(ignored inputs)COMMENT submitted by: Johannes Waldmann
Rewrite Rules:
   [ b(b(?x)) -> c(a(?x)),
     c(a(?x)) -> b(b(?x)),
     c(a(?x)) -> a(b(?x)),
     a(b(?x)) -> b(b(?x)),
     c(b(?x)) -> a(b(?x)),
     a(c(?x)) -> c(c(?x)),
     c(b(?x)) -> b(b(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ c(b(b(?x_3))) = b(b(b(?x_3))),
     c(c(c(?x_5))) = b(b(c(?x_5))),
     c(b(b(?x_3))) = a(b(b(?x_3))),
     c(c(c(?x_5))) = a(b(c(?x_5))),
     a(c(a(?x))) = b(b(b(?x))),
     c(c(a(?x))) = a(b(b(?x))),
     a(b(b(?x_1))) = c(c(a(?x_1))),
     a(a(b(?x_2))) = c(c(a(?x_2))),
     a(a(b(?x_4))) = c(c(b(?x_4))),
     a(b(b(?x_6))) = c(c(b(?x_6))),
     c(c(a(?x))) = b(b(b(?x))),
     b(c(a(?x))) = c(a(b(?x))) ]
Outer CPs:
   [ b(b(?x_1)) = a(b(?x_1)),
     a(b(?x_4)) = b(b(?x_4)) ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
inner CP cond (upside-parallel)
innter CP Cond (outside)
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ b(c(a(?x_1))) = c(a(b(?x_1))),
     c(a(c(a(?x_1)))) = b(c(a(b(?x_1)))),
     b(b(c(a(?x_1)))) = a(c(a(b(?x_1)))),
     a(b(c(a(?x_1)))) = c(c(a(b(?x_1)))),
     b(b(c(a(?x_1)))) = c(c(a(b(?x_1)))),
     c(a(b(?x))) = b(c(a(?x))),
     b(b(b(?x))) = a(c(a(?x))),
     a(b(b(?x))) = c(c(a(?x))),
     b(b(b(?x))) = c(c(a(?x))),
     a(b(?x)) = b(b(?x)),
     c(b(b(?x_4))) = b(b(b(?x_4))),
     c(c(c(?x_6))) = b(b(c(?x_6))),
     c(c(b(b(?x_4)))) = a(b(b(b(?x_4)))),
     c(c(c(c(?x_6)))) = a(b(b(c(?x_6)))),
     c(c(a(?x))) = a(b(b(?x))),
     b(b(?x)) = a(b(?x)),
     c(b(b(?x_4))) = a(b(b(?x_4))),
     c(c(c(?x_6))) = a(b(c(?x_6))),
     c(c(b(b(?x_4)))) = a(a(b(b(?x_4)))),
     c(c(c(c(?x_6)))) = a(a(b(c(?x_6)))),
     c(c(a(?x))) = a(a(b(?x))),
     a(c(a(?x_2))) = b(b(b(?x_2))),
     b(b(c(a(?x_2)))) = c(b(b(b(?x_2)))),
     a(b(c(a(?x_2)))) = c(b(b(b(?x_2)))),
     b(b(b(?x))) = c(b(b(?x))),
     a(b(b(?x))) = c(b(b(?x))),
     c(c(c(a(?x_2)))) = a(a(b(b(?x_2)))),
     c(c(b(?x))) = a(a(b(?x))),
     a(a(b(?x_4))) = c(c(a(?x_4))),
     a(a(b(?x_6))) = c(c(b(?x_6))),
     a(b(b(?x_7))) = c(c(b(?x_7))),
     b(b(b(b(?x_3)))) = c(c(c(a(?x_3)))),
     b(b(a(b(?x_4)))) = c(c(c(a(?x_4)))),
     b(b(a(b(?x_6)))) = c(c(c(b(?x_6)))),
     b(b(b(b(?x_7)))) = c(c(c(b(?x_7)))),
     a(b(b(b(?x_3)))) = c(c(c(a(?x_3)))),
     a(b(a(b(?x_4)))) = c(c(c(a(?x_4)))),
     a(b(a(b(?x_6)))) = c(c(c(b(?x_6)))),
     a(b(b(b(?x_7)))) = c(c(c(b(?x_7)))),
     b(b(c(?x))) = c(c(c(?x))),
     a(b(c(?x))) = c(c(c(?x))),
     c(c(a(?x_2))) = b(b(b(?x_2))),
     c(c(c(a(?x_2)))) = a(b(b(b(?x_2)))),
     c(c(b(?x))) = a(b(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 <c(b(b(?x_3))), b(b(b(?x_3)))> by Rules <3, 1> preceded by [(c,1)]
 joinable by a reduction of rules <[([],6)], []>
Critical Pair <c(c(c(?x_5))), b(b(c(?x_5)))> by Rules <5, 1> preceded by [(c,1)]
 joinable by a reduction of rules <[], [([],0),([(c,1)],5)]>
Critical Pair <c(b(b(?x_3))), a(b(b(?x_3)))> by Rules <3, 2> preceded by [(c,1)]
 joinable by a reduction of rules <[([],4)], []>
