MAYBE
(ignored inputs)COMMENT submitted by: Johannes Waldmann secret problem 2019 category: SRS
Rewrite Rules:
   [ b(?x) -> b(a(b(?x))),
     a(?x) -> c(a(a(?x))),
     b(?x) -> a(c(a(?x))),
     c(?x) -> c(a(c(?x))),
     b(?x) -> a(c(b(?x))) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ b(a(b(?x))) = a(c(a(?x))),
     b(a(b(?x))) = a(c(b(?x))),
     a(c(a(?x_2))) = a(c(b(?x_2))) ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ a(c(a(?x))) = b(a(b(?x))),
     a(c(b(?x))) = b(a(b(?x))),
     b(a(b(?x))) = a(c(a(?x))),
     a(c(b(?x))) = a(c(a(?x))),
     b(a(b(?x))) = a(c(b(?x))),
     a(c(a(?x))) = a(c(b(?x))) ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Root-E-Closed/Non-E-Overlapping
unknown Strongly Weight-Preserving & Root-E-Closed/Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <a(c(a(?x_2))), b(a(b(?x_2)))> by Rules <2, 0> preceded by []
unknown Diagram Decreasing
check Non-Confluence...
obtain 7 rules by 3 steps unfolding
obtain 100 candidates for checking non-joinability
check by TCAP-Approximation (failure)
check by Ordering(rpo), check by Tree-Automata Approximation (failure)
check by Interpretation(mod2) (failure)
check by Descendants-Approximation, check by Ordering(poly) (failure)
unknown Non-Confluence
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check by Ordered Rewriting...
remove redundants rules and split
R-part:
   [ b(?x) -> b(a(b(?x))),
     a(?x) -> c(a(a(?x))),
     b(?x) -> a(c(a(?x))),
     c(?x) -> c(a(c(?x))),
     b(?x) -> a(c(b(?x))) ]
E-part:
   [ ]
...failed to find a suitable LPO.
unknown Confluence by Ordered Rewriting
Direct Methods: Can't judge

Try Persistent Decomposition for...
   [ b(?x) -> b(a(b(?x))),
     a(?x) -> c(a(a(?x))),
     b(?x) -> a(c(a(?x))),
     c(?x) -> c(a(c(?x))),
     b(?x) -> a(c(b(?x))) ]
Sort Assignment:
 a : 12=>12
 b : 12=>12
 c : 12=>12
maximal types: {12}
Persistent Decomposition failed: Can't judge

Try Layer Preserving Decomposition for...
   [ b(?x) -> b(a(b(?x))),
     a(?x) -> c(a(a(?x))),
     b(?x) -> a(c(a(?x))),
     c(?x) -> c(a(c(?x))),
     b(?x) -> a(c(b(?x))) ]
Layer Preserving Decomposition failed: Can't judge

Try Commutative Decomposition for...
   [ b(?x) -> b(a(b(?x))),
     a(?x) -> c(a(a(?x))),
     b(?x) -> a(c(a(?x))),
     c(?x) -> c(a(c(?x))),
     b(?x) -> a(c(b(?x))) ]
Outside Critical Pair: <a(c(a(?x_2))), b(a(b(?x_2)))> by Rules <2, 0>
develop reducts from lhs term...
<{1,3}, c(a(a(c(a(c(c(a(a(?x_2)))))))))>
<{1,3}, c(a(a(c(a(c(a(?x_2)))))))>
<{1}, c(a(a(c(c(a(a(?x_2)))))))>
<{1}, c(a(a(c(a(?x_2)))))>
<{1,3}, a(c(a(c(c(a(a(?x_2)))))))>
<{3}, a(c(a(c(a(?x_2)))))>
<{1}, a(c(c(a(a(?x_2)))))>
<{}, a(c(a(?x_2)))>
develop reducts from rhs term...
