(ignored inputs)COMMENT submitted by: Johannes Waldmann Rewrite Rules: [ b(b(?x)) -> b(c(?x)), b(c(?x)) -> a(c(?x)), a(b(?x)) -> b(a(?x)), a(c(?x)) -> a(b(?x)), b(a(?x)) -> a(c(?x)), b(a(?x)) -> b(b(?x)), b(a(?x)) -> b(c(?x)), b(b(?x)) -> b(b(?x)), c(a(?x)) -> b(c(?x)) ] Apply Direct Methods... Inner CPs: [ b(a(c(?x_1))) = b(c(c(?x_1))), b(a(c(?x_4))) = b(c(a(?x_4))), b(b(b(?x_5))) = b(c(a(?x_5))), b(b(c(?x_6))) = b(c(a(?x_6))), b(b(b(?x_7))) = b(c(b(?x_7))), b(b(c(?x_8))) = a(c(a(?x_8))), a(b(c(?x))) = b(a(b(?x))), a(a(c(?x_1))) = b(a(c(?x_1))), a(a(c(?x_4))) = b(a(a(?x_4))), a(b(b(?x_5))) = b(a(a(?x_5))), a(b(c(?x_6))) = b(a(a(?x_6))), a(b(b(?x_7))) = b(a(b(?x_7))), a(b(c(?x_8))) = a(b(a(?x_8))), b(b(a(?x_2))) = a(c(b(?x_2))), b(a(b(?x_3))) = a(c(c(?x_3))), b(b(a(?x_2))) = b(b(b(?x_2))), b(a(b(?x_3))) = b(b(c(?x_3))), b(b(a(?x_2))) = b(c(b(?x_2))), b(a(b(?x_3))) = b(c(c(?x_3))), b(b(c(?x))) = b(b(b(?x))), b(a(c(?x_1))) = b(b(c(?x_1))), b(a(c(?x_4))) = b(b(a(?x_4))), b(b(b(?x_5))) = b(b(a(?x_5))), b(b(c(?x_6))) = b(b(a(?x_6))), c(b(a(?x_2))) = b(c(b(?x_2))), c(a(b(?x_3))) = b(c(c(?x_3))), b(b(c(?x))) = b(c(b(?x))), b(b(b(?x))) = b(b(b(?x))) ] Outer CPs: [ b(c(?x)) = b(b(?x)), a(c(?x_4)) = b(b(?x_4)), a(c(?x_4)) = b(c(?x_4)), b(b(?x_5)) = b(c(?x_5)) ] 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(b(?x)) = b(c(?x)), b(b(c(?x_1))) = b(c(b(?x_1))), b(a(c(?x_2))) = b(c(c(?x_2))), b(a(c(?x_5))) = b(c(a(?x_5))), b(b(b(?x_6))) = b(c(a(?x_6))), b(b(c(?x_7))) = b(c(a(?x_7))), b(b(b(?x_8))) = b(c(b(?x_8))), b(c(b(c(?x_1)))) = b(b(c(b(?x_1)))), b(c(a(c(?x_2)))) = b(b(c(c(?x_2)))), b(c(a(c(?x_5)))) = b(b(c(a(?x_5)))), b(c(b(b(?x_6)))) = b(b(c(a(?x_6)))), b(c(b(c(?x_7)))) = b(b(c(a(?x_7)))), b(c(b(b(?x_8)))) = b(b(c(b(?x_8)))), b(a(b(c(?x_1)))) = a(b(c(b(?x_1)))), b(a(a(c(?x_2)))) = a(b(c(c(?x_2)))), b(a(a(c(?x_5)))) = a(b(c(a(?x_5)))), b(a(b(b(?x_6)))) = a(b(c(a(?x_6)))), b(a(b(c(?x_7)))) = a(b(c(a(?x_7)))), b(a(b(b(?x_8)))) = a(b(c(b(?x_8)))), b(b(b(c(?x_1)))) = b(b(c(b(?x_1)))), b(b(a(c(?x_2)))) = b(b(c(c(?x_2)))), b(b(a(c(?x_5)))) = b(b(c(a(?x_5)))), b(b(b(b(?x_6)))) = b(b(c(a(?x_6)))), b(b(b(c(?x_7)))) = b(b(c(a(?x_7)))), b(b(b(b(?x_8)))) = b(b(c(b(?x_8)))), b(c(b(?x))) = b(b(c(?x))), b(a(b(?x))) = a(b(c(?x))), b(b(b(?x))) = b(b(c(?x))), b(b(c(?x_9))) = a(c(a(?x_9))), b(c(b(c(?x_9)))) = b(a(c(a(?x_9)))), b(a(b(c(?x_9)))) = a(a(c(a(?x_9)))), b(b(b(c(?x_9)))) = b(a(c(a(?x_9)))), b(c(c(?x))) = b(a(c(?x))), b(a(c(?x))) = a(a(c(?x))), b(b(c(?x))) = b(a(c(?x))), a(b(c(?x_2))) = b(a(b(?x_2))), a(a(c(?x_3))) = b(a(c(?x_3))), a(a(c(?x_5))) = b(a(a(?x_5))), a(b(b(?x_6))) = b(a(a(?x_6))), a(b(c(?x_7))) = b(a(a(?x_7