(ignored inputs)COMMENT submitted by: Johannes Waldmann
Rewrite Rules:
[ a(a(?x)) -> b(c(?x)),
b(c(?x)) -> a(b(?x)),
b(a(?x)) -> a(a(?x)),
c(c(?x)) -> a(a(?x)),
c(a(?x)) -> c(a(?x)),
a(a(?x)) -> a(a(?x)),
c(b(?x)) -> a(b(?x)),
b(b(?x)) -> b(a(?x)),
a(c(?x)) -> c(b(?x)) ]
Apply Direct Methods...
Inner CPs:
[ a(a(a(?x_5))) = b(c(a(?x_5))),
a(c(b(?x_8))) = b(c(c(?x_8))),
b(a(a(?x_3))) = a(b(c(?x_3))),
b(c(a(?x_4))) = a(b(a(?x_4))),
b(a(b(?x_6))) = a(b(b(?x_6))),
b(b(c(?x))) = a(a(a(?x))),
b(a(a(?x_5))) = a(a(a(?x_5))),
b(c(b(?x_8))) = a(a(c(?x_8))),
c(c(a(?x_4))) = a(a(a(?x_4))),
c(a(b(?x_6))) = a(a(b(?x_6))),
c(b(c(?x))) = c(a(a(?x))),
c(a(a(?x_5))) = c(a(a(?x_5))),
c(c(b(?x_8))) = c(a(c(?x_8))),
a(b(c(?x))) = a(a(a(?x))),
a(c(b(?x_8))) = a(a(c(?x_8))),
c(a(b(?x_1))) = a(b(c(?x_1))),
c(a(a(?x_2))) = a(b(a(?x_2))),
c(b(a(?x_7))) = a(b(b(?x_7))),
b(a(b(?x_1))) = b(a(c(?x_1))),
b(a(a(?x_2))) = b(a(a(?x_2))),
a(a(a(?x_3))) = c(b(c(?x_3))),
a(c(a(?x_4))) = c(b(a(?x_4))),
a(a(b(?x_6))) = c(b(b(?x_6))),
a(b(c(?x))) = b(c(a(?x))),
c(a(a(?x))) = a(a(c(?x))),
a(a(a(?x))) = a(a(a(?x))),
b(b(a(?x))) = b(a(b(?x))) ]
Outer CPs:
[ b(c(?x)) = a(a(?x)) ]
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:
[ a(a(?x)) = b(c(?x)),
a(b(c(?x_1))) = b(c(a(?x_1))),
a(a(a(?x_6))) = b(c(a(?x_6))),
a(c(b(?x_9))) = b(c(c(?x_9))),
b(c(b(c(?x_1)))) = a(b(c(a(?x_1)))),
b(c(a(a(?x_6)))) = a(b(c(a(?x_6)))),
b(c(c(b(?x_9)))) = a(b(c(c(?x_9)))),
a(a(b(c(?x_1)))) = b(b(c(a(?x_1)))),
a(a(a(a(?x_6)))) = b(b(c(a(?x_6)))),
a(a(c(b(?x_9)))) = b(b(c(c(?x_9)))),
c(a(b(c(?x_1)))) = c(b(c(a(?x_1)))),
c(a(a(a(?x_6)))) = c(b(c(a(?x_6)))),
c(a(c(b(?x_9)))) = c(b(c(c(?x_9)))),
a(a(b(c(?x_1)))) = a(b(c(a(?x_1)))),
a(a(a(a(?x_6)))) = a(b(c(a(?x_6)))),
a(a(c(b(?x_9)))) = a(b(c(c(?x_9)))),
b(c(a(?x))) = a(b(c(?x))),
a(a(a(?x))) = b(b(c(?x))),
c(a(a(?x))) = c(b(c(?x))),
a(a(a(?x))) = a(b(c(?x))),
b(a(a(?x_4))) = a(b(c(?x_4))),
b(c(a(?x_5))) = a(b(a(?x_5))),
b(a(b(?x_7))) = a(b(b(?x_7))),