<{1,4}, a(c(b(c(a(a(a(c(b(?x_2)))))))))>
<{1,2,4}, a(c(b(c(a(a(a(c(a(?x_2)))))))))>
<{0,1,4}, a(c(b(c(a(a(b(a(b(?x_2)))))))))>
<{1,4}, a(c(b(c(a(a(b(?x_2)))))))>
<{4}, a(c(b(a(a(c(b(?x_2)))))))>
<{2,4}, a(c(b(a(a(c(a(?x_2)))))))>
<{0,4}, a(c(b(a(b(a(b(?x_2)))))))>
<{4}, a(c(b(a(b(?x_2)))))>
<{1,2,4}, a(c(a(c(a(a(a(c(b(?x_2)))))))))>
<{1,2}, a(c(a(c(a(a(a(c(a(?x_2)))))))))>
<{0,1,2}, a(c(a(c(a(a(b(a(b(?x_2)))))))))>
<{1,2}, a(c(a(c(a(a(b(?x_2)))))))>
<{2,4}, a(c(a(a(a(c(b(?x_2)))))))>
<{2}, a(c(a(a(a(c(a(?x_2)))))))>
<{0,2}, a(c(a(a(b(a(b(?x_2)))))))>
<{2}, a(c(a(a(b(?x_2)))))>
<{0,1,4}, b(a(b(c(a(a(a(c(b(?x_2)))))))))>
<{0,1,2}, b(a(b(c(a(a(a(c(a(?x_2)))))))))>
<{0,1}, b(a(b(c(a(a(b(a(b(?x_2)))))))))>
<{0,1}, b(a(b(c(a(a(b(?x_2)))))))>
<{0,4}, b(a(b(a(a(c(b(?x_2)))))))>
<{0,2}, b(a(b(a(a(c(a(?x_2)))))))>
<{0}, b(a(b(a(b(a(b(?x_2)))))))>
<{1,4}, b(c(a(a(a(c(b(?x_2)))))))>
<{1,2}, b(c(a(a(a(c(a(?x_2)))))))>
<{0,1}, b(c(a(a(b(a(b(?x_2)))))))>
<{1}, b(c(a(a(b(?x_2)))))>
<{4}, b(a(a(c(b(?x_2)))))>
<{2}, b(a(a(c(a(?x_2)))))>
<{0}, b(a(b(a(b(?x_2)))))>
<{}, b(a(b(?x_2)))>
Outside Critical Pair: <a(c(b(?x_4))), b(a(b(?x_4)))> by Rules <4, 0>
develop reducts from lhs term...
<{1,3,4}, c(a(a(c(a(c(a(c(b(?x_4)))))))))>
<{1,2,3}, c(a(a(c(a(c(a(c(a(?x_4)))))))))>
<{0,1,3}, c(a(a(c(a(c(b(a(b(?x_4)))))))))>
<{1,3}, c(a(a(c(a(c(b(?x_4)))))))>
<{1,4}, c(a(a(c(a(c(b(?x_4)))))))>
<{1,2}, c(a(a(c(a(c(a(?x_4)))))))>
<{0,1}, c(a(a(c(b(a(b(?x_4)))))))>
<{1}, c(a(a(c(b(?x_4)))))>
<{3,4}, a(c(a(c(a(c(b(?x_4)))))))>
<{2,3}, a(c(a(c(a(c(a(?x_4)))))))>
<{0,3}, a(c(a(c(b(a(b(?x_4)))))))>
<{3}, a(c(a(c(b(?x_4)))))>
<{4}, a(c(a(c(b(?x_4)))))>
<{2}, a(c(a(c(a(?x_4)))))>
<{0}, a(c(b(a(b(?x_4)))))>
<{}, a(c(b(?x_4)))>
develop reducts from rhs term...
