(ignored inputs)COMMENT submitted by: Johannes Waldmann
Rewrite Rules:
[ a(b(?x)) -> a(b(?x)),
c(a(?x)) -> b(a(?x)),
a(b(?x)) -> b(c(?x)),
b(a(?x)) -> b(c(?x)),
c(b(?x)) -> a(c(?x)),
b(c(?x)) -> c(b(?x)),
a(b(?x)) -> b(a(?x)),
a(a(?x)) -> b(c(?x)) ]
Apply Direct Methods...
Inner CPs:
[ a(b(c(?x_3))) = a(b(a(?x_3))),
a(c(b(?x_5))) = a(b(c(?x_5))),
c(a(b(?x))) = b(a(b(?x))),
c(b(c(?x_2))) = b(a(b(?x_2))),
c(b(a(?x_6))) = b(a(b(?x_6))),
c(b(c(?x_7))) = b(a(a(?x_7))),
a(b(c(?x_3))) = b(c(a(?x_3))),
a(c(b(?x_5))) = b(c(c(?x_5))),
b(a(b(?x))) = b(c(b(?x))),
b(b(c(?x_2))) = b(c(b(?x_2))),
b(b(a(?x_6))) = b(c(b(?x_6))),
b(b(c(?x_7))) = b(c(a(?x_7))),
c(b(c(?x_3))) = a(c(a(?x_3))),
c(c(b(?x_5))) = a(c(c(?x_5))),
b(b(a(?x_1))) = c(b(a(?x_1))),
b(a(c(?x_4))) = c(b(b(?x_4))),
a(b(c(?x_3))) = b(a(a(?x_3))),
a(c(b(?x_5))) = b(a(c(?x_5))),
a(a(b(?x))) = b(c(b(?x))),
a(b(c(?x_2))) = b(c(b(?x_2))),
a(b(a(?x_6))) = b(c(b(?x_6))),
a(b(c(?x))) = b(c(a(?x))) ]
Outer CPs:
[ a(b(?x)) = b(c(?x)),
a(b(?x)) = b(a(?x)),
b(c(?x_2)) = b(a(?x_2)) ]
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(?x)) = a(b(?x)),
b(a(?x)) = a(b(?x)),
a(b(c(?x_4))) = a(b(a(?x_4))),
a(c(b(?x_6))) = a(b(c(?x_6))),
b(a(b(c(?x_4)))) = c(a(b(a(?x_4)))),
b(a(c(b(?x_6)))) = c(a(b(c(?x_6)))),
b(c(b(c(?x_4)))) = b(a(b(a(?x_4)))),
b(c(c(b(?x_6)))) = b(a(b(c(?x_6)))),
b(c(b(c(?x_4)))) = a(a(b(a(?x_4)))),
b(c(c(b(?x_6)))) = a(a(b(c(?x_6)))),
b(a(b(?x))) = c(a(b(?x))),
b(c(b(?x))) = b(a(b(?x))),
b(c(b(?x))) = a(a(b(?x))),
c(a(b(?x_2))) = b(a(b(?x_2))),
c(b(c(?x_3))) = b(a(b(?x_3))),
c(b(a(?x_7))) = b(a(b(?x_7))),
c(b(c(?x_8))) = b(a(a(?x_8))),
c(b(a(b(?x_2)))) = b(b(a(b(?x_2)))),
c(b(b(c(?x_3)))) = b(b(a(b(?x_3)))),
c(b(b(a(?x_7)))) = b(b(a(b(?x_7)))),
c(b(b(c(?x_8)))) = b(b(a(a(?x_8)))),
c(b(a(?x))) = b(b(a(?x))),
a(b(?x)) = b(c(?x)),
b(a(?x)) = b(c(?x)),
a(b(c(?x_4))) = b(c(a(?x_4))),
a(c(b(?x_6))) = b(c(c(?x_6))),
b(a(b(c(?x_4)))) = c(b(c(a(?x_4)))),
b(a(c(b(?x_6)))) = c(b(c(c(?x_6)))),
b(c(b(c(?x_4)))) = b(b(c(a(?x_4)))),
b(c(c(b(?x_6)))) = b(b(c(c(?x_6)))),
b(c(b(c(?x_4)))) = a(b(c(a(?x_4)))),
b(c(c(b(?x_6)))) = a(b(c(c(?x_6)))),
b(a(b(?x))) = c(b(c(?x))),
b(c(b(?x))) = b(b(c(?x))),
b(c(b(?x))) = a(b(c(?x))),
b(a(b(?x_2))) = b(c(b(?x_2))),
b(b(c(?x_4))) = b(c(b(?x_4))),
b(b(a(?x_7))) = b(c(b(?x_7))),
b(b(c(?x_8))) = b(c(a(?x_8))),
a(b(a(b(?x_2)))) = a(b(c(b(?x_2)))),
a(b(b(c(?x_4)))) = a(b(c(b(?x_4)))),
a(b(b(a(?x_7)))) = a(b(c(b(?x_7)))),
a(b(b(c(?x_8)))) = a(b(c(a(?x_8)))),
b(c(a(b(?x_2)))) = a(b(c(b(?x_2)))),
b(c(b(c(?x_4)))) = a(b(c(b(?x_4)))),
b(c(b(a(?x_7)))) = a(b(c(b(?x_7)))),
a(c(a(b(?x_2)))) = c(b(c(b(?x_2)))),
a(c(b(c(?x_4)))) = c(b(c(b(?x_4)))),
a(c(b(a(?x_7)))) = c(b(c(b(?x_7)))),
a(c(b(c(?x_8)))) = c(b(c(a(?x_8)))),