a(b(a(a(?x_4)))) = c(a(b(c(?x_4)))),
a(b(c(a(?x_5)))) = c(a(b(a(?x_5)))),
a(b(a(b(?x_7)))) = c(a(b(b(?x_7)))),
b(a(a(a(?x_4)))) = b(a(b(c(?x_4)))),
b(a(c(a(?x_5)))) = b(a(b(a(?x_5)))),
b(a(a(b(?x_7)))) = b(a(b(b(?x_7)))),
a(b(c(?x))) = c(a(b(?x))),
b(a(c(?x))) = b(a(b(?x))),
b(b(c(?x_2))) = a(a(a(?x_2))),
b(a(a(?x_6))) = a(a(a(?x_6))),
b(c(b(?x_9))) = a(a(c(?x_9))),
a(b(b(c(?x_2)))) = c(a(a(a(?x_2)))),
a(b(a(a(?x_6)))) = c(a(a(a(?x_6)))),
a(b(c(b(?x_9)))) = c(a(a(c(?x_9)))),
b(a(b(c(?x_2)))) = b(a(a(a(?x_2)))),
b(a(a(a(?x_6)))) = b(a(a(a(?x_6)))),
b(a(c(b(?x_9)))) = b(a(a(c(?x_9)))),
a(b(a(?x))) = c(a(a(?x))),
b(a(a(?x))) = b(a(a(?x))),
c(a(a(?x_1))) = a(a(c(?x_1))),
c(c(a(?x_5))) = a(a(a(?x_5))),
c(a(b(?x_7))) = a(a(b(?x_7))),
a(a(a(a(?x_1)))) = c(a(a(c(?x_1)))),
a(a(c(a(?x_5)))) = c(a(a(a(?x_5)))),
a(a(a(b(?x_7)))) = c(a(a(b(?x_7)))),
a(b(a(a(?x_1)))) = b(a(a(c(?x_1)))),
a(b(c(a(?x_5)))) = b(a(a(a(?x_5)))),
a(b(a(b(?x_7)))) = b(a(a(b(?x_7)))),
c(b(a(a(?x_1)))) = a(a(a(c(?x_1)))),
c(b(c(a(?x_5)))) = a(a(a(a(?x_5)))),
c(b(a(b(?x_7)))) = a(a(a(b(?x_7)))),
a(a(c(?x))) = c(a(a(?x))),
a(b(c(?x))) = b(a(a(?x))),
c(b(c(?x))) = a(a(a(?x))),
c(b(c(?x_2))) = c(a(a(?x_2))),
c(a(a(?x_6))) = c(a(a(?x_6))),
c(c(b(?x_9))) = c(a(c(?x_9))),
a(b(b(c(?x_2)))) = b(c(a(a(?x_2)))),
a(b(a(a(?x_6)))) = b(c(a(a(?x_6)))),
a(b(c(b(?x_9)))) = b(c(a(c(?x_9)))),
a(a(b(c(?x_2)))) = c(c(a(a(?x_2)))),
a(a(a(a(?x_6)))) = c(c(a(a(?x_6)))),
a(a(c(b(?x_9)))) = c(c(a(c(?x_9)))),
c(b(b(c(?x_2)))) = a(c(a(a(?x_2)))),
c(b(a(a(?x_6)))) = a(c(a(a(?x_6)))),
c(b(c(b(?x_9)))) = a(c(a(c(?x_9)))),
a(b(a(?x))) = b(c(a(?x))),
a(a(a(?x))) = c(c(a(?x))),
c(b(a(?x))) = a(c(a(?x))),
b(c(?x)) = a(a(?x)),
a(a(a(?x_1))) = a(a(a(?x_1))),
a(b(c(?x_2))) = a(a(a(?x_2))),
a(c(b(?x_9))) = a(a(c(?x_9))),
a(a(a(a(?x_1)))) = a(a(a(a(?x_1)))),
a(a(b(c(?x_2)))) = a(a(a(a(?x_2)))),
a(a(c(b(?x_9)))) = a(a(a(c(?x_9)))),
b(c(a(a(?x_1)))) = a(a(a(a(?x_1)))),
b(c(b(c(?x_2)))) = a(a(a(a(?x_2)))),
