WOLFRAM NOTEBOOK

In[]:=
EnumerateCombinators[4]
Out[]=
{s[s][s][s],s[s[s][s]],s[s[s]][s],s[s[s[s]]],s[s][s[s]],s[s][s][k],s[s[s][k]],s[s[s]][k],s[s[s[k]]],s[s][s[k]],s[s][k][s],s[s[k][s]],s[s[k]][s],s[s[k[s]]],s[s][k[s]],s[s][k][k],s[s[k][k]],s[s[k]][k],s[s[k[k]]],s[s][k[k]],s[k][s][s],s[k[s][s]],s[k[s]][s],s[k[s[s]]],s[k][s[s]],s[k][s][k],s[k[s][k]],s[k[s]][k],s[k[s[k]]],s[k][s[k]],s[k][k][s],s[k[k][s]],s[k[k]][s],s[k[k[s]]],s[k][k[s]],s[k][k][k],s[k[k][k]],s[k[k]][k],s[k[k[k]]],s[k][k[k]],k[s][s][s],k[s[s][s]],k[s[s]][s],k[s[s[s]]],k[s][s[s]],k[s][s][k],k[s[s][k]],k[s[s]][k],k[s[s[k]]],k[s][s[k]],k[s][k][s],k[s[k][s]],k[s[k]][s],k[s[k[s]]],k[s][k[s]],k[s][k][k],k[s[k][k]],k[s[k]][k],k[s[k[k]]],k[s][k[k]],k[k][s][s],k[k[s][s]],k[k[s]][s],k[k[s[s]]],k[k][s[s]],k[k][s][k],k[k[s][k]],k[k[s]][k],k[k[s[k]]],k[k][s[k]],k[k][k][s],k[k[k][s]],k[k[k]][s],k[k[k[s]]],k[k][k[s]],k[k][k][k],k[k[k][k]],k[k[k]][k],k[k[k[k]]],k[k][k[k]]}
​
In[]:=
Select[GatherBy[#->CombinatorFixedPoint[#]&/@EnumerateCombinators[5],Last],Length[#]>1&]
Out[]=
In[]:=
Select[CombinatorFixedPointList/@Complement[EnumerateCombinators[7],{s[s[s]][s][s][s][s],s[s][s][s[s]][s][s]}],Last[#]===s[k]&]
Out[]=
{{s[k[k][k[k[k[k]]]]],s[k]},{s[k[k][k[k[k[s]]]]],s[k]},{s[k[k][k[k[s[k]]]]],s[k]},{s[k[k][k[k[s[s]]]]],s[k]},{s[k[k][k[s[k[k]]]]],s[k]},{s[k[k][k[s[k[s]]]]],s[k]},
⋯701⋯
,{s[k][k][k][s][s][k],k[k][k[k]][s][s][k],k[s][s][k],s[k]},{s[k][s][k][s][k][k],k[k][s[k]][s][k][k],k[s][k][k],s[k]},{s[k][s][k][s][s][k],k[k][s[k]][s][s][k],k[s][s][k],s[k]},{s[s][k][s][k][s][k],s[s][k[s]][k][s][k],s[k][k[s][k]][s][k],k[s][k[s][k][s]][k],s[k]},{s[s][s][s][k][s][k],s[s][s[s]][k][s][k],s[k][s[s][k]][s][k],k[s][s[s][k][s]][k],s[k]}}
large output
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set size limit...
In[]:=
TakeLargestBy[%,Length,10]
Out[]=
In[]:=
Length/@%
Out[]=
{12,11,8,8,8,8,7,7,7,7}
In[]:=
ResourceFunction["ParallelMapMonitored"][(If[Last[#]===s[k],#,Nothing]&[CombinatorFixedPointList[#,"MaxSize"1000,"MaxSteps"1000]])&,Complement[EnumerateCombinators[8],{s[s[s[s][s]]][s][s][s],s[s[s[s[s][s]]]][s][s],s[s[s]][s][s][s][s][s],s[s[s[s]][s][s][s][s]],s[s[s[s]][s][s]][s][s],s[s[s[s]][s]][s][s][s],s[s[s[s[s]][s]]][s][s],s[s[s[s]]][s][s][s][s],s[s[s[s[s]]][s]][s][s],s[s[s[s[s]]]][s][s][s],s[s][s][s[s]][s][s][s],s[s[s][s][s[s]][s][s]],s[s[s][s][s[s]]][s][s],s[s[s]][s[s]][s][s][s],s[s[s[s]][s[s]]][s][s],s[s[s]][s][s[s]][s][s],s[s[s[s]]][s[s]][s][s],s[s][s][s[s][s]][s][s],s[s][s][s[s[s]]][s][s],s[s[s]][s[s[s]]][s][s],s[s[s][s]][s][s[s]][s],s[s[s[s][s]]][s[s]][s],s[s[s]][s][s][s[s]][s],s[s[s[s]]][s][s[s]][s],s[s][s][s[s]][s[s]][s],s[s][s][s[s[s][s]]][s],s[s[s]][s][s[s][s]][s],s[s][s][s[s[s]][s]][s],s[s[s]][s][s[s[s]]][s],s[s][s][s[s[s[s]]]][s],s[s[s[s][s]]][s][s[s]],s[s[s]][s][s][s][s[s]],s[s[s[s]][s]][s][s[s]],s[s[s[s]]][s][s][s[s]],s[s][s][s[s]][s][s[s]],s[s[s]][s[s]][s][s[s]],s[s[s]][s][s[s]][s[s]],s[s][s][s[s][s]][s[s]],s[s][s][s[s[s]]][s[s]],s[s[s][s]][s][s[s][s]],s[s[s][s]][s][s[s[s]]]}]]
