(ignored inputs)COMMENT submitted by: Johannes Waldmann Rewrite Rules: [ b(c(?x)) -> b(a(?x)), c(b(?x)) -> a(c(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> a(b(?x)), a(c(?x)) -> b(c(?x)), a(c(?x)) -> a(c(?x)), a(b(?x)) -> c(a(?x)), c(c(?x)) -> b(a(?x)) ] Apply Direct Methods... Inner CPs: [ b(a(c(?x_1))) = b(a(b(?x_1))), b(c(b(?x_2))) = b(a(a(?x_2))), b(b(a(?x_7))) = b(a(c(?x_7))), c(b(a(?x))) = a(c(c(?x))), c(a(b(?x_3))) = c(b(a(?x_3))), c(b(c(?x_4))) = c(b(c(?x_4))), c(a(c(?x_5))) = c(b(c(?x_5))), c(c(a(?x_6))) = c(b(b(?x_6))), a(b(c(?x_4))) = a(b(c(?x_4))), a(a(c(?x_5))) = a(b(c(?x_5))), a(c(a(?x_6))) = a(b(b(?x_6))), a(a(c(?x_1))) = b(c(b(?x_1))), a(c(b(?x_2))) = b(c(a(?x_2))), a(b(a(?x_7))) = b(c(c(?x_7))), a(a(c(?x_1))) = a(c(b(?x_1))), a(c(b(?x_2))) = a(c(a(?x_2))), a(b(a(?x_7))) = a(c(c(?x_7))), a(b(a(?x))) = c(a(c(?x))), c(a(c(?x_1))) = b(a(b(?x_1))), c(c(b(?x_2))) = b(a(a(?x_2))), a(a(b(?x))) = a(b(a(?x))), c(b(a(?x))) = b(a(c(?x))) ] Outer CPs: [ b(c(?x_4)) = a(c(?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(a(c(?x_2))) = b(a(b(?x_2))), b(c(b(?x_3))) = b(a(a(?x_3))), b(b(a(?x_8))) = b(a(c(?x_8))), a(c(a(c(?x_2)))) = c(b(a(b(?x_2)))), a(c(c(b(?x_3)))) = c(b(a(a(?x_3)))), a(c(b(a(?x_8)))) = c(b(a(c(?x_8)))), c(a(a(c(?x_2)))) = a(b(a(b(?x_2)))), c(a(c(b(?x_3)))) = a(b(a(a(?x_3)))), c(a(b(a(?x_8)))) = a(b(a(c(?x_8)))), a(c(c(?x))) = c(b(a(?x))), c(a(c(?x))) = a(b(a(?x))), c(b(a(?x_2))) = a(c(c(?x_2))), b(a(b(a(?x_2)))) = b(a(c(c(?x_2)))), b(c(b(a(?x_2)))) = a(a(c(c(?x_2)))), a(c(b(a(?x_2)))) = a(a(c(c(?x_2)))), b(a(b(a(?x_2)))) = c(a(c(c(?x_2)))), b(a(b(?x))) = b(a(c(?x))), b(c(b(?x))) = a(a(c(?x))), a(c(b(?x))) = a(a(c(?x))), b(a(b(?x))) = c(a(c(?x))), c(a(b(?x_4))) = c(b(a(?x_4))), c(b(c(?x_5))) = c(b(c(?x_5))), c(a(c(?x_6))) = c(b(c(?x_6))), c(c(a(?x_7))) = c(b(b(?x_7))), b(a(a(b(?x_4)))) = b(c(b(a(?x_4)))), b(a(b(c(?x_5)))) = b(c(b(c(?x_5)))), b(a(a(c(?x_6)))) = b(c(b(c(?x_6)))), b(a(c(a(?x_7)))) = b(c(b(b(?x_7)))), b(c(a(b(?x_4)))) = a(c(b(a(?x_4)))), b(c(b(c(?x_5)))) = a(c(b(c(?x_5)))), b(c(a(c(?x_6)))) = a(c(b(c(?x_6)))), b(c(c(a(?x_7)))) = a(c(b(b(?x_7)))), a(c(a(b(?x_4)))) = a(c(b(a(?x_4)))), a(c(b(c(?x_5)))) = a(c(b(c(?x_5)))), a(c(a(c(?x_6)))) = a(c(b(c(?x_6)))), a(c(c(a(?x_7)))) = a(c(b(b(?x_7)))), b(a(a(b(?x_4)))) = c(c(b(a(?x_4)))), b(a(b(c(?x_5)))) = c(c(b(c(?