<{1,4}, a(c(b(c(a(a(a(c(b(?x_4)))))))))>
<{1,2,4}, a(c(b(c(a(a(a(c(a(?x_4)))))))))>
<{0,1,4}, a(c(b(c(a(a(b(a(b(?x_4)))))))))>
<{1,4}, a(c(b(c(a(a(b(?x_4)))))))>
<{4}, a(c(b(a(a(c(b(?x_4)))))))>
<{2,4}, a(c(b(a(a(c(a(?x_4)))))))>
<{0,4}, a(c(b(a(b(a(b(?x_4)))))))>
<{4}, a(c(b(a(b(?x_4)))))>
<{1,2,4}, a(c(a(c(a(a(a(c(b(?x_4)))))))))>
<{1,2}, a(c(a(c(a(a(a(c(a(?x_4)))))))))>
<{0,1,2}, a(c(a(c(a(a(b(a(b(?x_4)))))))))>
<{1,2}, a(c(a(c(a(a(b(?x_4)))))))>
<{2,4}, a(c(a(a(a(c(b(?x_4)))))))>
<{2}, a(c(a(a(a(c(a(?x_4)))))))>
<{0,2}, a(c(a(a(b(a(b(?x_4)))))))>
<{2}, a(c(a(a(b(?x_4)))))>
<{0,1,4}, b(a(b(c(a(a(a(c(b(?x_4)))))))))>
<{0,1,2}, b(a(b(c(a(a(a(c(a(?x_4)))))))))>
<{0,1}, b(a(b(c(a(a(b(a(b(?x_4)))))))))>
<{0,1}, b(a(b(c(a(a(b(?x_4)))))))>
<{0,4}, b(a(b(a(a(c(b(?x_4)))))))>
<{0,2}, b(a(b(a(a(c(a(?x_4)))))))>
<{0}, b(a(b(a(b(a(b(?x_4)))))))>
<{1,4}, b(c(a(a(a(c(b(?x_4)))))))>
<{1,2}, b(c(a(a(a(c(a(?x_4)))))))>
<{0,1}, b(c(a(a(b(a(b(?x_4)))))))>
<{1}, b(c(a(a(b(?x_4)))))>
<{4}, b(a(a(c(b(?x_4)))))>
<{2}, b(a(a(c(a(?x_4)))))>
<{0}, b(a(b(a(b(?x_4)))))>
<{}, b(a(b(?x_4)))>
Outside Critical Pair: <a(c(b(?x_4))), a(c(a(?x_4)))> by Rules <4, 2>
develop reducts from lhs term...
<{1,3,4}, c(a(a(c(a(c(a(c(b(?x_4)))))))))>
<{1,2,3}, c(a(a(c(a(c(a(c(a(?x_4)))))))))>
<{0,1,3}, c(a(a(c(a(c(b(a(b(?x_4)))))))))>
<{1,3}, c(a(a(c(a(c(b(?x_4)))))))>
<{1,4}, c(a(a(c(a(c(b(?x_4)))))))>
<{1,2}, c(a(a(c(a(c(a(?x_4)))))))>
<{0,1}, c(a(a(c(b(a(b(?x_4)))))))>
<{1}, c(a(a(c(b(?x_4)))))>
<{3,4}, a(c(a(c(a(c(b(?x_4)))))))>
<{2,3}, a(c(a(c(a(c(a(?x_4)))))))>
<{0,3}, a(c(a(c(b(a(b(?x_4)))))))>
<{3}, a(c(a(c(b(?x_4)))))>
<{4}, a(c(a(c(b(?x_4)))))>
<{2}, a(c(a(c(a(?x_4)))))>
<{0}, a(c(b(a(b(?x_4)))))>
<{}, a(c(b(?x_4)))>
develop reducts from rhs term...