Out[]=
{{s[k[k][k[k[k[k[k]]]]]],s[k]},{s[k[k][k[k[k[k[s]]]]]],s[k]},{s[k[k][k[k[k[s[k]]]]]],s[k]},{s[k[k][k[k[k[s[s]]]]]],s[k]},
⋯4436⋯
,{s[s][s][k][k][s][k][k],s[k][s[k]][k][s][k][k],k[k][s[k][k]][s][k][k],k[s][k][k],s[k]},{s[s][s][k][k][s][s][k],s[k][s[k]][k][s][s][k],k[k][s[k][k]][s][s][k],k[s][s][k],s[k]},{s[s][s][s][s][k][s][k],s[s][s[s]][s][k][s][k],s[s][s[s][s]][k][s][k],s[k][s[s][s][k]][s][k],k[s][s[s][s][k][s]][k],s[k]}}
large output
show less
show more
show all
set size limit...
In[]:=
TakeLargestBy[%,Length,10]
Out[]=
In[]:=
Length/@%
Out[]=
{31,30,15,12,12,12,11,11,11,11}
​
In[]:=
Select[CombinatorFixedPointList/@Complement[EnumerateCombinators[6],{s[s[s]][s][s][s][s],s[s][s][s[s]][s][s]}],Last[#]===s[k]&]
Out[]=
In[]:=
Length/@%
Out[]=
{2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,3,3,3,3,3,3,3,3,3,3,3,3,4,6,6,4,4,4,7,7,3,3,3,3,4,4}
In[]:=
First/@Select[%399,Length[#]≥4&]
Out[]=
{s[s[s][k][k][k]],s[s[s][s][k][k]],s[k[s]][k[k]][k],s[k[s]][k[k]][s],s[s[s][k]][s][k],s[s[s][s]][s][k],s[s][k][k][s[k]],s[s][k][s[k]][k],s[s][k][s[k]][s],s[s][s][k][s[k]],s[k[k]][s][k][k],s[k[k]][s][k][s],s[s[s]][s][k][k],s[s[s]][s][k][s],s[s][k][k][s][k],s[s][s][k][s][k]}
In[]:=
RandomSample[%,4]
Out[]=
{s[s][k][k][s[k]],s[s[s]][s][k][k],s[s][s][k][s][k],s[k[s]][k[k]][k]}
In[]:=
CombinatorFixedPoint[s[s[s][s]][s][s[s]][k]]
Out[]=
$Aborted
In[]:=
CombinatorEvolveList[s[s[s][s]][s][s[s]][k],1959];
Out[]=
$Aborted
Last[%]
In[]:=
#->Length[CombinatorFixedPointList[First[#]]]&/@{s[s[k][s][k]]s[k],s[s[k]][s][k]s[k],s[s[k][k][k]]s[k],s[k][s][s][k]s[k],s[k][s][s[k]]s[k],s[k[k][s[s]]]s[k],s[k][k][s][k]s[k],s[k[k][s[k]]]s[k],s[k][k][s[k]]s[k],s[k[k][k[s]]]s[k],s[k[k][k[k]]]s[k],k[s][s[s]][k]s[k],k[s][s[k]][k]s[k],k[s[k]][s[s]]s[k],k[s][k[s]][k]s[k],k[s[k]][s[k]]s[k],k[s[k]][k[s]]s[k],k[s][k[k]][k]s[k],k[s[k]][k[k]]s[k]}
Out[]=
{(s[s[k][s][k]]s[k])3,(s[s[k]][s][k]s[k])4,(s[s[k][k][k]]s[k])3,(s[k][s][s][k]s[k])3,(s[k][s][s[k]]s[k])3,(s[k[k][s[s]]]s[k])2,(s[k][k][s][k]s[k])3,(s[k[k][s[k]]]s[k])2,(s[k][k][s[k]]s[k])3,(s[k[k][k[s]]]s[k])2,(s[k[k][k[k]]]s[k])2,(k[s][s[s]][k]s[k])2,(k[s][s[k]][k]s[k])2,(k[s[k]][s[s]]s[k])2,(k[s][k[s]][k]s[k])2,(k[s[k]][s[k]]s[k])2,(k[s[k]][k[s]]s[k])2,(k[s][k[k]][k]s[k])2,(k[s[k]][k[k]]s[k])2}
{s[s[k][s][k]]s[k],s[s[k]][s][k]s[k],s[s[k][k][k]]s[k],s[k][s][s][k]s[k],s[k][s][s[k]]s[k],s[k[k][s[s]]]s[k],s[k][k][s][k]s[k],s[k[k][s[k]]]s[k],s[k][k][s[k]]s[k],s[k[k][k[s]]]s[k],s[k[k][k[k]]]s[k],k[s][s[s]][k]s[k],k[s][s[k]][k]s[k],k[s[k]][s[s]]s[k],k[s][k[s]][k]s[k],k[s[k]][s[k]]s[k],k[s[k]][k[s]]s[k],k[s][k[k]][k]s[k],k[s[k]][k[k]]s[k]}
{s[s[s]][s][s][s][s],s[s][s][s[s]][s][s]}
In[]:=
In[]:=
ResourceFunction["ParallelMapMonitored"][#->TimeConstrained[Map[StringLength,ResourceFunction["MultiwayCombinator"][{s[x_][y_][z_]x[z][y[z]],k[x_][y_]x},#,15],{2}],15]&,skgrowers]
