YES (ignored inputs)COMMENT submitted by: Johannes Waldmann Rewrite Rules: [ b(b(?x)) -> a(a(?x)), b(b(?x)) -> c(b(?x)), a(b(?x)) -> c(b(?x)), b(c(?x)) -> a(a(?x)), b(a(?x)) -> a(a(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> c(b(?x)), c(a(?x)) -> c(a(?x)), c(c(?x)) -> a(a(?x)) ] Apply Direct Methods... Inner CPs: [ b(c(b(?x_1))) = a(a(b(?x_1))), b(a(a(?x_3))) = a(a(c(?x_3))), b(a(a(?x_4))) = a(a(a(?x_4))), b(a(a(?x))) = c(b(b(?x))), b(a(a(?x_3))) = c(b(c(?x_3))), b(a(a(?x_4))) = c(b(a(?x_4))), a(a(a(?x))) = c(b(b(?x))), a(c(b(?x_1))) = c(b(b(?x_1))), a(a(a(?x_3))) = c(b(c(?x_3))), a(a(a(?x_4))) = c(b(a(?x_4))), b(c(b(?x_5))) = a(a(a(?x_5))), b(c(a(?x_7))) = a(a(a(?x_7))), b(a(a(?x_8))) = a(a(c(?x_8))), b(c(b(?x_2))) = a(a(b(?x_2))), b(c(b(?x_6))) = a(a(a(?x_6))), c(c(b(?x_2))) = c(b(b(?x_2))), c(c(b(?x_6))) = c(b(a(?x_6))), a(c(b(?x_2))) = c(b(b(?x_2))), c(c(b(?x_2))) = c(a(b(?x_2))), c(c(b(?x_6))) = c(a(a(?x_6))), c(c(b(?x_5))) = a(a(a(?x_5))), c(c(a(?x_7))) = a(a(a(?x_7))), b(a(a(?x))) = a(a(b(?x))), b(c(b(?x))) = c(b(b(?x))), a(c(b(?x))) = c(b(a(?x))), c(a(a(?x))) = a(a(c(?x))) ] Outer CPs: [ a(a(?x)) = c(b(?x)), c(b(?x_5)) = c(a(?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 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: [ c(b(?x)) = a(a(?x)), b(a(a(?x_1))) = a(a(b(?x_1))), b(c(b(?x_2))) = a(a(b(?x_2))), b(a(a(?x_4))) = a(a(c(?x_4))), b(a(a(?x_5))) = a(a(a(?x_5))), a(a(a(a(?x_1)))) = b(a(a(b(?x_1)))), a(a(c(b(?x_2)))) = b(a(a(b(?x_2)))), a(a(a(a(?x_4)))) = b(a(a(c(?x_4)))), a(a(a(a(?x_5)))) = b(a(a(a(?x_5)))), c(b(a(a(?x_1)))) = b(a(a(b(?x_1)))), c(b(c(b(?x_2)))) = b(a(a(b(?x_2)))), c(b(a(a(?x_4)))) = b(a(a(c(?x_4)))), c(b(a(a(?x_5)))) = b(a(a(a(?x_5)))), c(b(a(a(?x_1)))) = a(a(a(b(?x_1)))), c(b(c(b(?x_2)))) = a(a(a(b(?x_2)))), c(b(a(a(?x_4)))) = a(a(a(c(?x_4)))), c(b(a(a(?x_5)))) = a(a(a(a(?x_5)))), a(a(b(?x))) = b(a(a(?x))), c(b(b(?x))) = b(a(a(?x))), c(b(b(?x))) = a(a(a(?x))), a(a(?x)) = c(b(?x)), b(c(b(?x_1))) = c(b(b(?x_1))), b(a(a(?x_2))) = c(b(b(?x_2))), b(a(a(?x_4))) = c(b(c(?x_4))), b(a(a(?x_5))) = c(b(a(?x_5))), c(b(c(b(?x_1)))) = b(c(b(b(?x_1)))), c(b(a(a(?x_2)))) = b(c(b(b(?x_2)))), c(b(a(a(?x_4)))) = b(c(b(c(?x_4)))), c(b(a(a(?x_5)))) = b(c(b(a(?x_5)))), a(a(c(b(?x_1)))) = b(c(b(b(?x_1)))), a(a(a(a(?x_2)))) = b(c(b(b(?x_2)))), a(a(a(a(?x_4)))) = b(c(b(c(?x_4)))), a(a(a(a(?x_5)))) = b(c(b(a(?x_5)))), c(b(c(b(?x_1)))) = a(c(b(b(?x_1)))), c(b(a(a(?x_2)))) = a(c(b(b(?x_2)))), c(b(a(a(?x_4)))) = a(c(b(c(?x_4)))), c(b(a(a(?x_5)))) = a(c(b(a(?x_5)))), c(b(b(?x))) = b(c(b(?x))), a(a(b(?x))) = b(c(b(?x))