YES (ignored inputs)COMMENT reduction failed Rewrite Rules: [ a(c(?x)) -> a(a(?x)), b(c(?x)) -> c(b(?x)), a(c(?x)) -> a(b(?x)), b(c(?x)) -> a(c(?x)), c(b(?x)) -> b(c(?x)), a(b(?x)) -> b(a(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> a(b(?x)) ] Apply Direct Methods... Inner CPs: [ a(b(c(?x_4))) = a(a(b(?x_4))), a(a(b(?x_8))) = a(a(a(?x_8))), b(b(c(?x_4))) = c(b(b(?x_4))), b(a(b(?x_8))) = c(b(a(?x_8))), a(b(c(?x_4))) = a(b(b(?x_4))), a(a(b(?x_8))) = a(b(a(?x_8))), b(b(c(?x_4))) = a(c(b(?x_4))), b(a(b(?x_8))) = a(c(a(?x_8))), c(c(b(?x_1))) = b(c(c(?x_1))), c(a(c(?x_3))) = b(c(c(?x_3))), a(c(b(?x_1))) = b(a(c(?x_1))), a(a(c(?x_3))) = b(a(c(?x_3))), a(c(b(?x_1))) = c(a(c(?x_1))), a(a(c(?x_3))) = c(a(c(?x_3))), a(a(a(?x))) = a(c(c(?x))), a(a(b(?x_2))) = a(c(c(?x_2))), a(b(a(?x_5))) = a(c(b(?x_5))), a(c(a(?x_6))) = a(c(b(?x_6))), c(a(a(?x))) = a(b(c(?x))), c(a(b(?x_2))) = a(b(c(?x_2))), c(b(a(?x_5))) = a(b(b(?x_5))), c(c(a(?x_6))) = a(b(b(?x_6))), c(a(c(?x_7))) = a(b(a(?x_7))), a(a(c(?x))) = a(c(a(?x))) ] Outer CPs: [ a(a(?x)) = a(b(?x)), c(b(?x_1)) = a(c(?x_1)), b(a(?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 unknown Upside-Parallel-Closed/Outside-Closed (inner) Parallel CPs: (not computed) unknown Toyama (Parallel CPs) Simultaneous CPs: [ a(b(?x)) = a(a(?x)), a(b(c(?x_5))) = a(a(b(?x_5))), a(a(b(?x_9))) = a(a(a(?x_9))), a(c(b(c(?x_5)))) = a(a(a(b(?x_5)))), a(c(a(b(?x_9)))) = a(a(a(a(?x_9)))), a(b(b(c(?x_5)))) = c(a(a(b(?x_5)))), a(b(a(b(?x_9)))) = c(a(a(a(?x_9)))), a(c(c(?x))) = a(a(a(?x))), a(b(c(?x))) = c(a(a(?x))), a(c(?x)) = c(b(?x)), b(b(c(?x_5))) = c(b(b(?x_5))), b(a(b(?x_9))) = c(b(a(?x_9))), b(c(b(c(?x_5)))) = c(c(b(b(?x_5)))), b(c(a(b(?x_9)))) = c(c(b(a(?x_9)))), b(a(b(c(?x_5)))) = a(c(b(b(?x_5)))), b(a(a(b(?x_9)))) = a(c(b(a(?x_9)))), c(a(b(c(?x_5)))) = a(c(b(b(?x_5)))), c(a(a(b(?x_9)))) = a(c(b(a(?x_9)))), b(c(c(?x))) = c(c(b(?x))), b(a(c(?x))) = a(c(b(?x))), c(a(c(?x))) = a(c(b(?x))), a(a(?x)) = a(b(?x)), a(b(c(?x_5))) = a(b(b(?x_5))), a(a(b(?x_9))) = a(b(a(?x_9))), a(c(b(c(?x_5)))) = a(a(b(b(?x_5)))), a(c(a(b(?x_9)))) = a(a(b(a(?x_9)))), a(b(b(c(?x_5)))) = c(a(b(b(?x_5)))), a(b(a(b(?x_9)))) = c(a(b(a(?x_9)))), a(c(c(?x))) = a(a(b(?x))), a(b(c(?x))) = c(a(b(?x))), c(b(?x)) = a(c(?x)), b(b(c(?x_5))) = a(c(b(?x_5))), b(a(b(?x_9))) = a(c(a(?x_9))), b(c(b(c(?x_5)))) = c(a(c(b(?x_5)))), b(c(a(b(?x_9)))) = c(a(c(a(?x_9)))), b(a(b(c(?x_5)))) = a(a(c(b(?x_5)))), b(a(a(b(?x_9)))) = a(a(c(a(?x_9)))), c(a(b(c(?x_5)))) = a(a(c(b(?x_5)))), c(a(a(b(?x_9)))) = a(a(c(a(?x_9)))), b(c(c(?x))) = c(a(c(?x))), b(a(c(?x))) = a(a(c(?x))), c(a(c(?x))) = a(a(c(?x))), c(c(b(?x_3))) = b(c(c(?x_3))), c(a(c(?x_5))) = b(c(c(?x_5))), a(a(c(b(?x_3)))) = a(b(c(c(?x_3)))), a(a(a(c(?x_5)))) = a(b(c(c(?x_5)))), c(b(c(b(?x_3)))) = b(b(c(c(?x_3)))), c(b(a(c(?x_5)))) = b(b(c(c(?x_5)))), a(b(c(b(?x_3)))) = a(b(c(c(?x_3)))), a(b(a(c(?x_5)))) = a(b(c(c(?x_5)))), a(c(c(b(?x_3)