 joinable by a reduction of rules <[([],6)], [([],3)]>
Critical Pair <c(c(c(?x_5))), a(b(c(?x_5)))> by Rules <5, 2> preceded by [(c,1)]
 joinable by a reduction of rules <[], [([],3),([],0),([(c,1)],5)]>
Critical Pair <a(c(a(?x))), b(b(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),([(c,1)],2)], [([],0)]>
 joinable by a reduction of rules <[([(a,1)],1)], [([],0),([],2)]>
 joinable by a reduction of rules <[([(a,1)],2),([(a,1)],3)], [([],0),([],2)]>
 joinable by a reduction of rules <[([],5),([(c,1)],1)], [([],0),([(c,1)],3)]>
Critical Pair <c(c(a(?x))), a(b(b(?x)))> by Rules <0, 4> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],2),([],2)], []>
 joinable by a reduction of rules <[([(c,1)],2),([],1)], [([],3)]>
 joinable by a reduction of rules <[([(c,1)],1),([],6)], [([],3)]>
 joinable by a reduction of rules <[([(c,1)],1),([],4)], []>
 joinable by a reduction of rules <[], [([(a,1)],0),([],5)]>
 joinable by a reduction of rules <[([(c,1)],2)], [([],3),([],0)]>
Critical Pair <a(b(b(?x_1))), c(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),([],0)], [([(c,1)],2)]>
 joinable by a reduction of rules <[], [([(c,1)],2),([],2)]>
 joinable by a reduction of rules <[], [([(c,1)],1),([],4)]>
 joinable by a reduction of rules <[([],3)], [([(c,1)],2),([],1)]>
 joinable by a reduction of rules <[([],3)], [([(c,1)],1),([],6)]>
Critical Pair <a(a(b(?x_2))), c(c(a(?x_2)))> by Rules <2, 5> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],3)], [([(c,1)],2),([],2)]>
 joinable by a reduction of rules <[([(a,1)],3)], [([(c,1)],1),([],4)]>
 joinable by a reduction of rules <[([(a,1)],3),([],3)], [([(c,1)],2),([],1)]>
 joinable by a reduction of rules <[([(a,1)],3),([],3)], [([(c,1)],1),([],6)]>
Critical Pair <a(a(b(?x_4))), c(c(b(?x_4)))> by Rules <4, 5> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],3)], [([(c,1)],6),([],4)]>
 joinable by a reduction of rules <[([(a,1)],3)], [([(c,1)],4),([],2)]>
 joinable by a reduction of rules <[([(a,1)],3),([],3)], [([(c,1)],6),([],6)]>
 joinable by a reduction of rules <[([(a,1)],3),([],3)], [([(c,1)],4),([],1)]>
Critical Pair <a(b(b(?x_6))), c(c(b(?x_6)))> by Rules <6, 5> preceded by [(a,1)]
 joinable by a reduction of rules <[([],3),([],0)], [([(c,1)],4)]>
 joinable by a reduction of rules <[], [([(c,1)],6),([],4)]>
 joinable by a reduction of rules <[], [([(c,1)],4),([],2)]>
 joinable by a reduction of rules <[([],3)], [([(c,1)],6),([],6)]>
 joinable by a reduction of rules <[([],3)], [([(c,1)],4),([],1)]>
 joinable by a reduction of rules <[([(a,1)],0),([],5)], [([(c,1)],6),([(c,1)],0)]>
Critical Pair <c(c(a(?x))), b(b(b(?x)))> by Rules <0, 6> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],2)], [([],0)]>
Critical Pair <b(c(a(?x))), c(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(?x_2)), b(b(?x_2))> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[([],3)], []>
Critical Pair <b(b(?x_6)), a(b(?x_6))> by Rules <6, 4> preceded by []
 joinable by a reduction of rules <[], [([],3)]>
unknown Diagram Decreasing
check Non-Confluence...