b(c(c(b(?x_9)))) = a(a(a(c(?x_9)))),
a(a(a(a(?x_1)))) = b(a(a(a(?x_1)))),
a(a(b(c(?x_2)))) = b(a(a(a(?x_2)))),
a(a(c(b(?x_9)))) = b(a(a(c(?x_9)))),
c(a(a(a(?x_1)))) = c(a(a(a(?x_1)))),
c(a(b(c(?x_2)))) = c(a(a(a(?x_2)))),
c(a(c(b(?x_9)))) = c(a(a(c(?x_9)))),
b(c(a(?x))) = a(a(a(?x))),
a(a(a(?x))) = b(a(a(?x))),
c(a(b(?x_3))) = a(b(c(?x_3))),
c(a(a(?x_4))) = a(b(a(?x_4))),
c(b(a(?x_8))) = a(b(b(?x_8))),
a(b(a(b(?x_3)))) = b(a(b(c(?x_3)))),
a(b(a(a(?x_4)))) = b(a(b(a(?x_4)))),
a(b(b(a(?x_8)))) = b(a(b(b(?x_8)))),
a(a(a(b(?x_3)))) = c(a(b(c(?x_3)))),
a(a(a(a(?x_4)))) = c(a(b(a(?x_4)))),
a(a(b(a(?x_8)))) = c(a(b(b(?x_8)))),
c(b(a(b(?x_3)))) = a(a(b(c(?x_3)))),
c(b(a(a(?x_4)))) = a(a(b(a(?x_4)))),
c(b(b(a(?x_8)))) = a(a(b(b(?x_8)))),
a(b(b(?x))) = b(a(b(?x))),
a(a(b(?x))) = c(a(b(?x))),
c(b(b(?x))) = a(a(b(?x))),
b(b(a(?x_1))) = b(a(b(?x_1))),
b(a(b(?x_3))) = b(a(c(?x_3))),
b(a(b(a(?x_1)))) = b(b(a(b(?x_1)))),
b(a(a(b(?x_3)))) = b(b(a(c(?x_3)))),
b(a(a(a(?x_4)))) = b(b(a(a(?x_4)))),
a(b(b(a(?x_1)))) = c(b(a(b(?x_1)))),
a(b(a(b(?x_3)))) = c(b(a(c(?x_3)))),
a(b(a(a(?x_4)))) = c(b(a(a(?x_4)))),
b(a(b(?x))) = b(b(a(?x))),
a(b(b(?x))) = c(b(a(?x))),
a(a(a(?x_5))) = c(b(c(?x_5))),
a(c(a(?x_6))) = c(b(a(?x_6))),
a(a(b(?x_8))) = c(b(b(?x_8))),
b(c(a(a(?x_5)))) = a(c(b(c(?x_5)))),
b(c(c(a(?x_6)))) = a(c(b(a(?x_6)))),
b(c(a(b(?x_8)))) = a(c(b(b(?x_8)))),
a(a(a(a(?x_5)))) = b(c(b(c(?x_5)))),
a(a(c(a(?x_6)))) = b(c(b(a(?x_6)))),
a(a(a(b(?x_8)))) = b(c(b(b(?x_8)))),
c(a(a(a(?x_5)))) = c(c(b(c(?x_5)))),
c(a(c(a(?x_6)))) = c(c(b(a(?x_6)))),
c(a(a(b(?x_8)))) = c(c(b(b(?x_8)))),
a(a(a(a(?x_5)))) = a(c(b(c(?x_5)))),
a(a(c(a(?x_6)))) = a(c(b(a(?x_6)))),
a(a(a(b(?x_8)))) = a(c(b(b(?x_8)))),
b(c(c(?x))) = a(c(b(?x))),
a(a(c(?x))) = b(c(b(?x))),
c(a(c(?x))) = c(c(b(?x))),
a(a(c(?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 by Rules <5, 0> preceded by [(a,1)]
joinable by a reduction of rules <[([],0)], []>
Critical Pair by Rules <8, 0> preceded by [(a,1)]
