(ignored inputs)COMMENT submitted by: Johannes Waldmann Rewrite Rules: [ a(c(?x)) -> c(a(?x)), a(a(?x)) -> a(a(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> c(b(?x)), b(b(?x)) -> a(c(?x)), b(a(?x)) -> a(c(?x)), c(b(?x)) -> a(b(?x)), a(a(?x)) -> c(a(?x)) ] Apply Direct Methods... Inner CPs: [ a(c(b(?x_4))) = c(a(a(?x_4))), a(a(b(?x_7))) = c(a(b(?x_7))), a(c(a(?x))) = a(a(c(?x))), a(c(a(?x_2))) = a(a(b(?x_2))), a(a(c(?x_3))) = a(a(a(?x_3))), a(c(a(?x_8))) = a(a(a(?x_8))), a(a(c(?x_5))) = c(a(b(?x_5))), a(a(c(?x_6))) = c(a(a(?x_6))), a(c(a(?x))) = a(c(c(?x))), a(a(a(?x_1))) = a(c(a(?x_1))), a(c(a(?x_2))) = a(c(b(?x_2))), a(c(a(?x_8))) = a(c(a(?x_8))), c(c(a(?x))) = c(b(c(?x))), c(a(a(?x_1))) = c(b(a(?x_1))), c(c(a(?x_2))) = c(b(b(?x_2))), c(a(c(?x_3))) = c(b(a(?x_3))), c(c(a(?x_8))) = c(b(a(?x_8))), b(a(c(?x_6))) = a(c(a(?x_6))), b(c(a(?x))) = a(c(c(?x))), b(a(a(?x_1))) = a(c(a(?x_1))), b(c(a(?x_2))) = a(c(b(?x_2))), b(a(c(?x_3))) = a(c(a(?x_3))), b(c(a(?x_8))) = a(c(a(?x_8))), c(a(c(?x_5))) = a(b(b(?x_5))), c(a(c(?x_6))) = a(b(a(?x_6))), a(c(a(?x))) = c(a(c(?x))), a(a(a(?x_1))) = c(a(a(?x_1))), a(c(a(?x_2))) = c(a(b(?x_2))), a(a(c(?x_3))) = c(a(a(?x_3))), a(a(a(?x))) = a(a(a(?x))), a(a(c(?x))) = a(c(a(?x))), b(a(c(?x))) = a(c(b(?x))), a(c(a(?x))) = c(a(a(?x))) ] Outer CPs: [ a(a(?x_1)) = a(c(?x_1)), a(a(?x_1)) = c(a(?x_1)), a(c(?x_3)) = c(a(?x_3)) ] 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(c(b(?x_5))) = c(a(a(?x_5))), a(a(b(?x_8))) = c(a(b(?x_8))), a(a(c(b(?x_5)))) = a(c(a(a(?x_5)))), a(a(a(b(?x_8)))) = a(c(a(b(?x_8)))), a(c(c(b(?x_5)))) = a(c(a(a(?x_5)))), a(c(a(b(?x_8)))) = a(c(a(b(?x_8)))), c(b(c(b(?x_5)))) = c(c(a(a(?x_5)))), c(b(a(b(?x_8)))) = c(c(a(b(?x_8)))), a(c(c(b(?x_5)))) = b(c(a(a(?x_5)))), a(c(a(b(?x_8)))) = b(c(a(b(?x_8)))), c(a(c(b(?x_5)))) = a(c(a(a(?x_5)))), c(a(a(b(?x_8)))) = a(c(a(b(?x_8)))), a(a(c(?x))) = a(c(a(?x))), a(c(c(?x))) = a(c(a(?x))), c(b(c(?x))) = c(c(a(?x))), a(c(c(?x))) = b(c(a(?x))), c(a(c(?x))) = a(c(a(?x))), a(c(?x)) = a(a(?x)), c(a(?x)) = a(a(?x)), a(a(a(?x_1))) = a(a(a(?x_1))), a(c(a(?x_2))) = a(a(c(?x_2))), a(c(a(?x_3))) = a(a(b(?x_3))), a(a(c(?x_4))) = a(a(a(?x_4))), a(c(a(?x_9))) = a(a(a(?x_9))), a(a(a(a(?x_1)))) = a(a(a(a(?x_1)))), a(a(c(a(?x_2)))) = a(a(a(c(?x_2)))), a(a(c(a(?x_3)))) = a(a(a(b(?x_3)))), a(a(a(c(?x_4)))) = a(a(a(a(?x_4)))), a(a(c(a(?x_9)))) = a(a(a(a(?x_9)))), a(c(a(a(?x_1)))) = a(a(a(a(?x_1)))), a(c(c(a(?x_2)))) = a(a(a(c(?x_2)))), a(c(c(a(?x_3)))) = a(a(a(b(?x_3)))), a(c(a(c(?x_4)))) = a(a(a(a(?x_4)))), a(c(c(a(?x_9)))) = a(a(a(a(?x_9)))), c(b(a(a(?x_1)))) = c(a(a(a(?x_1)))), c(b(c(a(?x_2)))) = c(a(a(c(?x_2)))), c(b(c(a(?x_3)))) = c(a(a(b(?x_3)))), c(b(a(c(?x_