(ignored inputs)COMMENT submitted by: Johannes Waldmann
Rewrite Rules:
   [ b(b(?x)) -> a(a(?x)),
     a(a(?x)) -> b(b(?x)),
     a(c(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     c(c(?x)) -> a(a(?x)),
     a(a(?x)) -> a(c(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ a(c(c(?x_2))) = b(b(c(?x_2))),
     a(c(a(?x_3))) = b(b(b(?x_3))),
     a(a(c(?x_5))) = b(b(a(?x_5))),
     a(a(a(?x_4))) = c(c(c(?x_4))),
     a(a(a(?x))) = c(a(b(?x))),
     a(b(b(?x_1))) = a(c(a(?x_1))),
     a(c(c(?x_2))) = a(c(c(?x_2))),
     a(c(a(?x_3))) = a(c(b(?x_3))),
     b(a(a(?x))) = a(a(b(?x))),
     a(b(b(?x))) = b(b(a(?x))),
     c(a(a(?x))) = a(a(c(?x))),
     a(a(c(?x))) = a(c(a(?x))) ]
Outer CPs:
   [ b(b(?x_1)) = a(c(?x_1)) ]
not Overlay, check Termination...
unknown/not 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(a(?x_1))) = a(a(b(?x_1))),
     a(a(a(a(?x_1)))) = b(a(a(b(?x_1)))),
     c(a(a(a(?x_1)))) = a(a(a(b(?x_1)))),
     a(a(b(?x))) = b(a(a(?x))),
     c(a(b(?x))) = a(a(a(?x))),
     a(c(?x)) = b(b(?x)),
     a(b(b(?x_1))) = b(b(a(?x_1))),
     a(c(c(?x_3))) = b(b(c(?x_3))),
     a(c(a(?x_4))) = b(b(b(?x_4))),
     a(a(c(?x_6))) = b(b(a(?x_6))),
     b(b(b(b(?x_1)))) = a(b(b(a(?x_1)))),
     b(b(c(c(?x_3)))) = a(b(b(c(?x_3)))),
     b(b(c(a(?x_4)))) = a(b(b(b(?x_4)))),
     b(b(a(c(?x_6)))) = a(b(b(a(?x_6)))),
     a(c(b(b(?x_1)))) = a(b(b(a(?x_1)))),
     a(c(c(c(?x_3)))) = a(b(b(c(?x_3)))),
     a(c(c(a(?x_4)))) = a(b(b(b(?x_4)))),
     a(c(a(c(?x_6)))) = a(b(b(a(?x_6)))),
     b(b(a(?x))) = a(b(b(?x))),
     a(c(a(?x))) = a(b(b(?x))),
     a(a(a(?x_5))) = c(c(c(?x_5))),
     b(b(a(a(?x_5)))) = a(c(c(c(?x_5)))),
     a(c(a(a(?x_5)))) = a(c(c(c(?x_5)))),
     b(b(c(?x))) = a(c(c(?x))),
     a(c(c(?x))) = a(c(c(?x))),
     a(a(a(?x_2))) = c(a(b(?x_2))),
     b(b(a(a(?x_2)))) = a(c(a(b(?x_2)))),
     a(c(a(a(?x_2)))) = a(c(a(b(?x_2)))),
     b(b(b(?x))) = a(c(a(?x))),
     a(c(b(?x))) = a(c(a(?x))),
     c(a(a(?x_1))) = a(a(c(?x_1))),
     a(a(a(a(?x_1)))) = c(a(a(c(?x_1)))),
     c(c(a(a(?x_1)))) = a(a(a(c(?x_1)))),
     a(a(c(?x))) = c(a(a(?x))),
     c(c(c(?x))) = a(a(a(?x))),
     b(b(?x)) = a(c(?x)),
     a(a(c(?x_1))) = a(c(a(?x_1))),
     a(b(b(?x_3))) = a(c(a(?x_3))),
     a(c(a(?x_5))) = a(c(b(?x_5))),
     a(c(a(c(?x_1)))) = a(a(c(a(?x_1)))),
     a(c(b(b(?x_3)))) = a(a(c(a(?x_3)))),
     a(c(c(c(?x_4)))) = a(a(c(c(?x_4)))),
     a(c(c(a(?x_5)))) = a(a(c(b(?x_5)))),
     b(b(a(c(?x_1)))) = a(a(c(a(?x_1)))),
     b(b(b(b(?x_3)))) = a(a(c(a(?x_3)))),
     b(b(c(c(?x_4)))) = a(a(c(c(?x_4)))),
     b(b(c(a(?x_5)))) = a(a(c(b(?x_5)))),
     a(c(a(?x))) = a(a(c(?x))),
     b(b(a(?x))) = a(a(c(?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(c(?x_2))), b(b(c(?x_2)))> by Rules <2, 1> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],4),([(a,1)],5)], [([],0)]>
 joinable by a reduction of rules <[([],2),([],4)], [([],0)]>
 joinable by a reduction of rules <[], [([],0),([(a,1)],2)]>
 joinable by a reduction of rules <[], [([],0),([],5)]>
Critical Pair <a(c(a(?x_3))), b(b(b(?x_3)))> by Rules <3, 1> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([],0),([(a,1)],3)]>
Critical Pair <a(a(c(?x_5))), b(b(a(?x_5)))> by Rules <5, 1> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],2),([(a,1)],4)], [([],0)]>
