248</span>,<span style="color:orange;">248</span>,<span style="color:orange;">247</span>,<span style="color:orange;">0</span>,<span style="color:orange;">247</span>,<span style="color:orange;">0</span>,<span style="color:orange;">258</span>,<span style="color:orange;">258</span>,<span style="color:orange;">247</span>,<span style="color:orange;">0</span>,<span style="color:orange;">258</span>,<span style="color:orange;">258</span>,<span style="color:orange;">245</span>,<span style="color:orange;">238</span>,<span style="color:orange;">234</span>,<span style="color:orange;">0</span>,<span style="color:orange;">247</span>,<span style="color:orange;">0</span>,<span style="color:orange;">270</span>,<span style="color:orange;">270</span>,<span style="color:orange;">247</span>,<span style="color:orange;">0</span>,<span style="color:orange;">270</span>,<span style="color:orange;">247</span>,<span style="color:orange;">0</span>,<span style="color:orange;">277</span>,<span style="color:orange;">277</span>,<span style="color:orange;">245</span>,<span style="color:orange;">240</span>,<span style="color:orange;">238</span>,<span style="color:orange;">0</span>,<span style="color:orange;">283</span>,<span style="color:orange;">275</span>,<span style="color:orange;">247</span>,<span style="color:orange;">0</span>,<span style="color:orange;">0</span>,<span style="color:orange;">277</span>,<span style="color:orange;">277</span>,<span style="color:orange;">245</span>,<span style="color:brown;">0</span>,<span style="color:brown;">292</span>,<span style="color:brown;">292</span>,<span style="color:brown;">0</span>,<span style="color:brown;">295</span>,<span style="color:brown;">292</span>,<span style="color:brown;">0</span>,<span style="color:brown;">298</span>,<span style="color:brown;">298</span>,<span style="color:brown;">296</span>,<span style="color:brown;">0</span>,<span style="color:brown;">302</span>,<span style="color:brown;">298</span>,<span style="color:brown;">296</span>,<span style="color:brown;">0</span>,<span style="color:brown;">306</span>,<span style="color:brown;">306</span>,<span style="color:brown;">305</span>,<span style="color:brown;">0</span>,<span style="color:brown;">306</span>,<span style="color:brown;">306</span>,<span style="color:brown;">305</span>,<span style="color:brown;">0</span>,<span style="color:brown;">305</span>,<span style="color:brown;">0</span>,<span style="color:brown;">316</span>,<span style="color:brown;">316</span>,<span style="color:brown;">305</span>,<span style="color:brown;">0</span>,<span style="color:brown;">316</span>,<span style="color:brown;">316</span>,<span style="color:brown;">303</span>,<span style="color:brown;">296</span>,<span style="color:brown;">292</span>,<span style="color:brown;">0</span>,<span style="color:brown;">305</span>,<span style="color:brown;">0</span>,<span style="color:brown;">328</span>,<span style="color:brown;">328</span>,<span style="color:brown;">305</span>,<span style="color:brown;">0</span>,<span style="color:brown;">328</span>,<span style="color:brown;">305</span>,<span style="color:brown;">0</span>,<span style="color:brown;">335</span>,<span style="color:brown;">335</span>,<span style="color:brown;">303</span>,<span style="color:brown;">298</span>,<span style="color:brown;">296</span>,<span style="color:brown;">0</span>,<span style="color:brown;">341</span>,<span style="color:brown;">333</span>,<span style="color:brown;">305</span>,<span style="color:brown;">0</span>,<span style="color:brown;">0</span>,<span style="color:brown;">335</span>,<span style="color:brown;">335</span>,...

对PPS无穷降链的寻找可使用[https://github.com/hzyhhzy/AutoGuogaoMachine/tree/master 自动果糕机]进行辅助。

后续phyrion在圣诞节前连夜修改了10+个版本，亦未能避免无穷降链，不过受部分启发提出了[[PRRS]]。

前排提示：PPS的行为极其复杂，是标准的“果糕“记号

=== 定义 ===
以下为PPS1的定义。PPS2和PPS3的问题较为明显且未能修改PPS1的不足之处，因此不在此给出定义。

PPS是形如0,1,0,3这样用逗号分隔的序列（序列首项是第1项）

极限表达式：0,1,2,3,4,5,......

记末项的值为x，坏根为第x项，坏根的值为b，末项是序列中的第y项，并令L=y-x

展开：

1.如果末项是0，则它是后继序数

2.末项之前的部分保持不变

3.替换末项：如果末项和坏根之间(两边都不含)存在一项，它的值等于b，那么将末项的值换成b；否则将末项的值减1

4.递归生成其他项(第i+L项的值由第i项确定)：对任意的i>x，如果第i项的值大于等于x，那么第i+L项的值等于第i项的值+L，否则第i+L项的值等于第i项的值

5.基本列[n]为展开到第y+n*L-1项
=== 分析 ===
另见[[PPS分析]]

PPS的分析是极为困难的，即便是一些分析力很强的googologist，如mtl、zcmx，都曾在PPS上折戟。

=== 地府段 ===
PPS有部分从表达式上看差距不大，但分析却极为困难且需要大量篇幅的段落，它们被称为地府段，简称地府。

==== 第一部分：0,1,0,2,0,4,4,3,0,4,3,0,11,9 ~ 0,1,0,2,0,4,4,3,0,4,3,0,11,10 ====
[[文件:pcf 25-07-22.jpg|缩略图|2025年7月22日，来自PCF]]
0,1,0,2,0,4,4,3,0,4,3,0,11,9 ~ 0,1,0,2,0,4,4,3,0,4,3,0,11,10是地府的第一层。