4)))) = c(a(a(a(?x_4)))), c(b(c(a(?x_9)))) = c(a(a(a(?x_9)))), a(c(a(a(?x_1)))) = b(a(a(a(?x_1)))), a(c(c(a(?x_2)))) = b(a(a(c(?x_2)))), a(c(c(a(?x_3)))) = b(a(a(b(?x_3)))), a(c(a(c(?x_4)))) = b(a(a(a(?x_4)))), a(c(c(a(?x_9)))) = b(a(a(a(?x_9)))), c(a(a(a(?x_1)))) = a(a(a(a(?x_1)))), c(a(c(a(?x_2)))) = a(a(a(c(?x_2)))), c(a(c(a(?x_3)))) = a(a(a(b(?x_3)))), c(a(a(c(?x_4)))) = a(a(a(a(?x_4)))), c(a(c(a(?x_9)))) = a(a(a(a(?x_9)))), c(b(a(?x))) = c(a(a(?x))), a(c(a(?x))) = b(a(a(?x))), c(a(a(?x))) = a(a(a(?x))), a(a(c(?x_6))) = c(a(b(?x_6))), a(a(c(?x_7))) = c(a(a(?x_7))), a(a(a(c(?x_6)))) = a(c(a(b(?x_6)))), a(a(a(c(?x_7)))) = a(c(a(a(?x_7)))), a(c(a(c(?x_6)))) = a(c(a(b(?x_6)))), a(c(a(c(?x_7)))) = a(c(a(a(?x_7)))), c(b(a(c(?x_6)))) = c(c(a(b(?x_6)))), c(b(a(c(?x_7)))) = c(c(a(a(?x_7)))), a(c(a(c(?x_6)))) = b(c(a(b(?x_6)))), a(c(a(c(?x_7)))) = b(c(a(a(?x_7)))), c(a(a(c(?x_6)))) = a(c(a(b(?x_6)))), c(a(a(c(?x_7)))) = a(c(a(a(?x_7)))), a(a(b(?x))) = a(c(a(?x))), a(c(b(?x))) = a(c(a(?x))), c(b(b(?x))) = c(c(a(?x))), a(c(b(?x))) = b(c(a(?x))), c(a(b(?x))) = a(c(a(?x))), a(a(?x)) = a(c(?x)), c(a(?x)) = a(c(?x)), a(c(a(?x_2))) = a(c(c(?x_2))), a(a(a(?x_3))) = a(c(a(?x_3))), a(c(a(?x_4))) = a(c(b(?x_4))), a(c(a(?x_9))) = a(c(a(?x_9))), a(c(a(c(?x_1)))) = a(a(c(a(?x_1)))), a(c(c(a(?x_2)))) = a(a(c(c(?x_2)))), a(c(a(a(?x_3)))) = a(a(c(a(?x_3)))), a(c(c(a(?x_4)))) = a(a(c(b(?x_4)))), a(c(c(a(?x_9)))) = a(a(c(a(?x_9)))), a(a(a(c(?x_1)))) = a(a(c(a(?x_1)))), a(a(c(a(?x_2)))) = a(a(c(c(?x_2)))), a(a(a(a(?x_3)))) = a(a(c(a(?x_3)))), a(a(c(a(?x_4)))) = a(a(c(b(?x_4)))), a(a(c(a(?x_9)))) = a(a(c(a(?x_9)))), c(b(a(c(?x_1)))) = c(a(c(a(?x_1)))), c(b(c(a(?x_2)))) = c(a(c(c(?x_2)))), c(b(a(a(?x_3)))) = c(a(c(a(?x_3)))), c(b(c(a(?x_4)))) = c(a(c(b(?x_4)))), c(b(c(a(?x_9)))) = c(a(c(a(?x_9)))), a(c(a(c(?x_1)))) = b(a(c(a(?x_1)))), a(c(c(a(?x_2)))) = b(a(c(c(?x_2)))), a(c(a(a(?x_3)))) = b(a(c(a(?x_3)))), a(c(c(a(?x_4)))) = b(a(c(b(?x_4)))), a(c(c(a(?x_9)))) = b(a(c(a(?x_9)))), c(a(a(c(?x_1)))) = a(a(c(a(?x_1)))), c(a(c(a(?x_2)))) = a(a(c(c(?x_2)))), c(a(a(a(?x_3)))) = a(a(c(a(?x_3)))), c(a(c(a(?x_4)))) = a(a(c(b(?x_4)))), c(a(c(a(?x_9)))) = a(a(c(a(?x_9)))), a(a(a(?x))) = a(a(c(?x))), c(b(a(?x))) = c(a(c(?x))), a(c(a(?x))) = b(a(c(?x))), c(a(a(?x))) = a(a(c(?x))), c(c(a(?x_2))) = c(b(c(?x_2))), c(a(a(?x_3))) = c(b(a(?x_3))), c(c(a(?x_4))) = c(b(b(?x_4))), c(a(c(?x_5))) = c(b(a(?x_5))), c(c(a(?x_9))) = c(b(a(?x_9))), c(a(c(a(?x_2)))) = a(c(b(c(?x_2)))), c(a(a(a(?x_3)))) = a(c(b(a(?x_3)))), c(a(c(a(?x_4)))) = a(c(b(b(?x_4)))), c(a(a(c(?x_5)))) = a(c(b(a(?x_5)))), c(a(c(a(?x_9)))) = a(c(b(a(?x_9)))), c(a(a(?x))) = a(c(b(?x))), b(a(c(?x_1))) = a(c(b(?x_1))), b(a(c(?x_