 joinable by a reduction of rules <[([],5),([(a,1)],4)], [([],0)]>
 joinable by a reduction of rules <[], [([],0),([(a,1)],5)]>
Critical Pair <a(a(a(?x_4))), c(c(c(?x_4)))> by Rules <4, 2> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],5)], [([],4)]>
Critical Pair <a(a(a(?x))), c(a(b(?x)))> by Rules <0, 3> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],1),([],3)], []>
 joinable by a reduction of rules <[([],5),([],2)], [([(c,1)],3)]>
 joinable by a reduction of rules <[], [([(c,1)],3),([],4)]>
Critical Pair <a(b(b(?x_1))), a(c(a(?x_1)))> by Rules <1, 5> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],0),([],5)], []>
 joinable by a reduction of rules <[([],3),([(c,1)],3)], [([],2)]>
 joinable by a reduction of rules <[([(a,1)],0)], [([],2),([],4)]>
Critical Pair <a(c(c(?x_2))), a(c(c(?x_2)))> by Rules <2, 5> preceded by [(a,1)]
 joinable by a reduction of rules <[], []>
Critical Pair <a(c(a(?x_3))), a(c(b(?x_3)))> by Rules <3, 5> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([],2),([],4),([(a,1)],3)]>
Critical Pair <b(a(a(?x))), a(a(b(?x)))> by Rules <0, 0> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],1)], [([],1)]>
Critical Pair <a(b(b(?x))), b(b(a(?x)))> by Rules <1, 1> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],0)], [([],0)]>
Critical Pair <c(a(a(?x))), a(a(c(?x)))> by Rules <4, 4> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],5),([(c,1)],2)], [([(a,1)],2),([],2)]>
 joinable by a reduction of rules <[([(c,1)],5),([(c,1)],2)], [([],5),([],2)]>
Critical Pair <a(a(c(?x))), a(c(a(?x)))> by Rules <5, 5> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],2),([(a,1)],4)], [([],2),([],4)]>
 joinable by a reduction of rules <[([],5),([(a,1)],4)], [([],2),([],4)]>
Critical Pair <a(c(?x_5)), b(b(?x_5))> by Rules <5, 1> preceded by []
 joinable by a reduction of rules <[([],2),([],4)], [([],0)]>
 joinable by a reduction of rules <[], [([],0),([],5)]>
unknown Diagram Decreasing
check Non-Confluence...
obtain 11 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...
STEP: 1 (parallel)
S:
   [ a(c(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     c(c(?x)) -> a(a(?x)),
     a(a(?x)) -> a(c(?x)) ]
P:
   [ b(b(?x)) -> a(a(?x)),
     a(a(?x)) -> b(b(?x)) ]
S: unknown termination
failure(Step 1)
STEP: 2 (linear)
S:
   [ a(c(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     c(c(?x)) -> a(a(?x)),
     a(a(?x)) -> a(c(?x)) ]
P:
   [ b(b(?x)) -> a(a(?x)),
     a(a(?x)) -> b(b(?x)) ]
S: unknown termination
failure(Step 2)
STEP: 3 (relative)
S:
   [ a(c(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     c(c(?x)) -> a(a(?x)),
     a(a(?x)) -> a(c(?x)) ]
P:
   [ b(b(?x)) -> a(a(?x)),
     a(a(?x)) -> b(b(?x)) ]
Check relative termination:
   [ a(c(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     c(c(?x)) -> a(a(?x)),
     a(a(?x)) -> a(c(?x)) ]
   [ b(b(?x)) -> a(a(?x)),
     a(a(?x)) -> b(b(?x)) ]
Polynomial Interpretation:
  a:= (1)*x1
  b:= (3)+(1)*x1
  c:= (1)+(1)*x1
retract a(b(?x)) -> c(a(?x))
retract c(c(?x)) -> a(a(?x))
retract b(b(?x)) -> a(a(?x))
Polynomial Interpretation:
  a:= (2)+(1)*x1*x1
  b:= (2)+(1)*x1*x1
  c:= (1)+(1)*x1*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
failure(no possibility remains)
unknown Reduction-Preserving Completion
Direct Methods: Can't judge

Try Persistent Decomposition for...