b(a(a(b(?x_2)))) = a(b(c(b(?x_2)))),
b(a(b(c(?x_4)))) = a(b(c(b(?x_4)))),
b(a(b(a(?x_7)))) = a(b(c(b(?x_7)))),
b(a(b(c(?x_8)))) = a(b(c(a(?x_8)))),
a(b(a(?x))) = a(b(c(?x))),
b(c(a(?x))) = a(b(c(?x))),
a(c(a(?x))) = c(b(c(?x))),
b(a(a(?x))) = a(b(c(?x))),
c(b(c(?x_5))) = a(c(a(?x_5))),
c(c(b(?x_6))) = a(c(c(?x_6))),
c(b(b(c(?x_5)))) = b(a(c(a(?x_5)))),
c(b(c(b(?x_6)))) = b(a(c(c(?x_6)))),
c(b(b(?x))) = b(a(c(?x))),
b(b(a(?x_3))) = c(b(a(?x_3))),
b(a(c(?x_6))) = c(b(b(?x_6))),
a(b(b(a(?x_3)))) = a(c(b(a(?x_3)))),
a(b(a(c(?x_6)))) = a(c(b(b(?x_6)))),
b(c(b(a(?x_3)))) = a(c(b(a(?x_3)))),
b(c(a(c(?x_6)))) = a(c(b(b(?x_6)))),
a(c(b(a(?x_3)))) = c(c(b(a(?x_3)))),
a(c(a(c(?x_6)))) = c(c(b(b(?x_6)))),
b(a(b(a(?x_3)))) = a(c(b(a(?x_3)))),
b(a(a(c(?x_6)))) = a(c(b(b(?x_6)))),
a(b(c(?x))) = a(c(b(?x))),
b(c(c(?x))) = a(c(b(?x))),
a(c(c(?x))) = c(c(b(?x))),
b(a(c(?x))) = a(c(b(?x))),
a(b(?x)) = b(a(?x)),
b(c(?x)) = b(a(?x)),
a(b(c(?x_5))) = b(a(a(?x_5))),
a(c(b(?x_7))) = b(a(c(?x_7))),
b(a(b(c(?x_5)))) = c(b(a(a(?x_5)))),
b(a(c(b(?x_7)))) = c(b(a(c(?x_7)))),
b(c(b(c(?x_5)))) = b(b(a(a(?x_5)))),
b(c(c(b(?x_7)))) = b(b(a(c(?x_7)))),
b(c(b(c(?x_5)))) = a(b(a(a(?x_5)))),
b(c(c(b(?x_7)))) = a(b(a(c(?x_7)))),
b(a(b(?x))) = c(b(a(?x))),
b(c(b(?x))) = b(b(a(?x))),
b(c(b(?x))) = a(b(a(?x))),
a(a(b(?x_2))) = b(c(b(?x_2))),
a(b(c(?x_4))) = b(c(b(?x_4))),
a(b(a(?x_8))) = b(c(b(?x_8))),
b(a(a(b(?x_2)))) = c(b(c(b(?x_2)))),
b(a(b(c(?x_4)))) = c(b(c(b(?x_4)))),
b(a(b(a(?x_8)))) = c(b(c(b(?x_8)))),
b(c(a(b(?x_2)))) = b(b(c(b(?x_2)))),
b(c(b(c(?x_4)))) = b(b(c(b(?x_4)))),
b(c(b(a(?x_8)))) = b(b(c(b(?x_8)))),
b(a(a(?x))) = c(b(c(?x))),
b(c(a(?x))) = b(b(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 by Rules <3, 0> preceded by [(a,1)]
joinable by a reduction of rules <[], [([(a,1)],3)]>
Critical Pair by Rules <5, 0> preceded by [(a,1)]
joinable by a reduction of rules <[], [([(a,1)],5)]>
Critical Pair by Rules <0, 1> preceded by [(c,1)]
joinable by a reduction of rules <[([],1)], []>
Critical Pair by Rules <2, 1> preceded by [(c,1)]
joinable by a reduction of rules <[([(c,1)],5),([(c,1)],4),([],1)], [([],3),([(b,1)],4)]>
joinable by a reduction of rules <[([(c,1)],5),([(c,1)],4),([],1)], [([(b,1)],2),([(b,1)],5),([(b,1)],4)]>
Critical Pair by Rules <6, 1> preceded by [(c,1)]
joinable by a reduction of rules <[([(c,1)],3),([(c,1)],5),([(c,1)],4),([],1)], [([],3),([(b,1)],4)]>
joinable by a reduction of rules <[([(c,1)],3),([(c,1)],5),([(c,1)],4),([],1)], [([(b,1)],2),([(b,1)],5),([(b,1)],4)]>
joinable by a reduction of rules <[([],4),([(a,1)],1),([(a,1)],3),([(a,1)],5)], [([],3),([],5),([],4)]>
joinable by a reduction of rules <[([],4),([(a,1)],1),([(a,1)],3),([],6)], [([],3),([(b,1)],4)]>