obtain 12 rules by 3 steps unfolding
obtain 90 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:
   [ c(a(?x)) -> a(b(?x)),
     a(b(?x)) -> b(b(?x)),
     c(b(?x)) -> a(b(?x)),
     a(c(?x)) -> c(c(?x)),
     c(b(?x)) -> b(b(?x)) ]
P:
   [ b(b(?x)) -> c(a(?x)),
     c(a(?x)) -> b(b(?x)) ]
S: terminating
CP(S,S):
<c(b(b(?x_1))), a(b(b(?x_1)))> --> <b(b(b(?x_1))), b(b(b(?x_1)))> => yes
<c(c(c(?x_3))), a(b(c(?x_3)))> --> <c(c(c(?x_3))), b(b(c(?x_3)))> => no
<b(b(?x_4)), a(b(?x_4))> --> <b(b(?x_4)), b(b(?x_4))> => yes
<a(a(b(?x))), c(c(a(?x)))> --> <b(b(b(?x))), b(b(b(?x)))> => yes
<a(a(b(?x_2))), c(c(b(?x_2)))> --> <b(b(b(?x_2))), b(b(b(?x_2)))> => yes
<a(b(b(?x_4))), c(c(b(?x_4)))> --> <b(b(b(?x_4))), b(b(b(?x_4)))> => yes
<a(b(?x_2)), b(b(?x_2))> --> <b(b(?x_2)), b(b(?x_2))> => yes
PCP_in(symP,S):
<a(c(a(?x_2))), b(b(b(?x_2)))> --> <b(b(b(?x_2))), b(b(b(?x_2)))> => yes
<c(c(a(?x_2))), a(b(b(?x_2)))> --> <b(b(b(?x_2))), b(b(b(?x_2)))> => yes
<a(b(b(?x_1))), c(c(a(?x_1)))> --> <b(b(b(?x_1))), b(b(b(?x_1)))> => yes
<c(c(a(?x_2))), b(b(b(?x_2)))> --> <b(b(b(?x_2))), b(b(b(?x_2)))> => yes
CP(S,symP):
<a(b(?x)), b(b(?x))> --> <b(b(?x)), b(b(?x))> => yes
<c(b(b(?x))), b(b(b(?x)))> --> <b(b(b(?x))), b(b(b(?x)))> => yes
<c(c(c(?x))), b(b(c(?x)))> --> <c(c(c(?x))), b(b(c(?x)))> => no
check joinability condition:
check modulo joinability of c(c(c(?x_3))) and b(b(c(?x_3))): joinable by {1}
check modulo reachablity from c(c(c(?x))) to b(b(c(?x))): maybe not reachable
failed
failure(Step 1)
   [ ]
Added S-Rules:
   [ ]
Added P-Rules:
   [ ]
STEP: 2 (linear)
S:
   [ c(a(?x)) -> a(b(?x)),
     a(b(?x)) -> b(b(?x)),
     c(b(?x)) -> a(b(?x)),
     a(c(?x)) -> c(c(?x)),
     c(b(?x)) -> b(b(?x)) ]
P:
   [ b(b(?x)) -> c(a(?x)),
     c(a(?x)) -> b(b(?x)) ]
S: terminating
CP(S,S):
<c(b(b(?x_1))), a(b(b(?x_1)))> --> <b(b(b(?x_1))), b(b(b(?x_1)))> => yes
<c(c(c(?x_3))), a(b(c(?x_3)))> --> <c(c(c(?x_3))), b(b(c(?x_3)))> => no
<b(b(?x_4)), a(b(?x_4))> --> <b(b(?x_4)), b(b(?x_4))> => yes
<a(a(b(?x))), c(c(a(?x)))> --> <b(b(b(?x))), b(b(b(?x)))> => yes
<a(a(b(?x_2))), c(c(b(?x_2)))> --> <b(b(b(?x_2))), b(b(b(?x_2)))> => yes
<a(b(b(?x_4))), c(c(b(?x_4)))> --> <b(b(b(?x_4))), b(b(b(?x_4)))> => yes
<a(b(?x_2)), b(b(?x_2))> --> <b(b(?x_2)), b(b(?x_2))> => yes
CP_in(symP,S):
<a(b(b(?x))), c(c(a(?x)))> --> <b(b(b(?x))), b(b(b(?x)))> => yes
<a(c(a(?x))), b(b(b(?x)))> --> <b(b(b(?x))), b(b(b(?x)))> => yes
<c(c(a(?x))), a(b(b(?x)))> --> <b(b(b(?x))), b(b(b(?x)))> => yes
<c(c(a(?x))), b(b(b(?x)))> --> <b(b(b(?x))), b(b(b(?x)))> => yes
CP(S,symP):