   [ b(b(?x)) -> a(a(?x)),
     a(a(?x)) -> b(b(?x)),
     a(c(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     c(c(?x)) -> a(a(?x)),
     a(a(?x)) -> a(c(?x)) ]
Sort Assignment:
 a : 13=>13
 b : 13=>13
 c : 13=>13
maximal types: {13}
Persistent Decomposition failed: Can't judge

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

Try Commutative Decomposition for...
   [ b(b(?x)) -> a(a(?x)),
     a(a(?x)) -> b(b(?x)),
     a(c(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     c(c(?x)) -> a(a(?x)),
     a(a(?x)) -> a(c(?x)) ]
Outside Critical Pair: <a(c(?x_5)), b(b(?x_5))> by Rules <5, 1>
develop reducts from lhs term...
<{2}, c(c(?x_5))>
<{}, a(c(?x_5))>
develop reducts from rhs term...
<{0}, a(a(?x_5))>
<{}, b(b(?x_5))>
Inside Critical Pair: <a(c(c(?x_2))), b(b(c(?x_2)))> by Rules <2, 1>
develop reducts from lhs term...
<{2}, c(c(c(?x_2)))>
<{4}, a(a(a(?x_2)))>
<{}, a(c(c(?x_2)))>
develop reducts from rhs term...
<{0}, a(a(c(?x_2)))>
<{}, b(b(c(?x_2)))>
Inside Critical Pair: <a(c(a(?x_3))), b(b(b(?x_3)))> by Rules <3, 1>
develop reducts from lhs term...
<{2}, c(c(a(?x_3)))>
<{}, a(c(a(?x_3)))>
develop reducts from rhs term...
<{0}, a(a(b(?x_3)))>
<{0}, b(a(a(?x_3)))>
<{}, b(b(b(?x_3)))>
Inside Critical Pair: <a(a(c(?x_5))), b(b(a(?x_5)))> by Rules <5, 1>
develop reducts from lhs term...
<{5}, a(c(c(?x_5)))>
<{1}, b(b(c(?x_5)))>
<{2}, a(c(c(?x_5)))>
<{}, a(a(c(?x_5)))>
develop reducts from rhs term...
<{0}, a(a(a(?x_5)))>
<{}, b(b(a(?x_5)))>
Inside Critical Pair: <a(a(a(?x_4))), c(c(c(?x_4)))> by Rules <4, 2>
develop reducts from lhs term...
<{5}, a(c(a(?x_4)))>
<{1}, b(b(a(?x_4)))>
<{5}, a(a(c(?x_4)))>
<{1}, a(b(b(?x_4)))>
<{}, a(a(a(?x_4)))>
develop reducts from rhs term...
<{4}, a(a(c(?x_4)))>
<{4}, c(a(a(?x_4)))>
<{}, c(c(c(?x_4)))>
Inside Critical Pair: <a(a(a(?x))), c(a(b(?x)))> by Rules <0, 3>
develop reducts from lhs term...
<{5}, a(c(a(?x)))>
<{1}, b(b(a(?x)))>
<{5}, a(a(c(?x)))>
<{1}, a(b(b(?x)))>
<{}, a(a(a(?x)))>
develop reducts from rhs term...
<{3}, c(c(a(?x)))>
<{}, c(a(b(?x)))>
Inside Critical Pair: <a(b(b(?x_1))), a(c(a(?x_1)))> by Rules <1, 5>
develop reducts from lhs term...
<{3}, c(a(b(?x_1)))>
<{0}, a(a(a(?x_1)))>
<{}, a(b(b(?x_1)))>
develop reducts from rhs term...
<{2}, c(c(a(?x_1)))>
<{}, a(c(a(?x_1)))>
Inside Critical Pair: <a(c(c(?x_2))), a(c(c(?x_2)))> by Rules <2, 5>
develop reducts from lhs term...
<{2}, c(c(c(?x_2)))>
<{4}, a(a(a(?x_2)))>
<{}, a(c(c(?x_2)))>
develop reducts from rhs term...
<{2}, c(c(c(?x_2)))>
<{4}, a(a(a(?x_2)))>
<{}, a(c(c(?x_2)))>
Inside Critical Pair: <a(c(a(?x_3))), a(c(b(?x_3)))> by Rules <3, 5>
develop reducts from lhs term...
<{2}, c(c(a(?x_3)))>
<{}, a(c(a(?x_3)))>
develop reducts from rhs term...