它们分别对应序数<math>\varepsilon_{\omega^{\omega^{\omega^{\varepsilon_0+1}\times 2}}}</math>

和<math>\varepsilon_{\omega^{\omega^{\omega^{\omega^{\varepsilon_{\varepsilon_0}+1}}}}}</math>。

几位扽西力较强的gggist合力也用了近五天才把它扽出来。在分析表格中，它们占用了超过两百行。

==== 第二部分：0,1,0,2,0,4,4,3,0,4,3,0,11,10,0,0,10,0,17,9 ~ 0,1,0,2,0,4,4,3,0,4,3,0,11,10,0,0,10,0,17,10 ====
0,1,0,2,0,4,4,3,0,4,3,0,11,10,0,0,10,0,17,9 ~ 0,1,0,2,0,4,4,3,0,4,3,0,11,10,0,0,10,0,17,10是地府的第二层。这之间还有不计其数的地府第一层的结构。分析它更是难上加难，'''四百行'''扽西也仅仅只能在第二层地府中踏出小而无力的一步。

截至目前，我们仍未得知0,1,0,2,0,4,4,3,0,4,3,0,11,10,0,0,10,0,17,10对应哪个序数，不过根据Phyrion的猜测，它有可能<math>\ge \zeta_0</math>。

=== 改版 ===

==== PPM 1.2 ====
Parented Predecessor Matrix 1,2

极限表达式：(0)(1,1,1,1,1,...)

LNZ：末列的最大非零行序号

坏根：第(末项的值)列LNZ行的元素，其中末项是末列LNZ行（首项的列标是1、首项是第1项）；如果末列是全0，则表示后继序数

记此时末项的列标减末项的值为L，坏根的值为b、列标为c，末项的值为x、列标为y

末列展开：把末列LNZ-1行的祖先链上（设第0行父项固定为前一项）所有列的LNZ行元素提取出来组成判断序列；在判断序列中，如果末项和坏根之间(两边都不含)存在一项，它的值等于b则弱展开，否则强展开。

弱展开：LNZ行之前的不变，LNZ行及之后的用坏根列的相同行元素替换

强展开：LNZ行之前的不变，LNZ行的值减一，LNZ行之后的行是(坏根列同行元素+展开后末项的值-坏根项的值)

其他项展开：对任意的i>y-L，如果第i项的值大于等于x，那么第i+L项的值等于第i项的值+L，否则第i+L项的值等于第i项的值

基本列[n]为展开到第y+nL-1项

==== PPM 2 ====
Parented Predecessor Matrix 2

极限表达式：(0)(1,1,1,1,1,...)

LNZ：末列的最大非零行序号

坏根：第(末项的值)列LNZ行的元素，其中末项是末列LNZ行（首项的列标是1、首项是第1项）；如果末列是全0，则表示后继序数

记此时末项的列标减末项的值为L，坏根的值为b、列标为c，末项的值为x、列标为y

末列展开：把末列LNZ-1行的祖先链上（设第0行父项固定为前一项）所有列的LNZ行元素提取出来组成判断序列；在判断序列中，如果末项和坏根之间(两边都不含)存在一项，它的值等于b，则比较[以这个项为首的判断序列]和[以坏根为首的判断序列]的字典序，如果前者大(只要存在一个)∨(坏根不在末项祖先链上∧这样的项存在)则弱展开，如果后者大∨这样的项不存在则强展开。

弱展开：LNZ行之前的不变，LNZ行及之后的用坏根列的相同行元素替换

强展开：LNZ行之前的不变，LNZ行的值减一，LNZ行之后的行是(坏根列同行元素+展开后末项的值-坏根项的值)

其他项展开：对任意的i>y-L+1，如果第i项的值大于等于x，那么第i+L项的值等于第i项的值+L，否则第i+L项的值等于第i项的值

基本列[n]为展开到第y+nL-1项

==== PPM 3 ====
Parented Predecessor Matrix 3

极限表达式：(0)(1,1,1,1,1,...)

LNZ：末列的最大非零行序号

坏根：第(末项的值)列LNZ行的元素，其中末项是末列LNZ行（首项的列标是1、首项是第1项）；如果末列是全0，则表示后继序数

记此时末项的列标减末项的值为L，坏根的值为b、列标为c，末项的值为x、列标为y

末列展开：{

把末列LNZ-1行的祖先链上（设第0行父项固定为前一项）所有列的LNZ行元素提取出来组成判断序列；在判断序列中，如果末项和坏根之间(两边都不含)存在一项，它的值等于b则弱展开，否则强展开。

弱展开：LNZ行之前的不变，LNZ行及之后的用坏根列的相同行元素替换

强展开：LNZ行之前的不变，LNZ行的值减一，LNZ行之后的行是(坏根列同行元素+展开后末项的值-坏根项的值)

}

末列以右整数个复制单元长度的列展开：强展开时LNZ行元素的值每次复制时增加一个复制单元长度，其他同“其他项展开”

其他项展开：对任意的i>y-L，如果第i项的值大于等于x，那么第i+L项的值等于第i项的值+L，否则第i+L项的值等于第i项的值

基本列[n]为展开到第y+nL-1项{{默认排序:序数记号}}
[[分类:记号]]