<a(b(?x)), b(b(?x))> --> <b(b(?x)), b(b(?x))> => yes
<c(b(b(?x))), b(b(b(?x)))> --> <b(b(b(?x))), b(b(b(?x)))> => yes
<c(c(c(?x))), b(b(c(?x)))> --> <c(c(c(?x))), b(b(c(?x)))> => no
check joinability condition:
check modulo joinability of c(c(c(?x_3))) and b(b(c(?x_3))): maybe not joinable
check modulo reachablity from c(c(c(?x))) to b(b(c(?x))): maybe not reachable
failed
failure(Step 2)
   [ ]
Added S-Rules:
   [ ]
Added P-Rules:
   [ ]
STEP: 3 (relative)
S:
   [ c(a(?x)) -> a(b(?x)),
     a(b(?x)) -> b(b(?x)),
     c(b(?x)) -> a(b(?x)),
     a(c(?x)) -> c(c(?x)),
     c(b(?x)) -> b(b(?x)) ]
P:
   [ b(b(?x)) -> c(a(?x)),
     c(a(?x)) -> b(b(?x)) ]
Check relative termination:
   [ c(a(?x)) -> a(b(?x)),
     a(b(?x)) -> b(b(?x)),
     c(b(?x)) -> a(b(?x)),
     a(c(?x)) -> c(c(?x)),
     c(b(?x)) -> b(b(?x)) ]
   [ b(b(?x)) -> c(a(?x)),
     c(a(?x)) -> b(b(?x)) ]
Polynomial Interpretation:
  a:= (2)+(2)*x1
  b:= (2)*x1
  c:= (2)*x1
retract c(a(?x)) -> a(b(?x))
retract a(b(?x)) -> b(b(?x))
retract a(c(?x)) -> c(c(?x))
retract c(a(?x)) -> b(b(?x))
Polynomial Interpretation:
  a:= (2)+(1)*x1
  b:= (2)+(3)*x1
  c:= (2)+(3)*x1
retract c(a(?x)) -> a(b(?x))
retract a(b(?x)) -> b(b(?x))
retract c(b(?x)) -> a(b(?x))
retract a(c(?x)) -> c(c(?x))
retract c(a(?x)) -> b(b(?x))
Polynomial Interpretation:
  a:= (1)*x1*x1
  b:= (1)*x1*x1
  c:= (2)+(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)) -> c(a(?x)),
     c(a(?x)) -> b(b(?x)),
     c(a(?x)) -> a(b(?x)),
     a(b(?x)) -> b(b(?x)),
     c(b(?x)) -> a(b(?x)),
     a(c(?x)) -> c(c(?x)),
     c(b(?x)) -> b(b(?x)) ]
Sort Assignment:
 a : 14=>14
 b : 14=>14
 c : 14=>14
maximal types: {14}
Persistent Decomposition failed: Can't judge

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

Try Commutative Decomposition for...
   [ b(b(?x)) -> c(a(?x)),
     c(a(?x)) -> b(b(?x)),
     c(a(?x)) -> a(b(?x)),
     a(b(?x)) -> b(b(?x)),
     c(b(?x)) -> a(b(?x)),
     a(c(?x)) -> c(c(?x)),
     c(b(?x)) -> b(b(?x)) ]
Outside Critical Pair: <a(b(?x_2)), b(b(?x_2))> by Rules <2, 1>
develop reducts from lhs term...
<{3}, b(b(?x_2))>
<{}, a(b(?x_2))>
develop reducts from rhs term...
<{0}, c(a(?x_2))>
<{}, b(b(?x_2))>
Outside Critical Pair: <b(b(?x_6)), a(b(?x_6))> by Rules <6, 4>
develop reducts from lhs term...
<{0}, c(a(?x_6))>
<{}, b(b(?x_6))>
develop reducts from rhs term...
<{3}, b(b(?x_6))>
<{}, a(b(?x_6))>
Inside Critical Pair: <c(b(b(?x_3))), b(b(b(?x_3)))> by Rules <3, 1>
develop reducts from lhs term...
<{6}, b(b(b(?x_3)))>
<{4}, a(b(b(?x_3)))>
<{0}, c(c(a(?x_3)))>
<{}, c(b(b(?x_3)))>
develop reducts from rhs term...
