(ignored inputs)COMMENT submitted by: Johannes Waldmann Rewrite Rules: [ b(a(?x)) -> b(a(?x)), c(c(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> c(a(?x)), c(b(?x)) -> c(b(?x)), c(a(?x)) -> a(c(?x)), a(a(?x)) -> a(a(?x)), a(a(?x)) -> b(b(?x)), b(b(?x)) -> a(a(?x)) ] Apply Direct Methods... Inner CPs: [ b(a(a(?x_6))) = b(a(a(?x_6))), b(b(b(?x_7))) = b(a(a(?x_7))), c(c(a(?x_2))) = a(a(a(?x_2))), c(c(b(?x_4))) = a(a(b(?x_4))), c(a(c(?x_5))) = a(a(a(?x_5))), c(a(a(?x_6))) = c(a(a(?x_6))), c(b(b(?x_7))) = c(a(a(?x_7))), b(a(a(?x_6))) = c(a(a(?x_6))), b(b(b(?x_7))) = c(a(a(?x_7))), c(b(a(?x))) = c(b(a(?x))), c(c(a(?x_3))) = c(b(a(?x_3))), c(a(a(?x_8))) = c(b(b(?x_8))), c(a(a(?x_6))) = a(c(a(?x_6))), c(b(b(?x_7))) = a(c(a(?x_7))), a(b(b(?x_7))) = a(a(a(?x_7))), a(a(a(?x_6))) = b(b(a(?x_6))), b(b(a(?x))) = a(a(a(?x))), b(c(a(?x_3))) = a(a(a(?x_3))), c(a(a(?x))) = a(a(c(?x))), a(a(a(?x))) = a(a(a(?x))), a(b(b(?x))) = b(b(a(?x))), b(a(a(?x))) = a(a(b(?x))) ] Outer CPs: [ b(a(?x)) = c(a(?x)), c(a(?x_2)) = a(c(?x_2)), a(a(?x_6)) = b(b(?x_6)) ] 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(a(?x)) = b(a(?x)), b(a(a(?x_7))) = b(a(a(?x_7))), b(b(b(?x_8))) = b(a(a(?x_8))), c(b(a(a(?x_7)))) = c(b(a(a(?x_7)))), c(b(b(b(?x_8)))) = c(b(a(a(?x_8)))), a(a(a(a(?x_7)))) = b(b(a(a(?x_7)))), a(a(b(b(?x_8)))) = b(b(a(a(?x_8)))), c(b(a(?x))) = c(b(a(?x))), a(a(a(?x))) = b(b(a(?x))), c(a(a(?x_1))) = a(a(c(?x_1))), c(c(a(?x_3))) = a(a(a(?x_3))), c(c(b(?x_5))) = a(a(b(?x_5))), c(a(c(?x_6))) = a(a(a(?x_6))), a(a(a(a(?x_1)))) = c(a(a(c(?x_1)))), a(a(c(a(?x_3)))) = c(a(a(a(?x_3)))), a(a(c(b(?x_5)))) = c(a(a(b(?x_5)))), a(a(a(c(?x_6)))) = c(a(a(a(?x_6)))), a(a(c(?x))) = c(a(a(?x))), a(c(?x)) = c(a(?x)), c(a(a(?x_7))) = c(a(a(?x_7))), c(b(b(?x_8))) = c(a(a(?x_8))), a(a(a(a(?x_7)))) = c(c(a(a(?x_7)))), a(a(b(b(?x_8)))) = c(c(a(a(?x_8)))), a(a(a(?x))) = c(c(a(?x))), b(a(?x)) = c(a(?x)), b(a(a(?x_7))) = c(a(a(?x_7))), b(b(b(?x_8))) = c(a(a(?x_8))), c(b(a(a(?x_7)))) = c(c(a(a(?x_7)))), c(b(b(b(?x_8)))) = c(c(a(a(?x_8)))), a(a(a(a(?x_7)))) = b(c(a(a(?x_7)))), a(a(b(b(?x_8)))) = b(c(a(a(?x_8)))), c(b(a(?x))) = c(c(a(?x))), a(a(a(?x))) = b(c(a(?x))), c(c(a(?x_5))) = c(b(a(?x_5))), c(a(a(?x_9))) = c(b(b(?x_9))), a(a(b(a(?x_2)))) = c(c(b(a(?x_2)))), a(a(c(a(?x_5)))) = c(c(b(a(?x_5)))), a(a(a(a(?x_9)))) = c(c(b(b(?x_9)))), a(a(b(?x))) = c(c(b(?x))), c(a(?x)) = a(c(?x)), c(a(a(?x_7))) = a(c(a(?x_7))), c(b(b(?x_8))) = a(c(a(?x_8))), a(a(a(a(?x_7)))) = c(a(c(a(?x_7)))), a(a(b(b(?x_8)))) = c(a(c(a(?x_8)))), a(a(a(?x))) = c(a(c(?x))), b(b(?x)) = a(a(?x)), a(a(a(?x_1))) = a(a(a(?x_1))), a(b(b(?x_8))) = a(a(a(?x_8))), a(a(a(a(?x_1)