), c(b(b(?x))) = a(c(b(?x))), a(a(a(?x_2))) = c(b(b(?x_2))), a(c(b(?x_3))) = c(b(b(?x_3))), a(a(a(?x_4))) = c(b(c(?x_4))), a(a(a(?x_5))) = c(b(a(?x_5))), c(b(a(a(?x_2)))) = c(c(b(b(?x_2)))), c(b(c(b(?x_3)))) = c(c(b(b(?x_3)))), c(b(a(a(?x_4)))) = c(c(b(c(?x_4)))), c(b(a(a(?x_5)))) = c(c(b(a(?x_5)))), c(a(a(a(?x_2)))) = c(c(b(b(?x_2)))), c(a(c(b(?x_3)))) = c(c(b(b(?x_3)))), c(a(a(a(?x_4)))) = c(c(b(c(?x_4)))), c(a(a(a(?x_5)))) = c(c(b(a(?x_5)))), c(b(b(?x))) = c(c(b(?x))), c(a(b(?x))) = c(c(b(?x))), b(c(b(?x_6))) = a(a(a(?x_6))), b(c(a(?x_8))) = a(a(a(?x_8))), a(a(c(b(?x_6)))) = b(a(a(a(?x_6)))), a(a(c(a(?x_8)))) = b(a(a(a(?x_8)))), c(b(c(b(?x_6)))) = b(a(a(a(?x_6)))), c(b(c(a(?x_8)))) = b(a(a(a(?x_8)))), c(b(c(b(?x_6)))) = a(a(a(a(?x_6)))), c(b(c(a(?x_8)))) = a(a(a(a(?x_8)))), a(a(c(?x))) = b(a(a(?x))), c(b(c(?x))) = b(a(a(?x))), c(b(c(?x))) = a(a(a(?x))), a(a(a(?x))) = b(a(a(?x))), c(b(a(?x))) = b(a(a(?x))), c(b(a(?x))) = a(a(a(?x))), c(a(?x)) = c(b(?x)), c(c(b(?x_4))) = c(b(b(?x_4))), c(c(b(?x_7))) = c(b(a(?x_7))), a(a(c(b(?x_7)))) = b(c(b(a(?x_7)))), a(a(c(b(?x_4)))) = c(c(b(b(?x_4)))), a(a(c(b(?x_7)))) = c(c(b(a(?x_7)))), a(a(a(?x))) = b(c(b(?x))), a(a(a(?x))) = c(c(b(?x))), a(c(b(?x_1))) = c(b(a(?x_1))), c(b(c(b(?x_1)))) = a(c(b(a(?x_1)))), c(b(c(b(?x_1)))) = c(c(b(a(?x_1)))), c(a(c(b(?x_1)))) = c(c(b(a(?x_1)))), c(b(a(?x))) = a(c(b(?x))), c(b(a(?x))) = c(c(b(?x))), c(a(a(?x))) = c(c(b(?x))), c(b(?x)) = c(a(?x)), c(c(b(?x_4))) = c(a(b(?x_4))), c(c(b(?x_8))) = c(a(a(?x_8))), a(a(c(b(?x_4)))) = b(c(a(b(?x_4)))), a(a(c(b(?x_8)))) = b(c(a(a(?x_8)))), a(a(c(b(?x_4)))) = c(c(a(b(?x_4)))), a(a(c(b(?x_8)))) = c(c(a(a(?x_8)))), a(a(a(?x))) = b(c(a(?x))), a(a(a(?x))) = c(c(a(?x))), c(a(a(?x_1))) = a(a(c(?x_1))), c(c(b(?x_7))) = a(a(a(?x_7))), c(c(a(?x_9))) = a(a(a(?x_9))), a(a(a(a(?x_1)))) = c(a(a(c(?x_1)))), a(a(c(b(?x_7)))) = c(a(a(a(?x_7)))), a(a(c(a(?x_9)))) = c(a(a(a(?x_9)))), a(a(c(?x))) = c(a(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 <[([],3)], []> Critical Pair by Rules <3, 0> preceded by [(b,1)] joinable by a reduction of rules <[([],4),([],6),([(c,1)],4)], [([],6),([(c,1)],3)]> joinable by a reduction of rules <[([],4),([],6)], [([],6),([(c,1)],3),([],5)]> Critical Pair by Rules <4, 0> preceded by [(b,1)] joinable by a reduction of rules <[([],4)], []> Critical Pair by Rules <0, 1> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],6),([],3)], [([(c,1)],1),([],8)]> joinable by a reduction of rules <[([],4),([],6)], [([(c,1)],0),([],5)]> Critical Pair by Rules <3, 1> preceded by [(b,1)] joinable by a reduction of rules <[([],4),([],6)], [([(c,1)],3),([],5)]> Critical Pair by Rules <4, 1> preceded by [(b,1)] joinable by a reduction of rules <[([],4),([],6)], []> Critical Pair by Rules <0, 2> preceded by [(a,1)] joinable by a reduction of rules <[([],6),([(c,1)],4)], [([(c,1)],0)]> joinable by a reduction of rules <[([],6)], [([(c,1)],0),([],5)]> Critical Pair by Rules <1, 2> preceded by [(a,1)] joinable by a reduction of rules <[], [([(c,1)],1),([],8),([(a,1)],2)]> Critical Pair by Rules <3, 2> preceded by [(a,1)] joinable by a reduction of rules <[([],6),([(c,1)],4)], [([(c,1)],3)]> joinable by a reduction of rules <[([],6)], [([(c,1)],3),([],5)]> Critical Pair by Rules <4, 2> preceded by [(a,1)] joinable by a reduction of rules <[([],6)], []> Critical Pair by Rules <5, 3> preceded by [(b,1)] joinable by a reduction of rules <[([],3),([(a,1)],2)], [([(a,1)],6)]> Critical Pair by Rules <7, 3> preceded by [(b,1)] joinable by a reduction of rules <[([],3)], []> Critical Pair by Rules <8, 3> preceded by [(b,1)] joinable by a reduction of rules <[([],4),([],6),([(c,1)],4)], [([],6),([(c,1)],3)]> joinable by a reduction of rules <[([],4),([],6)], [([],6),([(c,1)],3),([],5)]> Critical Pair by Rules <2, 4> preceded by [(b,1)] joinable by a reduction of rules <[([],3)], []> Critical Pair by Rules <6, 4> preceded by [(b,1)] joinable by a reduction of rules <[([],3),([(a,1)],2)], [([(a,1)],6)]> Critical Pair by Rules <2, 5> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],1)]> Critical Pair by Rules <6, 5> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],4),([(c,1)],6)]> Critical Pair by Rules <2, 6> preceded by [(a,1)] joinable by a reduction of rules <[], [([(c,1)],1),([],8),([(a,1)],2)]> Critical Pair by Rules <2, 7> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],2)]> Critical Pair by Rules <6, 7> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],6)]> Critical Pair by Rules <5, 8> preceded by [(c,1)] joinable by a reduction of rules <[([],8),([(a,1)],2)], [([(a,1)],6)]> Critical Pair by Rules <7, 8> preceded by [(c,1)] joinable by a reduction of rules <[([],8)], []> Critical Pair by Rules <0, 0> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],6),([],3)], []> joinable by a reduction of rules <[([],4),([(a,1)],6)], [([(a,1)],2)]> Critical Pair by Rules <1, 1> preceded by [(b,1)] joinable by a reduction of rules <[([],3),([],6)], []> joinable by a reduction of rules <[([],3)], [([(c,1)],1),([],8)]> Critical Pair by Rules <6, 6> preceded by [(a,1)] joinable by a reduction of rules <[], [([(c,1)],4),([(c,1)],6),([],8),([(a,1)],2)]> Critical