joinable by a reduction of rules <[([],4),([(a,1)],1),([(a,1)],3),([],6)], [([(b,1)],2),([(b,1)],5),([(b,1)],4)]>
joinable by a reduction of rules <[([],4),([(a,1)],1),([(a,1)],3),([],2)], [([],3),([(b,1)],4),([],3)]>
joinable by a reduction of rules <[([],4),([(a,1)],1),([],6),([(b,1)],7)], [([(b,1)],2)]>
joinable by a reduction of rules <[([],4),([(a,1)],1),([],6),([(b,1)],7)], [([(b,1)],6),([(b,1)],3)]>
joinable by a reduction of rules <[([],4),([(a,1)],1),([],2),([(b,1)],1)], [([(b,1)],6)]>
joinable by a reduction of rules <[([(c,1)],3)], [([],3),([(b,1)],4),([],3),([],5)]>
joinable by a reduction of rules <[([(c,1)],3),([(c,1)],5),([(c,1)],4),([],1)], [([(b,1)],6),([(b,1)],3),([(b,1)],5),([(b,1)],4)]>
joinable by a reduction of rules <[([],4),([(a,1)],1),([(a,1)],3),([(a,1)],5)], [([(b,1)],2),([(b,1)],5),([],5),([],4)]>
joinable by a reduction of rules <[([],4),([(a,1)],1),([(a,1)],3),([],6)], [([(b,1)],6),([(b,1)],3),([(b,1)],5),([(b,1)],4)]>
joinable by a reduction of rules <[([],4),([(a,1)],1),([(a,1)],3),([],2)], [([(b,1)],2),([(b,1)],5),([(b,1)],4),([],3)]>
Critical Pair by Rules <7, 1> preceded by [(c,1)]
joinable by a reduction of rules <[], [([],3),([],5),([(c,1)],3)]>
joinable by a reduction of rules <[([(c,1)],5),([(c,1)],4),([],1)], [([(b,1)],7),([(b,1)],5),([(b,1)],4)]>
Critical Pair by Rules <3, 2> preceded by [(a,1)]
joinable by a reduction of rules <[([],2),([],5)], [([],5),([(c,1)],3)]>
Critical Pair by Rules <5, 2> preceded by [(a,1)]
joinable by a reduction of rules <[([(a,1)],4),([],7)], []>
Critical Pair by Rules <0, 3> preceded by [(b,1)]
joinable by a reduction of rules <[([],3)], []>
Critical Pair by Rules <2, 3> preceded by [(b,1)]
joinable by a reduction of rules <[([(b,1)],5)], []>
Critical Pair by Rules <6, 3> preceded by [(b,1)]
joinable by a reduction of rules <[([(b,1)],3),([(b,1)],5)], []>
Critical Pair by Rules <7, 3> preceded by [(b,1)]
joinable by a reduction of rules <[], [([(b,1)],1),([(b,1)],3)]>
Critical Pair by Rules <3, 4> preceded by [(c,1)]
joinable by a reduction of rules <[([(c,1)],5),([(c,1)],4),([],1)], [([(a,1)],1),([(a,1)],3),([],6)]>
Critical Pair by Rules <5, 4> preceded by [(c,1)]
joinable by a reduction of rules <[([(c,1)],4),([],1),([],3),([],5),([],4)], []>
Critical Pair by Rules <1, 5> preceded by [(b,1)]
joinable by a reduction of rules <[], [([],4),([(a,1)],1),([],2),([(b,1)],1)]>
joinable by a reduction of rules <[([(b,1)],3)], [([],4),([(a,1)],1),([],6),([(b,1)],7)]>
joinable by a reduction of rules <[([(b,1)],3),([(b,1)],5),([(b,1)],4)], [([(c,1)],3),([(c,1)],5),([(c,1)],4),([],1)]>
joinable by a reduction of rules <[([(b,1)],3),([(b,1)],5),([(b,1)],4)], [([],4),([(a,1)],1),([(a,1)],3),([],6)]>