))), a(b(b(?x_8))) = b(a(b(?x_8))), a(c(b(c(?x_2)))) = b(b(a(b(?x_2)))), a(c(a(c(?x_3)))) = b(b(a(c(?x_3)))), a(c(a(c(?x_5)))) = b(b(a(a(?x_5)))), a(c(b(b(?x_6)))) = b(b(a(a(?x_6)))), a(c(b(c(?x_7)))) = b(b(a(a(?x_7)))), a(c(b(b(?x_8)))) = b(b(a(b(?x_8)))), b(b(b(c(?x_2)))) = b(b(a(b(?x_2)))), b(b(a(c(?x_3)))) = b(b(a(c(?x_3)))), b(b(a(c(?x_5)))) = b(b(a(a(?x_5)))), b(b(b(b(?x_6)))) = b(b(a(a(?x_6)))), b(b(b(c(?x_7)))) = b(b(a(a(?x_7)))), b(b(b(b(?x_8)))) = b(b(a(b(?x_8)))), b(c(b(c(?x_2)))) = b(b(a(b(?x_2)))), b(c(a(c(?x_3)))) = b(b(a(c(?x_3)))), b(c(a(c(?x_5)))) = b(b(a(a(?x_5)))), b(c(b(b(?x_6)))) = b(b(a(a(?x_6)))), b(c(b(c(?x_7)))) = b(b(a(a(?x_7)))), b(c(b(b(?x_8)))) = b(b(a(b(?x_8)))), b(c(b(c(?x_2)))) = c(b(a(b(?x_2)))), b(c(a(c(?x_3)))) = c(b(a(c(?x_3)))), b(c(a(c(?x_5)))) = c(b(a(a(?x_5)))), b(c(b(b(?x_6)))) = c(b(a(a(?x_6)))), b(c(b(c(?x_7)))) = c(b(a(a(?x_7)))), b(c(b(b(?x_8)))) = c(b(a(b(?x_8)))), a(c(b(?x))) = b(b(a(?x))), b(b(b(?x))) = b(b(a(?x))), b(c(b(?x))) = b(b(a(?x))), b(c(b(?x))) = c(b(a(?x))), a(b(c(?x_9))) = a(b(a(?x_9))), a(c(b(c(?x_9)))) = b(a(b(a(?x_9)))), b(b(b(c(?x_9)))) = b(a(b(a(?x_9)))), b(c(b(c(?x_9)))) = b(a(b(a(?x_9)))), b(c(b(c(?x_9)))) = c(a(b(a(?x_9)))), a(c(c(?x))) = b(a(b(?x))), b(b(c(?x))) = b(a(b(?x))), b(c(c(?x))) = b(a(b(?x))), b(c(c(?x))) = c(a(b(?x))), b(b(?x)) = a(c(?x)), b(c(?x)) = a(c(?x)), b(b(a(?x_4))) = a(c(b(?x_4))), b(a(b(?x_5))) = a(c(c(?x_5))), b(c(b(a(?x_4)))) = b(a(c(b(?x_4)))), b(c(a(b(?x_5)))) = b(a(c(c(?x_5)))), b(a(b(a(?x_4)))) = a(a(c(b(?x_4)))), b(a(a(b(?x_5)))) = a(a(c(c(?x_5)))), b(b(b(a(?x_4)))) = b(a(c(b(?x_4)))), b(b(a(b(?x_5)))) = b(a(c(c(?x_5)))), b(c(a(?x))) = b(a(c(?x))), b(a(a(?x))) = a(a(c(?x))), b(b(a(?x))) = b(a(c(?x))), a(c(?x)) = b(b(?x)), b(c(?x)) = b(b(?x)), b(b(a(?x_4))) = b(b(b(?x_4))), b(a(b(?x_5))) = b(b(c(?x_5))), b(c(b(a(?x_4)))) = b(b(b(b(?x_4)))), b(c(a(b(?x_5)))) = b(b(b(c(?x_5)))), b(a(b(a(?x_4)))) = a(b(b(b(?x_4)))), b(a(a(b(?x_5)))) = a(b(b(c(?x_5)))), b(b(b(a(?x_4)))) = b(b(b(b(?x_4)))), b(b(a(b(?x_5)))) = b(b(b(c(?x_5)))), b(c(a(?x))) = b(b(b(?x))), b(a(a(?x))) = a(b(b(?x))), a(c(?x)) = b(c(?x)), b(b(a(?x_4))) = b(c(b(?x_4))), b(a(b(?x_5))) = b(c(c(?x_5))), b(c(b(a(?x_4)))) = b(b(c(b(?x_4)))), b(c(a(b(?x_5)))) = b(b(c(c(?x_5)))), b(a(b(a(?x_4)))) = a(b(c(b(?x_4)))), b(a(a(b(?x_5)))) = a(b(c(c(?x_5)))), b(b(b(a(?x_4)))) = b(b(c(b(?x_4)))), b(b(a(b(?x_5)))) = b(b(c(c(?x_5)))), b(c(a(?x))) = b(b(c(?x))), b(a(a(?x))) = a(b(c(?x))), b(b(a(?x))) = b(b(c(?x))), b(b(b(?x_1))) = b(b(b(?x_1))), b(b(c(?x_2))) = b(b(b(?x_2))), b(a(c(?x_3))) = b(b(c(?x_3))), b(a(c(?x_6))) = b(b(a(?x_6))), b(b(c(?x_8))) = b(b(a(?x_8))), b(b(b