Out[]=
{s[s[s[s][s]]][s][s][s]{
⋯1⋯
},s[s[s[s[s][s]]]][s][s]$Aborted,s[s[s]][s][s][s][s][s]{{22},{22},{25},{25},{34},{34},{34},{37},{52},{76},{76,100},{76,100,124},{76,94,100,124,151},{76,94,94,94,100,118,124,139,151,178,178},{79,94,94,94,94,94,94,100,118,118,118,124,139,139,139,142,151,163,163,178,178,187,205,205,223,262},{82,94,94,94,97,97,97,97,97,97,100,103,118,118,118,118,118,118,124,139,139,139,139,139,139,142,142,142,151,163,163,163,163,163,163,169,178,178,187,187,187,187,187,187,205,205,211,211,223,232,232,235,250,250,250,262,277,289,289,316,346}},s[s[s[s]][s][s][s][s]]{{22},{22},
⋯13⋯
,{232,232,250,250,250,277,289,289,316,346}},
⋯65⋯
,
⋯1⋯
⋯8⋯
,s[s][k][s[s[s][k]]][k]$Aborted,k[s[s[s]][s][s][s][s]]{{22},{22},{25},{25},{34},{34},{34},{37},{52},{76},{100},{124},{151},{178,178},{205,205,223,262},{232,232,250,250,250,277,289,289,316,346}},k[s[s][s][s[s]][s][s]]{{22},{25},{25},{34},{34},{34},{37},{52},{76},{100},{124},{151},{178,178},{205,205,223,262},{232,232,250,250,250,277,289,289,316,346},{259,262,277,277,277,277,277,277,304,304,304,316,316,316,331,334,334,343,343,361,361,370,373,373,373,400,406,427,430,448}}}
large output
show less
show more
show all
set size limit...
In[]:=
Map[Min]/@Association[%]
Out[]=
In[]:=
ResourceFunction["ParallelMapMonitored"][#->TimeConstrained[Map[StringLength,ResourceFunction["MultiwayCombinator"][{s[x_][y_][z_]x[z][y[z]],k[x_][y_]x},#,15],{2}],30]&,{s[s[s]][s][s][s][k[s]]}]
Out[]=
{s[s[s]][s][s][s][k[s]]{{22},{22},{25},{25},{34},{37},{40},{46},{64,34},{97,52},{130,85,85,73,58},{163,118,91,118,106,118,73,94,91,58,46},{199,151,124,151,91,151,139,127,106,79,151,106,79,115,106,94,67,124,91,79,52,46,79,67},{238,187,160,187,124,187,139,112,187,175,139,112,163,151,139,127,100,79,187,139,112,139,79,139,139,127,115,88,94,67,160,124,112,85,112,79,52,112,100,88,79,67,40},{352,226,199,226,226,226,214,199,160,160,175,148,175,148,175,112,226,214,202,178,175,148,175,163,136,112,175,163,151,124,127,100,226,175,148,175,112,175,127,100,166,175,163,127,100,151,139,112,127,115,88,67,199,160,148,121,148,112,85,148,112,100,73,52,148,136,112,100,73,124,112,100,88,61,67,40},{340,313,340,340,340,313,340,328,316,313,274,235,340,328,340,328,199,214,187,265,214,187,214,214,187,214,202,214,202,199,187,160,187,148,187,148,148,148,163,136,340,316,292,256,214,187,214,202,175,241,214,202,190,163,214,202,202,190,205,190,178,151,187,148,175,136,163,151,124,148,163,136,163,136,163,151,124,100,340,214,187,214,214,214,202,187,148,148,163,136,163,136,163,100,244,214,202,190,193,166,139,139,112,163,136,163,151,124,100,163,151,139,112,115,88,313,199,187,160,187,187,187,175,160,121,148,121,148,136,109,85,148,136,109,136,100,73,187,175,163,139,148,136,109,136,124,97,100,73,136,124,112,85,100,88,61,40}}}

Algebraic Stuff

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Lemmas

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