x_5)))), b(a(a(c(?x_6)))) = c(c(b(c(?x_6)))), b(a(c(a(?x_7)))) = c(c(b(b(?x_7)))), b(a(a(?x))) = b(c(b(?x))), b(c(a(?x))) = a(c(b(?x))), a(c(a(?x))) = a(c(b(?x))), b(a(a(?x))) = c(c(b(?x))), a(a(b(?x_1))) = a(b(a(?x_1))), a(b(c(?x_5))) = a(b(c(?x_5))), a(a(c(?x_6))) = a(b(c(?x_6))), a(c(a(?x_7))) = a(b(b(?x_7))), a(b(a(b(?x_1)))) = a(a(b(a(?x_1)))), a(b(b(c(?x_5)))) = a(a(b(c(?x_5)))), a(b(a(c(?x_6)))) = a(a(b(c(?x_6)))), a(b(c(a(?x_7)))) = a(a(b(b(?x_7)))), c(b(a(b(?x_1)))) = c(a(b(a(?x_1)))), c(b(b(c(?x_5)))) = c(a(b(c(?x_5)))), c(b(a(c(?x_6)))) = c(a(b(c(?x_6)))), c(b(c(a(?x_7)))) = c(a(b(b(?x_7)))), a(b(a(?x))) = a(a(b(?x))), c(b(a(?x))) = c(a(b(?x))), a(c(?x)) = b(c(?x)), a(a(c(?x_3))) = b(c(b(?x_3))), a(c(b(?x_4))) = b(c(a(?x_4))), a(b(a(?x_8))) = b(c(c(?x_8))), c(b(a(c(?x_3)))) = c(b(c(b(?x_3)))), c(b(c(b(?x_4)))) = c(b(c(a(?x_4)))), c(b(b(a(?x_8)))) = c(b(c(c(?x_8)))), a(b(a(c(?x_3)))) = a(b(c(b(?x_3)))), a(b(c(b(?x_4)))) = a(b(c(a(?x_4)))), a(b(b(a(?x_8)))) = a(b(c(c(?x_8)))), b(c(?x)) = a(c(?x)), a(a(c(?x_3))) = a(c(b(?x_3))), a(c(b(?x_4))) = a(c(a(?x_4))), a(b(a(?x_8))) = a(c(c(?x_8))), c(b(a(c(?x_3)))) = c(a(c(b(?x_3)))), c(b(c(b(?x_4)))) = c(a(c(a(?x_4)))), c(b(b(a(?x_8)))) = c(a(c(c(?x_8)))), a(b(a(c(?x_3)))) = a(a(c(b(?x_3)))), a(b(c(b(?x_4)))) = a(a(c(a(?x_4)))), a(b(b(a(?x_8)))) = a(a(c(c(?x_8)))), c(b(c(?x))) = c(a(c(?x))), a(b(c(?x))) = a(a(c(?x))), a(b(a(?x_2))) = c(a(c(?x_2))), c(b(b(a(?x_2)))) = c(c(a(c(?x_2)))), a(b(b(a(?x_2)))) = a(c(a(c(?x_2)))), c(b(b(?x))) = c(c(a(?x))), a(b(b(?x))) = a(c(a(?x))), c(b(a(?x_1))) = b(a(c(?x_1))), c(a(c(?x_3))) = b(a(b(?x_3))), c(c(b(?x_4))) = b(a(a(?x_4))), b(a(b(a(?x_1)))) = c(b(a(c(?x_1)))), b(a(a(c(?x_3)))) = c(b(a(b(?x_3)))), b(a(c(b(?x_4)))) = c(b(a(a(?x_4)))), b(a(b(a(?x_1)))) = b(b(a(c(?x_1)))), b(a(a(c(?x_3)))) = b(b(a(b(?x_3)))), b(a(c(b(?x_4)))) = b(b(a(a(?x_4)))), b(c(b(a(?x_1)))) = a(b(a(c(?x_1)))), b(c(a(c(?x_3)))) = a(b(a(b(?x_3)))), b(c(c(b(?x_4)))) = a(b(a(a(?x_4)))), a(c(b(a(?x_1)))) = a(b(a(c(?x_1)))), a(c(a(c(?x_3)))) = a(b(a(b(?x_3)))), a(c(c(b(?x_4)))) = a(b(a(a(?x_4)))), b(a(c(?x))) = c(b(a(?x))), b(a(c(?x))) = b(b(a(?x))), b(c(c(?x))) = a(b(a(?x))), a(c(c(?x))) = a(b(a(?