Pair by Rules <8, 8> preceded by [(c,1)] joinable by a reduction of rules <[], [([],6),([(c,1)],3)]> Critical Pair by Rules <1, 0> preceded by [] joinable by a reduction of rules <[], [([],6)]> Critical Pair by Rules <7, 5> preceded by [] joinable by a reduction of rules <[([],5)], []> unknown Diagram Decreasing check Non-Confluence... obtain 11 rules by 3 steps unfolding obtain 92 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)) -> a(a(?x)), b(b(?x)) -> c(b(?x)), a(b(?x)) -> c(b(?x)), b(c(?x)) -> a(a(?x)), b(a(?x)) -> a(a(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> c(b(?x)), c(c(?x)) -> a(a(?x)) ] P: [ c(a(?x)) -> c(a(?x)) ] S: unknown termination failure(Step 1) STEP: 2 (linear) S: [ b(b(?x)) -> a(a(?x)), b(b(?x)) -> c(b(?x)), a(b(?x)) -> c(b(?x)), b(c(?x)) -> a(a(?x)), b(a(?x)) -> a(a(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> c(b(?x)), c(c(?x)) -> a(a(?x)) ] P: [ c(a(?x)) -> c(a(?x)) ] S: unknown termination failure(Step 2) STEP: 3 (relative) S: [ b(b(?x)) -> a(a(?x)), b(b(?x)) -> c(b(?x)), a(b(?x)) -> c(b(?x)), b(c(?x)) -> a(a(?x)), b(a(?x)) -> a(a(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> c(b(?x)), c(c(?x)) -> a(a(?x)) ] P: [ c(a(?x)) -> c(a(?x)) ] Check relative termination: [ b(b(?x)) -> a(a(?x)), b(b(?x)) -> c(b(?x)), a(b(?x)) -> c(b(?x)), b(c(?x)) -> a(a(?x)), b(a(?x)) -> a(a(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> c(b(?x)), c(c(?x)) -> a(a(?x)) ] [ c(a(?x)) -> c(a(?x)) ] Polynomial Interpretation: a:= (1)+(2)*x1+(1)*x1*x1 b:= (1)+(2)*x1+(1)*x1*x1 c:= (2)+(2)*x1+(1)*x1*x1 retract b(c(?x)) -> a(a(?x)) retract c(c(?x)) -> a(a(?x)) Polynomial Interpretation: a:= (2)+(2)*x1 b:= (2)*x1 c:= (1)*x1 retract a(b(?x)) -> c(b(?x)) retract b(c(?x)) -> a(a(?x)) retract c(a(?x)) -> c(b(?x)) retract a(a(?x)) -> c(b(?x)) retract c(c(?x)) -> a(a(?x)) Polynomial Interpretation: a:= (2)*x1 b:= (1)+(2)*x1+(1)*x1*x1 c:= (4)*x1 retract b(b(?x)) -> a(a(?x)) retract a(b(?x)) -> c(b(?x)) retract b(c(?x)) -> a(a(?x)) retract b(a(?x)) -> a(a(?x)) retract c(a(?x)) -> c(b(?x)) retract a(a(?x)) -> c(b(?x)) retract c(c(?x)) -> a(a(?x)) Polynomial Interpretation: a:= (2)+(2)*x1 b:= (1)+(2)*x1*x1 c:= (2)*x1 relatively terminating S/P: relatively terminating check CP condition: success S: [ b(b(?x)) -> a(a(?x)), b(b(?x)) -> c(b(?x)), a(b(?x)) -> c(b(?x)), b(c(?x)) -> a(a(?x)), b(a(?x)) -> a(a(?x)), c(a(?x)) -> c(b(?x)), a(a(?x)) -> c(b(?x)), c(c(?x)) -> a(a(?x)) ] P: [ c(a(?x)) -> c(a(?x)) ] Success Reduction-Preserving Completion Direct Methods: CR Combined result: CR 1021.trs: Success(CR) (2940 msec.)