<{1,3}, c(a(a(c(a(c(c(a(a(?x_4)))))))))>
<{1,3}, c(a(a(c(a(c(a(?x_4)))))))>
<{1}, c(a(a(c(c(a(a(?x_4)))))))>
<{1}, c(a(a(c(a(?x_4)))))>
<{1,3}, a(c(a(c(c(a(a(?x_4)))))))>
<{3}, a(c(a(c(a(?x_4)))))>
<{1}, a(c(c(a(a(?x_4)))))>
<{}, a(c(a(?x_4)))>
Try A Minimal Decomposition {0,2}{3,4}{1}
{0,2}
(cm)Rewrite Rules:
   [ b(?x) -> b(a(b(?x))),
     b(?x) -> a(c(a(?x))) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ b(a(b(?x))) = a(c(a(?x))) ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ a(c(a(?x))) = b(a(b(?x))),
     b(a(b(?x))) = a(c(a(?x))) ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Root-E-Closed/Non-E-Overlapping
unknown Strongly Weight-Preserving & Root-E-Closed/Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <a(c(a(?x_1))), b(a(b(?x_1)))> by Rules <1, 0> preceded by []
unknown Diagram Decreasing
check Non-Confluence...
obtain 5 rules by 3 steps unfolding
obtain 100 candidates for checking non-joinability
check by TCAP-Approximation (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (failure)
check by Ordering(rpo), check by Tree-Automata Approximation (success)
Witness for Non-Confluence: <b(a(b(c_1))) <- b(c_1) -> a(c(a(c_1)))>
Direct Methods: not CR
{3,4}
(cm)Rewrite Rules:
   [ c(?x) -> c(a(c(?x))),
     b(?x) -> a(c(b(?x))) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [  ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth & Bendix
Linear
Development Closed
Direct Methods: CR
{1}
(cm)Rewrite Rules:
   [ a(?x) -> c(a(a(?x))) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [  ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth & Bendix
Linear
Development Closed
Direct Methods: CR
Try A Minimal Decomposition {4,0,2}{1}{3}
{4,0,2}
(cm)Rewrite Rules:
   [ b(?x) -> a(c(b(?x))),
     b(?x) -> b(a(b(?x))),
     b(?x) -> a(c(a(?x))) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ a(c(b(?x))) = b(a(b(?x))),
     a(c(b(?x))) = a(c(a(?x))),
     b(a(b(?x_1))) = a(c(a(?x_1))) ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ b(a(b(?x))) = a(c(b(?x))),
     a(c(a(?x))) = a(c(b(?x))),
     a(c(b(?x))) = b(a(b(?x))),
     a(c(a(?x))) = b(a(b(?x))),
     a(c(b(?x))) = a(c(a(?x))),
     b(a(b(?x))) = a(c(a(?x))) ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Root-E-Closed/Non-E-Overlapping
unknown Strongly Weight-Preserving & Root-E-Closed/Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <b(a(b(?x_1))), a(c(b(?x_1)))> by Rules <1, 0> preceded by []
 joinable by a reduction of rules <[([],0)], [([(a,1),(c,1)],1)]>
Critical Pair <a(c(a(?x_2))), a(c(b(?x_2)))> by Rules <2, 0> preceded by []
unknown Diagram Decreasing
check Non-Confluence...
obtain 10 rules by 3 steps unfolding
obtain 100 candidates for checking non-joinability
check by TCAP-Approximation (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (failure)
check by Ordering(rpo), check by Tree-Automata Approximation (success)
Witness for Non-Confluence: <a(c(b(c_1))) <- b(c_1) -> a(c(a(c_1)))>
Direct Methods: not CR
{1}
(cm)Rewrite Rules:
   [ a(?x) -> c(a(a(?x))) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [  ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth & Bendix
Linear
Development Closed
Direct Methods: CR
{3}
(cm)Rewrite Rules:
   [ c(?x) -> c(a(c(?x))) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [  ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth & Bendix
Linear
Development Closed
Direct Methods: CR
Commutative Decomposition failed: Can't judge
No further decomposition possible


Combined result: Can't judge
1131.trs: Failure(unknown CR)
(63835 msec.)