7))) = a(c(a(?x_7))), a(c(a(c(?x_1)))) = b(a(c(b(?x_1)))), c(a(a(c(?x_1)))) = a(a(c(b(?x_1)))), a(b(a(c(?x_1)))) = c(a(c(b(?x_1)))), a(b(a(c(?x_7)))) = c(a(c(a(?x_7)))), a(c(b(?x))) = b(a(c(?x))), c(a(b(?x))) = a(a(c(?x))), a(b(b(?x))) = c(a(c(?x))), b(c(a(?x_2))) = a(c(c(?x_2))), b(a(a(?x_3))) = a(c(a(?x_3))), b(c(a(?x_4))) = a(c(b(?x_4))), b(c(a(?x_9))) = a(c(a(?x_9))), a(b(c(a(?x_2)))) = c(a(c(c(?x_2)))), a(b(a(a(?x_3)))) = c(a(c(a(?x_3)))), a(b(c(a(?x_4)))) = c(a(c(b(?x_4)))), a(b(c(a(?x_9)))) = c(a(c(a(?x_9)))), a(b(a(?x))) = c(a(c(?x))), c(a(c(?x_7))) = a(b(b(?x_7))), c(a(c(?x_8))) = a(b(a(?x_8))), c(a(a(c(?x_7)))) = a(a(b(b(?x_7)))), c(a(a(c(?x_8)))) = a(a(b(a(?x_8)))), c(a(b(?x))) = a(a(b(?x))), a(a(?x)) = c(a(?x)), a(c(?x)) = c(a(?x)), a(c(a(?x_1))) = c(a(a(?x_1))), a(c(a(?x_2))) = c(a(c(?x_2))), a(a(a(?x_3))) = c(a(a(?x_3))), a(c(a(?x_4))) = c(a(b(?x_4))), c(a(c(a(?x_1)))) = a(c(a(a(?x_1)))), c(a(c(a(?x_2)))) = a(c(a(c(?x_2)))), c(a(a(a(?x_3)))) = a(c(a(a(?x_3)))), c(a(c(a(?x_4)))) = a(c(a(b(?x_4)))), a(a(c(a(?x_1)))) = a(c(a(a(?x_1)))), a(a(c(a(?x_2)))) = a(c(a(c(?x_2)))), a(a(a(a(?x_3)))) = a(c(a(a(?x_3)))), a(a(c(a(?x_4)))) = a(c(a(b(?x_4)))), a(c(c(a(?x_1)))) = a(c(a(a(?x_1)))), a(c(c(a(?x_2)))) = a(c(a(c(?x_2)))), a(c(a(a(?x_3)))) = a(c(a(a(?x_3)))), a(c(c(a(?x_4)))) = a(c(a(b(?x_4)))), c(b(c(a(?x_1)))) = c(c(a(a(?x_1)))), c(b(c(a(?x_2)))) = c(c(a(c(?x_2)))), c(b(a(a(?x_3)))) = c(c(a(a(?x_3)))), c(b(c(a(?x_4)))) = c(c(a(b(?x_4)))), a(c(c(a(?x_1)))) = b(c(a(a(?x_1)))), a(c(c(a(?x_2)))) = b(c(a(c(?x_2)))), a(c(a(a(?x_3)))) = b(c(a(a(?x_3)))), a(c(c(a(?x_4)))) = b(c(a(b(?x_4)))), c(a(a(?x))) = a(c(a(?x))), c(b(a(?x))) = c(c(a(?x))), a(c(a(?x))) = b(c(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 <4, 0> preceded by [(a,1)] joinable by a reduction of rules <[([],0),([(c,1)],2)], [([(c,1)],8)]> joinable by a reduction of rules <[([],0),([(c,1)],2)], [([(c,1)],3),([(c,1)],0)]> Critical Pair by Rules <7, 0> preceded by [(a,1)] joinable by a reduction of rules <[([],8)], []> Critical Pair by Rules <0, 1> preceded by [(a,1)] joinable by a reduction of rules <[], [([(a,1)],0)]> Critical Pair by Rules <2, 1> preceded by [(a,1)] joinable by a reduction of rules <[], [([(a,1)],2)]> joinable by a reduction of rules <[([(a,1)],4)], [([],3)]> Critical Pair by Rules <3, 1> preceded by [(a,1)] joinable by a reduction of rules <[], [([(a,1)],3)]> joinable by a reduction of rules <[([(a,1)],0)], [([(a,1)],8)]> joinable by a reduction of rules <[([(a,1)],0)], [([],3)]> Critical Pair by Rules <8, 1> preceded by [(a,1)] joinable by a reduction of rules <[], [([(a,1)],8)]> joinable by a reduction of rules <[], [([],3)]> joinable by a reduction