(b(?x_1)))) = b(b(b(b(?x_1)))), b(b(b(c(?x_2)))) = b(b(b(b(?x_2)))), b(b(a(c(?x_3)))) = b(b(b(c(?x_3)))), b(b(a(c(?x_6)))) = b(b(b(a(?x_6)))), b(b(b(b(?x_7)))) = b(b(b(a(?x_7)))), b(b(b(c(?x_8)))) = b(b(b(a(?x_8)))), b(c(b(b(?x_1)))) = b(b(b(b(?x_1)))), b(c(b(c(?x_2)))) = b(b(b(b(?x_2)))), b(c(a(c(?x_3)))) = b(b(b(c(?x_3)))), b(c(a(c(?x_6)))) = b(b(b(a(?x_6)))), b(c(b(b(?x_7)))) = b(b(b(a(?x_7)))), b(c(b(c(?x_8)))) = b(b(b(a(?x_8)))), b(a(b(b(?x_1)))) = a(b(b(b(?x_1)))), b(a(b(c(?x_2)))) = a(b(b(b(?x_2)))), b(a(a(c(?x_3)))) = a(b(b(c(?x_3)))), b(a(a(c(?x_6)))) = a(b(b(a(?x_6)))), b(a(b(b(?x_7)))) = a(b(b(a(?x_7)))), b(a(b(c(?x_8)))) = a(b(b(a(?x_8)))), b(c(b(?x))) = b(b(b(?x))), b(a(b(?x))) = a(b(b(?x))), c(b(a(?x_4))) = b(c(b(?x_4))), c(a(b(?x_5))) = b(c(c(?x_5))), a(c(b(a(?x_4)))) = b(b(c(b(?x_4)))), a(c(a(b(?x_5)))) = b(b(c(c(?x_5)))), a(b(b(a(?x_4)))) = a(b(c(b(?x_4)))), a(b(a(b(?x_5)))) = a(b(c(c(?x_5)))), a(c(a(?x))) = b(b(c(?x))), a(b(a(?x))) = a(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 <1, 0> preceded by [(b,1)] joinable by a reduction of rules <[([],6)], []> joinable by a reduction of rules <[([],4)], [([],1)]> Critical Pair by Rules <4, 0> preceded by [(b,1)] joinable by a reduction of rules <[([],5)], [([(b,1)],8)]> Critical Pair by Rules <5, 0> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],0)], [([(b,1)],8)]> Critical Pair by Rules <6, 0> preceded by [(b,1)] joinable by a reduction of rules <[], [([(b,1)],8)]> Critical Pair by Rules <7, 0> preceded by [(b,1)] joinable by a reduction of rules <[([],0)], []> Critical Pair by Rules <8, 1> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],1)], [([(a,1)],8),([],2)]> Critical Pair by Rules <0, 2> preceded by [(a,1)] joinable by a reduction of rules <[([],2),([(b,1)],3)], []> joinable by a reduction of rules <[([],2)], [([(b,1)],2),([(b,1)],4)]> joinable by a reduction of rules <[([],2),([],5)], [([(b,1)],2),([(b,1)],6)]> joinable by a reduction of rules <[([],2),([],5)], [([],5),([(b,1)],0)]> Critical Pair by Rules <1, 2> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],3),([(a,1)],2),([(a,1)],6)], [([],4),([],3)]> joinable by a reduction of rules <[], [([],4),([],3),([(a,1)],1)]> joinable by a reduction of rules <[([(a,1)],3),([(a,1)],2),([(a,1)],6)], [([],6),([],1),([],3)]> joinable by a reduction of rules <[([(a,1)],3),([(a,1)],2),([(a,1)],5)], [([(b,1)],3),([],4),([],3)]> Critical Pair by Rules <4, 2> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],3),([(a,1)],2)], [([],4),([],3)]> Critical Pair by Rules <5, 2> preceded by [(a,1)] joinable by a reduction of rules <[([],2),([(b,1)],2)], [([],5)]> joinable by a