<{2}, c(c(b(?x_3)))>
<{}, a(c(b(?x_3)))>
Try A Minimal Decomposition {0,5,3,1,2}{4}
{0,5,3,1,2}
(cm)Rewrite Rules:
   [ b(b(?x)) -> a(a(?x)),
     a(a(?x)) -> a(c(?x)),
     a(b(?x)) -> c(a(?x)),
     a(a(?x)) -> b(b(?x)),
     a(c(?x)) -> c(c(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ a(c(a(?x_2))) = a(c(b(?x_2))),
     a(b(b(?x_3))) = a(c(a(?x_3))),
     a(c(c(?x_4))) = a(c(c(?x_4))),
     a(a(a(?x))) = c(a(b(?x))),
     a(a(c(?x_1))) = b(b(a(?x_1))),
     a(c(a(?x_2))) = b(b(b(?x_2))),
     a(c(c(?x_4))) = b(b(c(?x_4))),
     b(a(a(?x))) = a(a(b(?x))),
     a(a(c(?x))) = a(c(a(?x))),
     a(b(b(?x))) = b(b(a(?x))) ]
Outer CPs:
   [ a(c(?x_1)) = b(b(?x_1)) ]
not Overlay, check Termination...
unknown/not 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(a(?x_1))) = a(a(b(?x_1))),
     a(a(a(a(?x_1)))) = b(a(a(b(?x_1)))),
     c(a(a(a(?x_1)))) = a(a(a(b(?x_1)))),
     a(a(b(?x))) = b(a(a(?x))),
     c(a(b(?x))) = a(a(a(?x))),
     b(b(?x)) = a(c(?x)),
     a(a(c(?x_1))) = a(c(a(?x_1))),
     a(c(a(?x_3))) = a(c(b(?x_3))),
     a(b(b(?x_4))) = a(c(a(?x_4))),
     a(c(c(?x_5))) = a(c(c(?x_5))),
     a(c(a(c(?x_1)))) = a(a(c(a(?x_1)))),
     a(c(c(a(?x_3)))) = a(a(c(b(?x_3)))),
     a(c(b(b(?x_4)))) = a(a(c(a(?x_4)))),
     a(c(c(c(?x_5)))) = a(a(c(c(?x_5)))),
     b(b(a(c(?x_1)))) = a(a(c(a(?x_1)))),
     b(b(c(a(?x_3)))) = a(a(c(b(?x_3)))),
     b(b(b(b(?x_4)))) = a(a(c(a(?x_4)))),
     b(b(c(c(?x_5)))) = a(a(c(c(?x_5)))),
     a(c(a(?x))) = a(a(c(?x))),
     b(b(a(?x))) = a(a(c(?x))),
     a(a(a(?x_2))) = c(a(b(?x_2))),
     a(c(a(a(?x_2)))) = a(c(a(b(?x_2)))),
     b(b(a(a(?x_2)))) = a(c(a(b(?x_2)))),
     a(c(b(?x))) = a(c(a(?x))),
     b(b(b(?x))) = a(c(a(?x))),
     a(c(?x)) = b(b(?x)),
     a(b(b(?x_1))) = b(b(a(?x_1))),
     a(a(c(?x_3))) = b(b(a(?x_3))),
     a(c(a(?x_4))) = b(b(b(?x_4))),
     a(c(c(?x_5))) = b(b(c(?x_5))),
     b(b(b(b(?x_1)))) = a(b(b(a(?x_1)))),
     b(b(a(c(?x_3)))) = a(b(b(a(?x_3)))),
     b(b(c(a(?x_4)))) = a(b(b(b(?x_4)))),
     b(b(c(c(?x_5)))) = a(b(b(c(?x_5)))),
     a(c(b(b(?x_1)))) = a(b(b(a(?x_1)))),
     a(c(a(c(?x_3)))) = a(b(b(a(?x_3)))),
     a(c(c(a(?x_4)))) = a(b(b(b(?x_4)))),
     a(c(c(c(?x_5)))) = a(b(b(c(?x_5)))),
     b(b(a(?x))) = a(b(b(?x))),
     a(c(a(?x))) = a(b(b(?x))),
     b(b(c(?x))) = a(c(c(?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))), a(c(b(?x_2)))> by Rules <2, 1> preceded by [(a,1)]
unknown Diagram Decreasing
check Non-Confluence...
obtain 10 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)Witness for Non-Confluence: <a(a(a(c_1))) <- a(a(a(c_1))) -> a(a(c(c_1)))>
Direct Methods: not CR
{4}
(cm)Rewrite Rules:
   [ c(c(?x)) -> a(a(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ c(a(a(?x))) = a(a(c(?x))) ]
Outer CPs:
   [  ]
not Overlay, check Termination...
Terminating, not WCR
Knuth & Bendix
Direct Methods: not CR
Commutative Decomposition failed: Can't judge
No further decomposition possible


Combined result: Can't judge
990.trs: Failure(unknown CR)
MAYBE
(7880 msec.)