))) = b(b(c(c(?x_3)))), a(c(a(c(?x_5)))) = b(b(c(c(?x_5)))), a(a(b(?x))) = a(b(c(?x))), c(b(b(?x))) = b(b(c(?x))), a(b(b(?x))) = a(b(c(?x))), a(c(b(?x))) = b(b(c(?x))), c(a(?x)) = b(a(?x)), a(c(b(?x_3))) = b(a(c(?x_3))), a(a(c(?x_5))) = b(a(c(?x_5))), a(c(c(b(?x_3)))) = a(b(a(c(?x_3)))), a(c(a(c(?x_5)))) = a(b(a(c(?x_5)))), a(b(c(b(?x_3)))) = c(b(a(c(?x_3)))), a(b(a(c(?x_5)))) = c(b(a(c(?x_5)))), a(c(b(?x))) = a(b(a(?x))), a(b(b(?x))) = c(b(a(?x))), b(a(?x)) = c(a(?x)), a(c(b(?x_3))) = c(a(c(?x_3))), a(a(c(?x_5))) = c(a(c(?x_5))), a(c(c(b(?x_3)))) = a(c(a(c(?x_3)))), a(c(a(c(?x_5)))) = a(c(a(c(?x_5)))), a(b(c(b(?x_3)))) = c(c(a(c(?x_3)))), a(b(a(c(?x_5)))) = c(c(a(c(?x_5)))), a(c(b(?x))) = a(c(a(?x))), a(b(b(?x))) = c(c(a(?x))), a(a(c(?x_1))) = a(c(a(?x_1))), a(a(a(?x_2))) = a(c(c(?x_2))), a(a(b(?x_4))) = a(c(c(?x_4))), a(b(a(?x_7))) = a(c(b(?x_7))), a(c(a(?x_8))) = a(c(b(?x_8))), a(c(a(c(?x_1)))) = a(a(c(a(?x_1)))), a(c(a(a(?x_2)))) = a(a(c(c(?x_2)))), a(c(a(b(?x_4)))) = a(a(c(c(?x_4)))), a(c(b(a(?x_7)))) = a(a(c(b(?x_7)))), a(c(c(a(?x_8)))) = a(a(c(b(?x_8)))), a(b(a(c(?x_1)))) = c(a(c(a(?x_1)))), a(b(a(a(?x_2)))) = c(a(c(c(?x_2)))), a(b(a(b(?x_4)))) = c(a(c(c(?x_4)))), a(b(b(a(?x_7)))) = c(a(c(b(?x_7)))), a(b(c(a(?x_8)))) = c(a(c(b(?x_8)))), a(c(a(?x))) = a(a(c(?x))), a(b(a(?x))) = c(a(c(?x))), c(a(a(?x_2))) = a(b(c(?x_2))), c(a(b(?x_4))) = a(b(c(?x_4))), c(b(a(?x_7))) = a(b(b(?x_7))), c(c(a(?x_8))) = a(b(b(?x_8))), c(a(c(?x_9))) = a(b(a(?x_9))), a(a(a(a(?x_2)))) = a(a(b(c(?x_2)))), a(a(a(b(?x_4)))) = a(a(b(c(?x_4)))), a(a(b(a(?x_7)))) = a(a(b(b(?x_7)))), a(a(c(a(?x_8)))) = a(a(b(b(?x_8)))), a(a(a(c(?x_9)))) = a(a(b(a(?x_9)))), c(b(a(a(?x_2)))) = b(a(b(c(?x_2)))), c(b(a(b(?x_4)))) = b(a(b(c(?x_4)))), c(b(b(a(?x_7)))) = b(a(b(b(?x_7)))), c(b(c(a(?x_8)))) = b(a(b(b(?x_8)))), c(b(a(c(?x_9)))) = b(a(b(a(?x_9)))), a(b(a(a(?x_2)))) = a(a(b(c(?x_2)))), a(b(a(b(?x_4)))) = a(a(b(c(?x_4)))), a(b(b(a(?x_7)))) = a(a(b(b(?x_7)))), a(b(c(a(?x_8)))) = a(a(b(b(?x_8)))), a(b(a(c(?x_9)))) = a(a(b(a(?x_9)))), a(c(a(a(?x_2)))) = b(a(b(c(?x_2)))), a(c(a(b(?x_4)))) = b(a(b(c(?x_4)))), a(c(b(a(?x_7)))) = b(a(b(b(?x_7)))), a(c(c(a(?x_8)))) = b(a(b(b(?x_8)))), a(c(a(c(?x_9)))) = b(a(b(a(?x_9)))), a(a(a(?x))) = a(a(b(?x))), c(b(a(?x))) = b(a(b(?x))), a(b(a(?x))) = a(a(b(?x))), a(c(a(?x))) = b(a(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 <4, 0> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],1)], [([],7)]> Critical Pair by Rules <8, 0> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],6)], [([],7)]> Critical Pair by Rules <4, 1> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],1)], [([],4)]> Critical Pair by Rules <8, 1> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],6)], [([],4)]> Critical Pair by Rules <4, 2> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],1),([],2)], []> joinable by a reduction of rules <[([],6),([(c,1)],2)], [([],6)]> joinable