reduction of rules <[([(a,1)],0)], [([],4),([(a,1)],8)]> joinable by a reduction of rules <[([(a,1)],0),([],2)], [([],5),([(b,1)],4)]> joinable by a reduction of rules <[([],2),([],5)], [([],5),([(b,1)],5)]> Critical Pair by Rules <6, 2> preceded by [(a,1)] joinable by a reduction of rules <[], [([],4),([(a,1)],8)]> joinable by a reduction of rules <[([],2)], [([],5),([(b,1)],4)]> joinable by a reduction of rules <[([],2),([],5)], [([],6),([(b,1)],8)]> joinable by a reduction of rules <[([],2),([],5)], [([],5),([(b,1)],6)]> Critical Pair by Rules <7, 2> preceded by [(a,1)] joinable by a reduction of rules <[([],2)], []> Critical Pair by Rules <8, 3> preceded by [(a,1)] joinable by a reduction of rules <[], [([(a,1)],6)]> joinable by a reduction of rules <[([(a,1)],1)], [([(a,1)],4)]> Critical Pair by Rules <2, 4> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],4),([(b,1)],3)], [([],3),([],2)]> Critical Pair by Rules <3, 4> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],2),([(b,1)],4)], [([],3),([],2)]> Critical Pair by Rules <2, 5> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],5)], []> joinable by a reduction of rules <[([(b,1)],6)], [([(b,1)],0)]> Critical Pair by Rules <3, 5> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],2),([(b,1)],6)], []> joinable by a reduction of rules <[([(b,1)],2),([(b,1)],4)], [([(b,1)],1)]> joinable by a reduction of rules <[([],5),([(b,1)],0)], []> joinable by a reduction of rules <[], [([(b,1)],1),([(b,1)],3)]> Critical Pair by Rules <2, 6> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],5),([],0)], []> Critical Pair by Rules <3, 6> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],2),([(b,1)],6),([],0)], []> joinable by a reduction of rules <[([(b,1)],2),([(b,1)],4),([],6)], []> joinable by a reduction of rules <[([(b,1)],2),([(b,1)],4),([],4)], [([],1)]> joinable by a reduction of rules <[([],5),([(b,1)],0),([],0)], []> joinable by a reduction of rules <[([],4),([],3),([(a,1)],0)], [([],1),([],3)]> joinable by a reduction of rules <[([(b,1)],2),([(b,1)],4)], [([],1),([],3),([],2)]> joinable by a reduction of rules <[([(b,1)],2),([(b,1)],6),([(b,1)],1)], [([],1),([],3),([],2)]> joinable by a reduction of rules <[([],5),([(b,1)],0),([(b,1)],1)], [([],1),([],3),([],2)]> Critical Pair by Rules <0, 7> preceded by [(b,1)] joinable by a reduction of rules <[], [([(b,1)],0)]> Critical Pair by Rules <1, 7> preceded by [(b,1)] joinable by a reduction of rules <[([],5)], []> joinable by a reduction of rules <[], [([(b,1)],1)]> joinable by a reduction of rules <[([],6)], [([],0)]> Critical Pair by Rules <4, 7> preceded by [(b,1)] joinable by a