joinable by a reduction of rules <[([(b,1)],3),([(b,1)],5),([(b,1)],4),([],3)], [([],4),([(a,1)],1),([(a,1)],3),([],2)]>
joinable by a reduction of rules <[([(b,1)],3),([(b,1)],5),([],5),([],4)], [([],4),([(a,1)],1),([(a,1)],3),([(a,1)],5)]>
Critical Pair by Rules <4, 5> preceded by [(b,1)]
joinable by a reduction of rules <[([],3)], [([],4),([(a,1)],4),([],7)]>
Critical Pair by Rules <3, 6> preceded by [(a,1)]
joinable by a reduction of rules <[([],6)], [([(b,1)],7),([(b,1)],5),([(b,1)],4)]>
joinable by a reduction of rules <[([],2),([],5)], [([],3),([],5),([(c,1)],3)]>
joinable by a reduction of rules <[([],6),([],3),([],5)], [([],3),([],5),([(c,1)],3)]>
Critical Pair by Rules <5, 6> preceded by [(a,1)]
joinable by a reduction of rules <[([(a,1)],4),([],7)], [([],3)]>
Critical Pair by Rules <0, 7> preceded by [(a,1)]
joinable by a reduction of rules <[([],7)], []>
Critical Pair by Rules <2, 7> preceded by [(a,1)]
joinable by a reduction of rules <[([],6)], [([(b,1)],4)]>
Critical Pair by Rules <6, 7> preceded by [(a,1)]
joinable by a reduction of rules <[([(a,1)],3),([],6)], [([(b,1)],4)]>
joinable by a reduction of rules <[([(a,1)],3),([(a,1)],5)], [([],5),([],4)]>
joinable by a reduction of rules <[([(a,1)],3),([],2)], [([(b,1)],4),([],3)]>
Critical Pair by Rules <7, 7> preceded by [(a,1)]
joinable by a reduction of rules <[([],2),([],5)], [([],5),([(c,1)],3)]>
Critical Pair by Rules <2, 0> preceded by []
joinable by a reduction of rules <[], [([],2)]>
Critical Pair by Rules <6, 0> preceded by []
joinable by a reduction of rules <[], [([],6)]>
joinable by a reduction of rules <[([],3)], [([],2)]>
Critical Pair by Rules <6, 2> preceded by []
joinable by a reduction of rules <[([],3)], []>
unknown Diagram Decreasing
check Non-Confluence...
obtain 13 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:
[ c(a(?x)) -> b(a(?x)),
a(b(?x)) -> b(c(?x)),
b(a(?x)) -> b(c(?x)),
c(b(?x)) -> a(c(?x)),
b(c(?x)) -> c(b(?x)),
a(b(?x)) -> b(a(?x)),
a(a(?x)) -> b(c(?x)) ]
P:
[ a(b(?x)) -> a(b(?x)) ]
S: unknown termination
failure(Step 1)
STEP: 2 (linear)
S:
[ c(a(?x)) -> b(a(?x)),
a(b(?x)) -> b(c(?x)),
b(a(?x)) -> b(c(?x)),
c(b(?x)) -> a(c(?x)),
b(c(?x)) -> c(b(?x)),
a(b(?x)) -> b(a(?x)),
a(a(?x)) -> b(c(?x)) ]
P:
[ a(b(?x)) -> a(b(?x)) ]
S: unknown termination
failure(Step 2)
STEP: 3 (relative)
S:
[ c(a(?x)) -> b(a(?x)),
a(b(?x)) -> b(c(?x)),
b(a(?x)) -> b(c(?x)),
c(b(?x)) -> a(c(?x)),
b(c(?x)) -> c(b(?x)),
a(b(?x)) -> b(a(?x)),
a(a(?x)) -> b(c(?x)) ]
P:
[ a(b(?x)) -> a(b(?x)) ]
Check relative termination:
[ c(a(?x)) -> b(a(?x)),
a(b(?x)) -> b(c(?x)),
b(a(?x)) -> b(c(?x)),
c(b(?x)) -> a(c(?x)),
b(c(?x)) -> c(b(?x)),
a(b(?x)) -> b(a(?x)),
a(a(?x)) -> b(c(?x)) ]
[ a(b(?x)) -> a(b(?x)) ]