by a reduction of rules <[([],5),([(b,1)],2)], [([],5)]> Critical Pair by Rules <8, 2> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],5)], []> Critical Pair by Rules <4, 3> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],1),([],3)], []> joinable by a reduction of rules <[([(b,1)],3)], [([(a,1)],4),([],5)]> joinable by a reduction of rules <[([(b,1)],3),([(b,1)],2)], [([],2),([],5)]> Critical Pair by Rules <8, 3> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],6),([],3)], []> Critical Pair by Rules <1, 4> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],4)], [([],1)]> Critical Pair by Rules <3, 4> preceded by [(c,1)] joinable by a reduction of rules <[], [([],1),([(c,1)],3)]> joinable by a reduction of rules <[([],8)], [([],3),([],2)]> joinable by a reduction of rules <[([],8),([(a,1)],3)], [([],3),([],0)]> Critical Pair by Rules <1, 5> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],4),([],5)], []> joinable by a reduction of rules <[([],2),([],5)], [([(b,1)],2)]> Critical Pair by Rules <3, 5> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],2),([(a,1)],5),([],5)], [([(b,1)],0)]> joinable by a reduction of rules <[([],7),([],2),([],5)], []> joinable by a reduction of rules <[([(a,1)],2),([(a,1)],6)], [([(b,1)],2),([(b,1)],6),([],3)]> joinable by a reduction of rules <[([(a,1)],0),([],7)], [([(b,1)],2),([(b,1)],6),([],3)]> Critical Pair by Rules <1, 6> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],4)], [([],8)]> Critical Pair by Rules <3, 6> preceded by [(a,1)] joinable by a reduction of rules <[([],7),([],2)], [([],8)]> joinable by a reduction of rules <[], [([],8),([(a,1)],3)]> joinable by a reduction of rules <[([(a,1)],2),([(a,1)],5)], [([(c,1)],0),([],8)]> joinable by a reduction of rules <[([(a,1)],2),([],7)], [([],8),([(a,1)],1)]> Critical Pair by Rules <0, 7> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],7)], [([],0)]> Critical Pair by Rules <2, 7> preceded by [(a,1)] joinable by a reduction of rules <[([],7),([(a,1)],4)], [([],2)]> joinable by a reduction of rules <[], [([],0),([(a,1)],2)]> joinable by a reduction of rules <[([],7)], [([],2),([(a,1)],1)]> joinable by a reduction of rules <[([(a,1)],6),([],0)], [([],0),([(a,1)],0)]> Critical Pair by Rules <5, 7> preceded by [(a,1)] joinable by a reduction of rules <[], [([],0),([(a,1)],5)]> joinable by a reduction of rules <[([],6),([(c,1)],7)], [([(a,1)],4),([],6)]> joinable by a reduction of rules <[([],5),([(b,1)],7)], [([(a,1)],4),([],5)]> Critical Pair by Rules <6, 7> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],8)], [([],0)]> Critical Pair by Rules <0, 8> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],7)], [([],6)]> Critical Pair by Rules <2, 8> preceded by [(c,1)] joinable by a reduction of rules <[], [([],6),([(c,1)],2)]> joinable by a reduction of rules <[([],8)], [([(a,1)],1),([],2)]> joinable by a reduction of rules <[([],8),([],5)], [([],5),([(b,1)],2)]> Critical Pair by Rules <5, 8> preceded by [(c,1)] joinable by a reduction of rules <[([],4),([(b,1)],8)], [([],5)]> joinable by a reduction of rules <[], [([],6),([(c,1)],5)]> joinable by a reduction of rules <[([],4)], [([],5),([(b,1)],6)]> Critical Pair by Rules <6, 8> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],8)], [([],6)]> Critical Pair by Rules <7, 8> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],0)], [([],6)]> Critical Pair by Rules <7, 7> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],2)], [([(a,1)],8)]> joinable by a reduction of rules <[([(a,1)],0)], [([],0)]> Critical Pair by Rules <2, 0> preceded by [] joinable by a reduction of rules <[], [([],7),([],2)]> Critical Pair by Rules <3, 1> preceded by [] joinable by a reduction of rules <[], [([],4),([],3)]> Critical Pair by Rules <6, 5> preceded by [] joinable by a reduction of rules <[([],8),([],5)], []> 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)) -> a(b(?x)), b(c(?x)) -> a(c(?x)), a(b(?x)) -> b(a(?x)) ] P: [ a(c(?x)) -> a(a(?x)), b(c(?x)) -> c(b(?x)), c(b(?x)) -> b(c(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> a(b(?x)) ] S: terminating CP(S,S): --> => no PCP_in(symP,S): --> => yes --> => no --> => no --> => yes --> => yes CP(S,symP): --> => yes --> => no --> => no --> => no --> => no --> => no --> => no --> => no --> => no check joinability condition: check modulo joinability of b(a(a(?x_1))) and b(b(a(?x_1))): joinable by {0} check modulo joinability of b(a(a(?x_2))) and b(b(a(?x_2))): joinable by {0} check modulo joinability of b(b(a(?x_4))) and b(a(a(?x_4))): joinable by {0} check modulo joinability of c(b(a(?x))) and b(a(a(?x))): joinable by {1} check modulo reachablity from b(a(?x)) to a(a(?x)): maybe not reachable check modulo joinability of c(b(a(?x))) and b(a(a(?x))): joinable by {1} check modulo reachablity from b(a(?x)) to c(b(?x)): maybe not reachable check modulo joinability of b(a(a(?x))) and c(b(a(?x))): joinable by {1} check modulo joinability of b(a(a(?x))) and b(b(a(?x))): joinable by {0} check modulo joinability of c(b(a(?x))) and b(b(a(?x))): joinable by {0,1} check modulo reachablity from b(a(?x)) to c(a(?x)): maybe not reachable failed failure(Step 1) [ a(a(?x)) -> b(a(?x)), c(b(?x)) -> b(a(?x)), c(a(?x)) -> b(a(?x)) ] Added S-Rules: [ a(a(?x)) -> b(a(?x)), c(b(?x)) -> b(a(?x)), c(a(?x)) -> b(a(?x)) ] Added P-Rules: [ ] replace: b(c(?x)) -> a(c(?x)) => b(c(?x)) -> a(a(?x)) replace: a(c(?x)) -> a(b(?x)) => a(c(?x)) -> c(a(?x)) STEP: 2 (linear) S: [ a(c(?x)) -> a(b(?x)), b(c(?x)) -> a(c(?x)), a(b(?x)) -> b(a(?x)) ] P: [ a(c(?x)) -> a(a(?x)), b(c(?x)) -> c(b(?x)), c(b(?x)) -> b(c(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> a(b(?x)) ] S: terminating CP(S,S): --> => no CP_in(symP,S): --> => no --> => yes --> => yes --> => yes --> => no CP(S,symP): --> => yes --> => no --> => no --> => no --> => no --> => no --> => no --> => no --> => no check joinability condition: check modulo joinability of b(a(a(?x_1))) and b(b(a(?x_1))): maybe not joinable check modulo joinability of b(a(a(?x))) and b(b(a(?x))): maybe not joinable check modulo joinability of b(b(a(?x))) and b(a(a(?x))): maybe not joinable check modulo joinability of c(b(a(?x))) and b(a(a(?x))): joinable by {0,1} check modulo reachablity from b(a(?x)) to a(a(?x)): maybe not reachable check modulo joinability of c(b(a(?x))) and b(a(a(?x))): joinable by {0,1} check modulo reachablity from b(a(?x)) to c(b(?x)): maybe not reachable check modulo joinability of b(a(a(?x))) and c(b(a(?x))): joinable by {0,1} check modulo joinability of b(a(a(?x))) and b(b(a(?x))): maybe not joinable check modulo joinability of c(b(a(?x))) and b(b(a(?x))): maybe not joinable check modulo reachablity from b(a(?x)) to c(a(?x)): maybe not reachable failed failure(Step 2) [ a(a(?x)) -> b(a(?x)), c(b(?x)) -> b(a(?x)), c(a(?x)) -> b(a(?x)) ] Added S-Rules: [ a(a(?x)) -> b(a(?x)), c(b(?x)) -> b(a(?x)), c(a(?x)) -> b(a(?x)) ] Added P-Rules: [ ] replace: b(c(?x)) -> a(c(?x)) => b(c(?x)) -> a(a(?x)) replace: a(c(?x)) -> a(b(?x)) => a(c(?x)) -> c(a(?x)) STEP: 3 (relative) S: [ a(c(?x)) -> a(b(?x)), b(c(?x)) -> a(c(?x)), a(b(?x)) -> b(a(?x)) ] P: [ a(c(?x)) -> a(a(?x)), b(c(?x)) -> c(b(?x)), c(b(?x)) -> b(c(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> a(b(?x)) ] Check relative termination: [ a(c(?x)) -> a(b(?x)), b(c(?x)) -> a(c(?x)), a(b(?x)) -> b(a(?x)) ] [ a(c(?x)) -> a(a(?x)), b(c(?x)) -> c(b(?x)), c(b(?x)) -> b(c(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> a(b(?x)) ] Polynomial Interpretation: a:= (2)*x1 b:= (1)+(2)*x1 c:= (2)*x1 retract b(c(?x)) -> a(c(?x)) retract a(b(?x)) -> b(a(?x)) retract c(b(?x)) -> b(c(?x)) retract a(b(?x)) -> c(a(?x)) Polynomial Interpretation: a:= (1)+(2)*x1*x1 b:= (1)*x1 c:= (1)+(2)*x1*x1 relatively terminating S/P: relatively terminating check CP condition: failed failure(Step 3) STEP: 4 (parallel) S: [ a(c(?x)) -> a(b(?x)), b(c(?x)) -> a(c(?x)), a(b(?x)) -> b(a(?x)), a(a(?x)) -> b(a(?x)), c(b(?x)) -> b(a(?x)), c(a(?x)) -> b(a(?x)) ] P: [ a(c(?x)) -> a(a(?x)), b(c(?x)) -> c(b(?x)), c(b(?x)) -> b(c(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> a(b(?x)) ] S: terminating CP(S,S): --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes PCP_in(symP,S): --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes CP(S,symP): --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes --> => yes S: [ a(c(?x)) -> a(b(?x)), b(c(?x)) -> a(c(?x)), a(b(?x)) -> b(a(?x)), a(a(?x)) -> b(a(?x)), c(b(?x)) -> b(a(?x)), c(a(?x)) -> b(a(?x)) ] P: [ a(c(?x)) -> a(a(?x)), b(c(?x)) -> c(b(?x)), c(b(?x)) -> b(c(?x)), a(b(?x)) -> c(a(?x)), a(a(?x)) -> a(c(?x)), c(a(?x)) -> a(b(?x)) ] Success Reduction-Preserving Completion Direct Methods: CR Combined result: CR /tmp/file7SDjK0.trs: Success(CR) (2868 msec.)