{"ABA""BAAB","BBBB""AA"}
{"ABA""BAAB","BBBB""AA"}
NOTE: for more on this rule, see MW-Time-StringOverlaps-1.nb
MapIndexed[Show[LabelWrapper[GraphicsRow[{SMWXGraphicsStreamed[SMWEvolveListX[#,"ABAAB",50],2.5],SMWXGraphicsStreamed[SMWEvolveListX[#,"ABAAB",50,Backward],2.5]}],First[#2]]]&,{{"BBBB""AA","ABA""BAAB"}}];
MapIndexed[Show[LabelWrapper[GraphicsRow[{SMWSimpleGraphics[SMWEvolveList[#,"ABAAB",50]],SMWSimpleGraphics[SMWEvolveList[#,"ABAAB",50,Backward]]}],First[#2]]]&,{{"BBBB""AA","ABA""BAAB"}}];
MapIndexed[Show[LabelWrapper[GraphicsRow[{SMWSimpleGraphics[SMWEvolveList[#,"ABAAABA",50]],SMWSimpleGraphics[SMWEvolveList[#,"ABAAABA",50,Backward]]}],First[#2]]]&,{{"BBBB""AA","ABA""BAAB"}}];
These are not the same, starting at step 2. But there is an attractor which probably leads to identical behavior.
MapIndexed[Show[LabelWrapper[GraphicsRow[{SMWSimpleGraphics[SMWEvolveList[#,"BBBBBAB",50]],SMWSimpleGraphics[SMWEvolveList[#,"BBBBBAB",50,Backward]]}],First[#2]]]&,{{"BBBB""AA","ABA""BAAB"}}];
SMWEvolveList[{"BBBB""AA","ABA""BAAB"},"BBBBBAB",10]
{BBBBBAB,AABAB,ABAABB,BAABABB,BABAABBB,BBAABABBB,BBABAABBBB,BBBAABAAA,BBBABAABAA,BBBBAABBAABA,AAAABBABAAB}
tryit[init_]:=Show[GraphicsRow[{SMWSimpleGraphics[SMWEvolveList[{"BBBB""AA","ABA""BAAB"},init,50]],SMWSimpleGraphics[SMWEvolveList[{"BBBB""AA","ABA""BAAB"},init,50,Backward]]}]]
tryit[Last[%52]];
tryit[%52[[2]]]
⁃Graphics⁃
RuleOverlapQ[{"BBBB""AA","ABA""BAAB"},#]&/@SMWEvolveList[{"BBBB""AA","ABA""BAAB"},"BBBBBAB",10]
{False,True,True,True,True,True,True,True,True,True,True}
I.e. There is no possibility of overlap after the first step.
Overlap can occur if there is BBBBB or ABABA.
SMWEvolveList[{"BBBB""AA","ABA""BAAB"},"BBBBBAB",10]
{BBBBBAB,AABAB,ABAABB,BAABABB,BABAABBB,BBAABABBB,BBABAABBBB,BBBAABAAA,BBBABAABAA,BBBBAABBAABA,AAAABBABAAB}
Position[RuleOverlapQ[{"BBBB""AA","ABA""BAAB"},#]&/@SMWEvolveList[{"BBBB""AA","ABA""BAAB"},"BBBBBAB",1000],False]
{{1}}
Select[AllStrings[6],(!RuleOverlapQ[{"BBBB""AA","ABA""BAAB"},#])&]
{ABABA,BBBBB,AABABA,ABABAA,ABABAB,ABBBBB,BABABA,BBBBBA,BBBBBB}
Select[AllStrings[7],(!RuleOverlapQ[{"BBBB""AA","ABA""BAAB"},#])&]
{ABABA,BBBBB,AABABA,ABABAA,ABABAB,ABBBBB,BABABA,BBBBBA,BBBBBB,AAABABA,AABABAA,AABABAB,AABBBBB,ABABAAA,ABABAAB,ABABABA,ABABABB,ABBBBBA,ABBBBBB,BAABABA,BABABAA,BABABAB,BABBBBB,BBABABA,BBBBBAA,BBBBBAB,BBBBBBA,BBBBBBB}
Can these occur except in initial conditions?
Look for initial conditions that evolve to them after one step.
Intersection[SMWStep[{"BBBB""AA","ABA""BAAB"},#]&/@AllStrings[9],Select[AllStrings[7],(!RuleOverlapQ[{"BBBB""AA","ABA""BAAB"},#])&]]
{}
Intersection[SMWStep[{"BBBB""AA","ABA""BAAB"},#]&/@AllStrings[10],Select[AllStrings[8],(!RuleOverlapQ[{"BBBB""AA","ABA""BAAB"},#])&]]
{AABABAAB}
Intersection[SMWStep[{"BBBB""AA","ABA""BAAB"},#]&/@AllStrings[11],Select[AllStrings[9],(!RuleOverlapQ[{"BBBB""AA","ABA""BAAB"},#])&]]
{AAABABAAB,AABABAAB,AABABAABA,AABABAABB,BAABABAAB}
Show[mwp[{{"ABA""BAAB","BBBB""AA"},"ABAAB"},5]];
{"BBA""ABA","AA""BAB"}
{"BBA""ABA","AA""BAB"}
Implies overall repetition period is 4.
---> exponential growth....
Appears to be equivalent to a neighbor-dependent substitution system...