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 <[], [([(b,1)],6),([(b,1)],2),([(b,1)],1)]> Critical Pair by Rules <2, 0> preceded by [(b,1)] joinable by a reduction of rules <[([],0)], [([(b,1)],3)]> Critical Pair by Rules <7, 0> preceded by [(b,1)] joinable by a reduction of rules <[], [([(b,1)],4),([(b,1)],0)]> Critical Pair by Rules <0, 1> preceded by [(c,1)] joinable by a reduction of rules <[], [([(a,1)],7),([],6),([],2)]> Critical Pair by Rules <3, 2> preceded by [(c,1)] joinable by a reduction of rules <[([],2),([],1)], [([],1),([(a,1)],2)]> Critical Pair by Rules <4, 2> preceded by [(c,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <5, 2> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],4)], []> joinable by a reduction of rules <[([],2)], []> Critical Pair by Rules <6, 2> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],2),([],7)], [([],1),([],4),([],0)]> joinable by a reduction of rules <[([],7),([(b,1)],3)], [([],1),([],4),([],0)]> Critical Pair by Rules <4, 3> preceded by [(a,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <5, 3> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],4)], []> joinable by a reduction of rules <[([],3)], []> Critical Pair by Rules <6, 3> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],2)], [([],6),([],2),([],1)]> joinable by a reduction of rules <[([],4),([],0)], [([],6),([(c,1)],6),([],7)]> Critical Pair by Rules <1, 4> preceded by [(a,1)] unknown Diagram Decreasing check Non-Confluence... obtain 16 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(c(?x)) -> b(a(?x)), c(b(?x)) -> a(c(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> a(b(?x)), a(c(?x)) -> b(c(?x)), a(b(?x)) -> c(a(?x)), c(c(?x)) -> b(a(?x)) ] P: [ a(c(?x)) -> a(c(?x)) ] S: unknown termination failure(Step 1) STEP: 2 (linear) S: [ b(c(?x)) -> b(a(?x)), c(b(?x)) -> a(c(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> a(b(?x)), a(c(?x)) -> b(c(?x)), a(b(?x)) -> c(a(?x)), c(c(?x)) -> b(a(?x)) ] P: [ a(c(?x)) -> a(c(?x)) ] S: unknown termination failure(Step 2) STEP: 3 (relative) S: [ b(c(?x)) -> b(a(?x)), c(b(?x)) -> a(c(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> a(b(?x)), a(c(?x)) -> b(c(?x)), a(b(?x)) -> c(a(?x)), c(c(?x)) -> b(a(?x)) ] P: [ a(c(?x)) -> a(c(?x)) ] Check relative termination: [ b(c(?x)) -> b(a(?x)), c(b(?x)) -> a(c(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> a(b(?x)), a(c(?x)) -> b(c(?x)), a(b(?x)) -> c(a(?x)), c(c(?x)) -> b(a(?x)) ] [ a(c(?x)) -> a(c(?x)) ]