MAYBE
(ignored inputs)COMMENT Cops #647 - #721: generated ground TRSs; evenly distributed in the UNR/UNC/NFP/CR hierarchy submitted by: Bertram Felgenhauer
Rewrite Rules:
   [ f(h(f(b),a)) -> b,
     f(c) -> b,
     b -> f(b) ]
Apply Direct Methods...
Inner CPs:
   [ f(h(f(f(b)),a)) = b ]
Outer CPs:
   [  ]
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:
   [ f(h(f(f(b)),a)) = b,
     b = f(h(f(f(b)),a)) ]
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 <f(h(f(f(b)),a)), b> by Rules <2, 0> preceded by [(f,1),(h,1),(f,1)]
unknown Diagram Decreasing
check Non-Confluence...
obtain 11 rules by 3 steps unfolding
obtain 66 candidates for checking non-joinability
check by TCAP-Approximation (failure)
check by Ordering(rpo), check by Tree-Automata Approximation (success)
<F,Q,Q_f,Delta> where
   F = {(a,0),(b,0),(c,0),(f,1),(h,2)}
   Q = {q_0,q_1,q_{h(f(f(b)),a)},q_{a},q_{b},q_{f(b)},q_{f(f(b))}}
   Qf = {q_0,q_1}
   Delta = 
   [ a -> q_{a},
     b -> q_1,
     b -> q_{b},
     f(q_1) -> q_1,
     f(q_1) -> q_{b},
     f(q_{h(f(f(b)),a)}) -> q_0,
     f(q_{b}) -> q_{f(b)},
     f(q_{f(b)}) -> q_{f(f(b))},
     h(q_{f(f(b))},q_{a}) -> q_{h(f(f(b)),a)} ]
 (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:
   [ f(h(f(b),a)) -> b,
     f(c) -> b,
     b -> f(b) ]
E-part:
   [ ]
...failed to find a suitable LPO.
unknown Confluence by Ordered Rewriting
Direct Methods: Can't judge

Try Persistent Decomposition for...
   [ f(h(f(b),a)) -> b,
     f(c) -> b,
     b -> f(b) ]
Sort Assignment:
 a : =>7
 b : =>9
 c : =>9
 f : 9=>9
 h : 9*7=>9
maximal types: {7,9}
Persistent Decomposition failed: Can't judge

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

Try Commutative Decomposition for...
   [ f(h(f(b),a)) -> b,
     f(c) -> b,
     b -> f(b) ]
Inside Critical Pair: <f(h(f(f(b)),a)), b> by Rules <2, 0>
develop reducts from lhs term...
<{2}, f(h(f(f(f(b))),a))>
<{}, f(h(f(f(b)),a))>
develop reducts from rhs term...
<{2}, f(b)>
<{}, b>
Try A Minimal Decomposition {0,2}{1}
{0,2}
(cm)Rewrite Rules:
   [ f(h(f(b),a)) -> b,
     b -> f(b) ]
Apply Direct Methods...
Inner CPs:
   [ f(h(f(f(b)),a)) = b ]
Outer CPs:
   [  ]
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:
   [ f(h(f(f(b)),a)) = b,
     b = f(h(f(f(b)),a)) ]
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 <f(h(f(f(b)),a)), b> by Rules <1, 0> preceded by [(f,1),(h,1),(f,1)]
unknown Diagram Decreasing
check Non-Confluence...
obtain 10 rules by 3 steps unfolding
obtain 86 candidates for checking non-joinability
check by TCAP-Approximation (failure)
check by Ordering(rpo), check by Tree-Automata Approximation (success)
<F,Q,Q_f,Delta> where
   F = {(a,0),(b,0),(f,1),(h,2)}
   Q = {q_0,q_1,q_{h(f(f(b)),a)},q_{a},q_{b},q_{f(b)},q_{f(f(b))}}
   Qf = {q_0,q_1}
   Delta = 
   [ a -> q_{a},
     b -> q_1,
     b -> q_{b},
     f(q_1) -> q_1,
     f(q_1) -> q_{b},
     f(q_{h(f(f(b)),a)}) -> q_0,
     f(q_{b}) -> q_{f(b)},
     f(q_{f(b)}) -> q_{f(f(b))},
     h(q_{f(f(b))},q_{a}) -> q_{h(f(f(b)),a)} ]
 (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:
   [ f(h(f(b),a)) -> b,
     b -> f(b) ]
E-part:
   [ ]
...failed to find a suitable LPO.
unknown Confluence by Ordered Rewriting
Direct Methods: Can't judge

Try Persistent Decomposition for...
   [ f(h(f(b),a)) -> b,
     b -> f(b) ]
Sort Assignment:
 a : =>6
 b : =>8
 f : 8=>8
 h : 8*6=>8
maximal types: {6,8}
Persistent Decomposition failed: Can't judge

Try Layer Preserving Decomposition for...
   [ f(h(f(b),a)) -> b,
     b -> f(b) ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible

{1}
(cm)Rewrite Rules:
   [ f(c) -> b ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [  ]
Overlay, check Innermost Termination...
Innermost Terminating (hence Terminating), WCR
Knuth & Bendix
Direct Methods: CR
Commutative Decomposition failed: Can't judge
No further decomposition possible


Combined result: Can't judge
711.trs: Failure(unknown CR)
(1496 msec.)