<{0}, c(a(b(?x_3)))>
<{0}, b(c(a(?x_3)))>
<{}, b(b(b(?x_3)))>
Inside Critical Pair: <c(c(c(?x_5))), b(b(c(?x_5)))> by Rules <5, 1>
develop reducts from lhs term...
<{}, c(c(c(?x_5)))>
develop reducts from rhs term...
<{0}, c(a(c(?x_5)))>
<{}, b(b(c(?x_5)))>
Inside Critical Pair: <c(b(b(?x_3))), a(b(b(?x_3)))> by Rules <3, 2>
develop reducts from lhs term...
<{6}, b(b(b(?x_3)))>
<{4}, a(b(b(?x_3)))>
<{0}, c(c(a(?x_3)))>
<{}, c(b(b(?x_3)))>
develop reducts from rhs term...
<{3}, b(b(b(?x_3)))>
<{0}, a(c(a(?x_3)))>
<{}, a(b(b(?x_3)))>
Inside Critical Pair: <c(c(c(?x_5))), a(b(c(?x_5)))> by Rules <5, 2>
develop reducts from lhs term...
<{}, c(c(c(?x_5)))>
develop reducts from rhs term...
<{3}, b(b(c(?x_5)))>
<{}, a(b(c(?x_5)))>
Inside Critical Pair: <a(c(a(?x))), b(b(b(?x)))> by Rules <0, 3>
develop reducts from lhs term...
<{5}, c(c(a(?x)))>
<{2}, a(a(b(?x)))>
<{1}, a(b(b(?x)))>
<{}, a(c(a(?x)))>
develop reducts from rhs term...
<{0}, c(a(b(?x)))>
<{0}, b(c(a(?x)))>
<{}, b(b(b(?x)))>
Inside Critical Pair: <c(c(a(?x))), a(b(b(?x)))> by Rules <0, 4>
develop reducts from lhs term...
<{2}, c(a(b(?x)))>
<{1}, c(b(b(?x)))>
<{}, c(c(a(?x)))>
develop reducts from rhs term...
<{3}, b(b(b(?x)))>
<{0}, a(c(a(?x)))>
<{}, a(b(b(?x)))>
Inside Critical Pair: <a(b(b(?x_1))), c(c(a(?x_1)))> by Rules <1, 5>
develop reducts from lhs term...
<{3}, b(b(b(?x_1)))>
<{0}, a(c(a(?x_1)))>
<{}, a(b(b(?x_1)))>
develop reducts from rhs term...
<{2}, c(a(b(?x_1)))>
<{1}, c(b(b(?x_1)))>
<{}, c(c(a(?x_1)))>
Inside Critical Pair: <a(a(b(?x_2))), c(c(a(?x_2)))> by Rules <2, 5>
develop reducts from lhs term...
<{3}, a(b(b(?x_2)))>
<{}, a(a(b(?x_2)))>
develop reducts from rhs term...
<{2}, c(a(b(?x_2)))>
<{1}, c(b(b(?x_2)))>
<{}, c(c(a(?x_2)))>
Inside Critical Pair: <a(a(b(?x_4))), c(c(b(?x_4)))> by Rules <4, 5>
develop reducts from lhs term...
<{3}, a(b(b(?x_4)))>
<{}, a(a(b(?x_4)))>
develop reducts from rhs term...
<{6}, c(b(b(?x_4)))>
<{4}, c(a(b(?x_4)))>
<{}, c(c(b(?x_4)))>
Inside Critical Pair: <a(b(b(?x_6))), c(c(b(?x_6)))> by Rules <6, 5>
develop reducts from lhs term...
<{3}, b(b(b(?x_6)))>
<{0}, a(c(a(?x_6)))>
<{}, a(b(b(?x_6)))>
develop reducts from rhs term...
<{6}, c(b(b(?x_6)))>
<{4}, c(a(b(?x_6)))>
<{}, c(c(b(?x_6)))>
Inside Critical Pair: <c(c(a(?x))), b(b(b(?x)))> by Rules <0, 6>
develop reducts from lhs term...
<{2}, c(a(b(?x)))>
<{1}, c(b(b(?x)))>
<{}, c(c(a(?x)))>
develop reducts from rhs term...
<{0}, c(a(b(?x)))>
<{0}, b(c(a(?x)))>
<{}, b(b(b(?x)))>
Commutative Decomposition failed: Can't judge
No further decomposition possible


Combined result: Can't judge
997.trs: Failure(unknown CR)
MAYBE
(5713 msec.)