reduction of rules <[], [([(b,1)],4)]> joinable by a reduction of rules <[([],5)], [([(b,1)],6)]> Critical Pair by Rules <5, 7> preceded by [(b,1)] joinable by a reduction of rules <[], [([(b,1)],5)]> joinable by a reduction of rules <[([(b,1)],0)], [([(b,1)],6)]> Critical Pair by Rules <6, 7> preceded by [(b,1)] joinable by a reduction of rules <[], [([(b,1)],6)]> joinable by a reduction of rules <[([(b,1)],1)], [([(b,1)],4)]> Critical Pair by Rules <2, 8> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],4),([(c,1)],3),([],8)], []> Critical Pair by Rules <3, 8> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],2),([(c,1)],4),([],8)], []> Critical Pair by Rules <0, 0> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],1),([(b,1)],3),([],6)], []> joinable by a reduction of rules <[([(b,1)],1),([(b,1)],3),([],4)], [([],1)]> joinable by a reduction of rules <[([(b,1)],1),([(b,1)],3)], [([],1),([],3),([],2)]> joinable by a reduction of rules <[([(b,1)],1),([],4),([],3)], [([],1),([],3),([(a,1)],0)]> joinable by a reduction of rules <[([],0),([],1),([],3)], [([],1),([],3),([(a,1)],0)]> Critical Pair by Rules <7, 7> preceded by [(b,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <7, 0> preceded by [] joinable by a reduction of rules <[([],0)], []> Critical Pair by Rules <5, 4> preceded by [] joinable by a reduction of rules <[([],0),([],1)], []> Critical Pair by Rules <6, 4> preceded by [] joinable by a reduction of rules <[([],1)], []> Critical Pair by Rules <6, 5> preceded by [] joinable by a reduction of rules <[], [([],0)]> unknown Diagram Decreasing check Non-Confluence... obtain 17 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: [ b(b(?x)) -> b(c(?x)), b(c(?x)) -> a(c(?x)), a(b(?x)) -> b(a(?x)), a(c(?x)) -> a(b(?x)), b(a(?x)) -> a(c(?x)), b(a(?x)) -> b(b(?x)), b(a(?x)) -> b(c(?x)), c(a(?x)) -> b(c(?x)) ] P: [ b(b(?x)) -> b(b(?x)) ] S: unknown termination failure(Step 1) STEP: 2 (linear) S: [ b(b(?x)) -> b(c(?x)), b(c(?x)) -> a(c(?x)), a(b(?x)) -> b(a(?x)), a(c(?x)) -> a(b(?x)), b(a(?x)) -> a(c(?x)), b(a(?x)) -> b(b(?x)), b(a(?x)) -> b(c(?x)), c(a(?x)) -> b(c(?x)) ] P: [ b(b(?x)) -> b(b(?x)) ] S: unknown termination failure(Step 2) STEP: 3 (relative) S: [ b(b(?x)) -> b(c(?x)), b(c(?x)) -> a(c(?x)), a(b(?x)) -> b(a(?x)), a(c(?x)) -> a(b(?x)), b(a(?x)) -> a(c(?x)), b(a(?x)) -> b(b(?x)), b(a(?x)) -> b(c(?x)), c(a(?x)) -> b(c(?x)) ] P: [ b(b(?x)) -> b(b(?x)) ] Check relative termination: [ b(b(?x)) -> b(c(?x)), b(c(?x)) -> a(c(?x)), a(b(?x)) -> b(a(?x)), a(c(?x)) -> a(b(?x)), b(a(?x)) -> a(c(?x)), b(a(?x)) -> b(b(?x)), b(a(?x)) -> b(c(?x)), c(a(?x)) -> b(c(?x)) ] [ b(b(?x)) -> b(b(?x)) ]