joinable by a reduction of rules <[([(a,1)],6)], [([],1),([(a,1)],1)]>
Critical Pair by Rules <3, 1> preceded by [(b,1)]
joinable by a reduction of rules <[([],2),([(a,1)],0)], []>
Critical Pair by Rules <4, 1> preceded by [(b,1)]
joinable by a reduction of rules <[([],1)], []>
Critical Pair by Rules <6, 1> preceded by [(b,1)]
joinable by a reduction of rules <[([],2),([],0),([],1)], []>
Critical Pair by Rules <0, 2> preceded by [(b,1)]
joinable by a reduction of rules <[([(b,1)],1),([],2)], [([(a,1)],0),([(a,1)],1)]>
Critical Pair by Rules <5, 2> preceded by [(b,1)]
joinable by a reduction of rules <[([],2)], []>
Critical Pair by Rules <8, 2> preceded by [(b,1)]
joinable by a reduction of rules <[([(b,1)],6),([],2)], [([(a,1)],8),([(a,1)],6)]>
Critical Pair by Rules <4, 3> preceded by [(c,1)]
joinable by a reduction of rules <[([],3)], []>
Critical Pair by Rules <6, 3> preceded by [(c,1)]
unknown Diagram Decreasing
check Non-Confluence...
obtain 15 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(a(?x)) -> b(c(?x)),
b(c(?x)) -> a(b(?x)),
b(a(?x)) -> a(a(?x)),
c(c(?x)) -> a(a(?x)),
c(b(?x)) -> a(b(?x)),
b(b(?x)) -> b(a(?x)),
a(c(?x)) -> c(b(?x)) ]
P:
[ c(a(?x)) -> c(a(?x)),
a(a(?x)) -> a(a(?x)) ]
S: unknown termination
failure(Step 1)
STEP: 2 (linear)
S:
[ a(a(?x)) -> b(c(?x)),
b(c(?x)) -> a(b(?x)),
b(a(?x)) -> a(a(?x)),
c(c(?x)) -> a(a(?x)),
c(b(?x)) -> a(b(?x)),
b(b(?x)) -> b(a(?x)),
a(c(?x)) -> c(b(?x)) ]
P:
[ c(a(?x)) -> c(a(?x)),
a(a(?x)) -> a(a(?x)) ]
S: unknown termination
failure(Step 2)
STEP: 3 (relative)
S:
[ a(a(?x)) -> b(c(?x)),
b(c(?x)) -> a(b(?x)),
b(a(?x)) -> a(a(?x)),
c(c(?x)) -> a(a(?x)),
c(b(?x)) -> a(b(?x)),
b(b(?x)) -> b(a(?x)),
a(c(?x)) -> c(b(?x)) ]
P:
[ c(a(?x)) -> c(a(?x)),
a(a(?x)) -> a(a(?x)) ]
Check relative termination:
[ a(a(?x)) -> b(c(?x)),
b(c(?x)) -> a(b(?x)),
b(a(?x)) -> a(a(?x)),
c(c(?x)) -> a(a(?x)),
c(b(?x)) -> a(b(?x)),
b(b(?x)) -> b(a(?x)),
a(c(?x)) -> c(b(?x)) ]
[ c(a(?x)) -> c(a(?x)),
a(a(?x)) -> a(a(?x)) ]