of rules <[([],0)], [([],8)]> Critical Pair by Rules <5, 2> preceded by [(a,1)] joinable by a reduction of rules <[([],8),([(c,1)],0)], [([(c,1)],2)]> joinable by a reduction of rules <[([],8)], [([],4),([(c,1)],5)]> joinable by a reduction of rules <[([],3),([],0)], [([],4),([(c,1)],5)]> Critical Pair by Rules <6, 2> preceded by [(a,1)] joinable by a reduction of rules <[([],8)], [([(c,1)],3)]> Critical Pair by Rules <0, 3> preceded by [(a,1)] joinable by a reduction of rules <[([],0),([(c,1)],3)], [([],0)]> joinable by a reduction of rules <[([],0),([(c,1)],8)], [([],0),([(c,1)],0)]> Critical Pair by Rules <1, 3> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],8)], []> joinable by a reduction of rules <[([],3)], []> joinable by a reduction of rules <[([],8)], [([],0)]> Critical Pair by Rules <2, 3> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],4)], []> Critical Pair by Rules <8, 3> preceded by [(a,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <0, 4> preceded by [(c,1)] joinable by a reduction of rules <[], [([],7),([],2),([(c,1)],0)]> Critical Pair by Rules <1, 4> preceded by [(c,1)] joinable by a reduction of rules <[([],4)], []> joinable by a reduction of rules <[([(c,1)],3)], [([(c,1)],6)]> Critical Pair by Rules <2, 4> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],5),([(c,1)],0)]> joinable by a reduction of rules <[([(c,1)],4),([(c,1)],7)], [([],7),([],2)]> Critical Pair by Rules <3, 4> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],6)]> Critical Pair by Rules <8, 4> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],6),([(c,1)],0)]> Critical Pair by Rules <6, 5> preceded by [(b,1)] joinable by a reduction of rules <[([],6),([],0)], [([],0),([(c,1)],3)]> Critical Pair by Rules <0, 6> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],4),([(b,1)],7),([],6),([],0)], [([],0),([(c,1)],0),([(c,1)],4),([(c,1)],7)]> Critical Pair by Rules <1, 6> preceded by [(b,1)] joinable by a reduction of rules <[([],6)], []> Critical Pair by Rules <2, 6> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],4),([(b,1)],7),([],6)], []> Critical Pair by Rules <3, 6> preceded by [(b,1)] joinable by a reduction of rules <[([],6),([],0)], [([],0),([(c,1)],3)]> Critical Pair by Rules <8, 6> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],4),([(b,1)],7),([],6)], [([(a,1)],4)]> Critical Pair by Rules <5, 7> preceded by [(c,1)] joinable by a reduction of rules <[], [([(a,1)],5),([],8)]> joinable by a reduction of rules <[([(c,1)],0)], [([],2),([(c,1)],2)]> Critical Pair by Rules <6, 7> preceded by [(c,1)] joinable by a reduction of rules <[], [([(a,1)],6),([],8)]> joinable by a reduction of rules <[], [([],2),([(c,1)],3)]> joinable by a reduction of rules <[([(c,1)],0)], [([],2),([(c,1)],8)]> Critical Pair by Rules <0, 8> preceded by [(a,1)] joinable by a reduction of rules <[([],0),([(c,1)],8)], [([(c,1)],0)]> joinable by a reduction of rules <[([],0),([(c,1)],3)], []> Critical Pair by Rules <1, 8> preceded by [(a,1)] joinable by a reduction of rules <[([],8)], []> Critical Pair by Rules <2, 8> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],4),([],0)], []> joinable by a reduction of rules <[([],0),([(c,1)],8)], [([(c,1)],2)]> joinable by a reduction of rules <[([],0),([(c,1)],3)], [([],4),([(c,1)],5)]> Critical Pair by Rules <3, 8> preceded by [(a,1)] joinable by a reduction of rules <[([],8)], [([(c,1)],3)]> Critical Pair by Rules <1, 1> preceded by [(a,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <3, 3> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],0)], []> Critical Pair by Rules <5, 5> preceded by [(b,1)] joinable by a reduction of rules <[([],6),([],0),([(c,1)],0)], [([],0),([(c,1)],2)]> joinable by a reduction of rules <[([],6),([],0)], [([],0),([],4),([(c,1)],5)]> joinable by a reduction of rules <[([],6),([],0),([(c,1)],0)], [([(a,1)],7),([],8),([(c,1)],2)]> Critical Pair by Rules <8, 8> preceded by [(a,1)] joinable by a reduction of rules <[([],0)], []> Critical Pair by Rules <3, 1> preceded by [] joinable by a reduction of rules <[], [([],3)]> joinable by a reduction of rules <[([],0)], [([],8)]> Critical Pair by Rules <8, 1> preceded by [] joinable by a reduction of rules <[], [([],8)]> Critical Pair by Rules <8, 3> preceded by [] joinable by a reduction of rules <[], [([],0)]> unknown Diagram Decreasing check Non-Confluence... obtain 14 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(c(?x)) -> c(a(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> c(b(?x)), b(b(?x)) -> a(c(?x)), b(a(?x)) -> a(c(?x)), c(b(?x)) -> a(b(?x)), a(a(?x)) -> c(a(?x)) ] P: [ a(a(?x)) -> a(a(?x)) ] S: unknown termination failure(Step 1) STEP: 2 (linear) S: [ a(c(?x)) -> c(a(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> c(b(?x)), b(b(?x)) -> a(c(?x)), b(a(?x)) -> a(c(?x)), c(b(?x)) -> a(b(?x)), a(a(?x)) -> c(a(?x)) ] P: [ a(a(?x)) -> a(a(?x)) ] S: unknown termination failure(Step 2) STEP: 3 (relative) S: [ a(c(?x)) -> c(a(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> c(b(?x)), b(b(?x)) -> a(c(?x)), b(a(?x)) -> a(c(?x)), c(b(?x)) -> a(b(?x)), a(a(?x)) -> c(a(?x)) ] P: [ a(a(?x)) -> a(a(?x)) ] Check relative termination: [ a(c(?x)) -> c(a(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> c(b(?x)), b(b(?x)) -> a(c(?x)), b(a(?x)) -> a(c(?x)), c(b(?x)) -> a(b(?x)), a(a(?x)) -> c(a(?x)) ] [ a(a(?x)) -> a(a(?x)) ]