))) = a(a(a(a(?x_1)))), a(a(b(b(?x_8)))) = a(a(a(a(?x_8)))), b(a(a(a(?x_1)))) = b(a(a(a(?x_1)))), b(a(b(b(?x_8)))) = b(a(a(a(?x_8)))), c(a(a(a(?x_1)))) = c(a(a(a(?x_1)))), c(a(b(b(?x_8)))) = c(a(a(a(?x_8)))), c(a(a(a(?x_1)))) = b(a(a(a(?x_1)))), c(a(b(b(?x_8)))) = b(a(a(a(?x_8)))), a(c(a(a(?x_1)))) = c(a(a(a(?x_1)))), a(c(b(b(?x_8)))) = c(a(a(a(?x_8)))), b(b(a(a(?x_1)))) = a(a(a(a(?x_1)))), b(b(b(b(?x_8)))) = a(a(a(a(?x_8)))), c(a(a(?x))) = b(a(a(?x))), a(c(a(?x))) = c(a(a(?x))), b(b(a(?x))) = a(a(a(?x))), a(a(?x)) = b(b(?x)), a(b(b(?x_1))) = b(b(a(?x_1))), b(b(b(b(?x_1)))) = a(b(b(a(?x_1)))), b(b(a(a(?x_8)))) = a(b(b(a(?x_8)))), b(a(b(b(?x_1)))) = b(b(b(a(?x_1)))), b(a(a(a(?x_8)))) = b(b(b(a(?x_8)))), c(a(b(b(?x_1)))) = c(b(b(a(?x_1)))), c(a(a(a(?x_8)))) = c(b(b(a(?x_8)))), c(a(b(b(?x_1)))) = b(b(b(a(?x_1)))), c(a(a(a(?x_8)))) = b(b(b(a(?x_8)))), a(c(b(b(?x_1)))) = c(b(b(a(?x_1)))), a(c(a(a(?x_8)))) = c(b(b(a(?x_8)))), a(a(b(b(?x_1)))) = a(b(b(a(?x_1)))), a(a(a(a(?x_8)))) = a(b(b(a(?x_8)))), b(b(a(?x))) = a(b(b(?x))), b(a(a(?x))) = b(b(b(?x))), c(a(a(?x))) = b(b(b(?x))), a(c(a(?x))) = c(b(b(?x))), a(a(a(?x))) = a(b(b(?x))), b(a(a(?x_1))) = a(a(b(?x_1))), b(c(a(?x_5))) = a(a(a(?x_5))), a(a(a(a(?x_1)))) = b(a(a(b(?x_1)))), a(a(b(a(?x_2)))) = b(a(a(a(?x_2)))), a(a(c(a(?x_5)))) = b(a(a(a(?x_5)))), c(b(a(a(?x_1)))) = c(a(a(b(?x_1)))), c(b(b(a(?x_2)))) = c(a(a(a(?x_2)))), c(b(c(a(?x_5)))) = c(a(a(a(?x_5)))), a(a(b(?x))) = b(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 <6, 0> preceded by [(b,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <7, 0> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],8)], []> joinable by a reduction of rules <[], [([(b,1)],7)]> Critical Pair by Rules <2, 1> preceded by [(c,1)] joinable by a reduction of rules <[([],1)], []> Critical Pair by Rules <4, 1> preceded by [(c,1)] joinable by a reduction of rules <[([],1)], []> Critical Pair by Rules <5, 1> preceded by [(c,1)] joinable by a reduction of rules <[([],5),([(a,1)],1)], []> Critical Pair by Rules <6, 2> preceded by [(c,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <7, 2> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],8)], []> joinable by a reduction of rules <[], [([(c,1)],7)]> Critical Pair by Rules <6, 3> preceded by [(b,1)] joinable by a reduction of rules <[([],3)], []> Critical Pair by Rules <7, 3> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],8),([],3)], []> Critical Pair by Rules <0, 4> preceded by [(c,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <3, 4> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],3)]> Critical Pair by Rules <8, 4> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],7)], []> joinable by a reduction of rules <[], [([(c,1)],8)]> Critical Pair by Rules <6, 5> preceded by [(c,1)] joinable by a reduction of rules <[([],5)], []> Critical Pair by Rules <7, 5> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],8),([],5)], []> Critical Pair by Rules <7, 6> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],8)], []> joinable by a reduction of rules <[], [([(a,1)],7)]> Critical Pair by Rules <6, 7> preceded by [(a,1)] joinable by a reduction of rules <[([],7)], []> joinable by a reduction of rules <[], [([],8)]> Critical Pair by Rules <0, 8> preceded by [(b,1)] joinable by a reduction of rules <[([],8)], []> joinable by a reduction of rules <[], [([],7)]> Critical Pair by Rules <3, 8> preceded by [(b,1)] joinable by a reduction of rules <[], [([],7),([(b,1)],3)]> Critical Pair by Rules <1, 1> preceded by [(c,1)] joinable by a reduction of rules <[([],5),([(a,1)],5)], []> Critical Pair by Rules <6, 6> preceded by [(a,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <7, 7> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],8)], [([],8)]> Critical Pair by Rules <8, 8> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],7)], [([],7)]> Critical Pair by Rules <3, 0> preceded by [] joinable by a reduction of rules <[], [([],3)]> Critical Pair by Rules <5, 2> preceded by [] joinable by a reduction of rules <[], [([],5)]> Critical Pair by Rules <7, 6> preceded by [] joinable by a reduction of rules <[([],8)], []> joinable by a reduction of rules <[], [([],7)]> unknown Diagram Decreasing check Non-Confluence... obtain 11 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: [ c(c(?x)) -> a(a(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> a(c(?x)) ] P: [ b(a(?x)) -> b(a(?x)), c(a(?x)) -> c(a(?x)), c(b(?x)) -> c(b(?x)), a(a(?x)) -> a(a(?x)), a(a(?x)) -> b(b(?x)), b(b(?x)) -> a(a(?x)) ] S: terminating CP(S,S): --> => yes --> => yes PCP_in(symP,S): --> => yes --> => yes --> => no --> => yes --> => no --> => yes CP(S,symP): --> => yes --> => yes --> => yes --> => yes check joinability condition: check modulo reachablity from a(a(c(?x_6))) to b(b(b(?x_6))): maybe not reachable check modulo reachablity from a(a(c(?x_6))) to c(b(b(?x_6))): maybe not reachable failed failure(Step 1) [ ] Added S-Rules: [ ] Added P-Rules: [ ] STEP: 2 (linear) S: [ c(c(?x)) -> a(a(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> a(c(?x)) ] P: [ b(a(?x)) -> b(a(?x)), c(a(?x)) -> c(a(?x)), c(b(?x)) -> c(b(?x)), a(a(?x)) -> a(a(?x)), a(a(?x)) -> b(b(?x)), b(b(?x)) -> a(a(?x)) ] S: terminating CP(S,S): --> => yes --> => yes CP_in(symP,S): --> => yes --> => yes --> => yes --> => yes --> => no --> => no CP(S,symP): --> => yes --> => yes --> => yes --> => yes check joinability condition: check modulo reachablity from a(a(c(?x))) to b(b(b(?x))): maybe not reachable check modulo reachablity from a(a(c(?x))) to c(b(b(?x))): maybe not reachable failed failure(Step 2) [ ] Added S-Rules: [ ] Added P-Rules: [ ] STEP: 3 (relative) S: [ c(c(?x)) -> a(a(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> a(c(?x)) ] P: [ b(a(?x)) -> b(a(?x)), c(a(?x)) -> c(a(?x)), c(b(?x)) -> c(b(?x)), a(a(?x)) -> a(a(?x)), a(a(?x)) -> b(b(?x)), b(b(?x)) -> a(a(?x)) ] Check relative termination: [ c(c(?x)) -> a(a(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> a(c(?x)) ] [ b(a(?x)) -> b(a(?x)), c(a(?x)) -> c(a(?x)), c(b(?x)) -> c(b(?x)), a(a(?x)) -> a(a(?x)), a(a(?x)) -> b(b(?x)), b(b(?x)) -> a(a(?x)) ] Polynomial Interpretation: a:= (1)*x1 b:= (1)+(1)*x1 c:= (1)*x1 retract b(a(?x)) -> c(a(?x)) retract b(b(?x)) -> a(a(?x)) Polynomial Interpretation: a:= (1)+(1)*x1 b:= (1)*x1 c:= (1)*x1+(4)*x1*x1 retract b(a(?x)) -> c(a(?x)) retract c(a(?x)) -> a(c(?x)) retract a(a(?x)) -> b(b(?x)) retract b(b(?x)) -> a(a(?x)) Polynomial Interpretation: a:= (1)+(2)*x1 b:= (1)*x1 c:= (2)+(2)*x1+(2)*x1*x1 relatively terminating S/P: relatively terminating check CP condition: failed failure(Step 3) failure(no possibility remains) unknown Reduction-Preserving Completion Direct Methods: Can't judge Try Persistent Decomposition for... [ b(a(?x)) -> b(a(?x)), c(c(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> c(a(?x)), c(b(?x)) -> c(b(?x)), c(a(?x)) -> a(c(?x)), a(a(?x)) -> a(a(?x)), a(a(?x)) -> b(b(?x)), b(b(?x)) -> a(a(?x)) ] Sort Assignment: a : 16=>16 b : 16=>16 c : 16=>16 maximal types: {16} Persistent Decomposition failed: Can't judge Try Layer Preserving Decomposition for... [ b(a(?x)) -> b(a(?x)), c(c(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> c(a(?x)), c(b(?x)) -> c(b(?x)), c(a(?x)) -> a(c(?x)), a(a(?x)) -> a(a(?x)), a(a(?x)) -> b(b(?x)), b(b(?x)) -> a(a(?x)) ] Layer Preserving Decomposition failed: Can't judge Try Commutative Decomposition for... [ b(a(?x)) -> b(a(?x)), c(c(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> c(a(?x)), c(b(?x)) -> c(b(?x)), c(a(?x)) -> a(c(?x)), a(a(?x)) -> a(a(?x)), a(a(?x)) -> b(b(?x)), b(b(?x)) -> a(a(?x)) ] Outside Critical Pair: by Rules <3, 0> develop reducts from lhs term... <{5}, a(c(?x_3))> <{2}, c(a(?x_3))> <{}, c(a(?x_3))> develop reducts from rhs term... <{3}, c(a(?x_3))> <{0}, b(a(?x_3))> <{}, b(a(?x_3))> Outside Critical Pair: by Rules <5, 2> develop reducts from lhs term... <{}, a(c(?x_5))> develop reducts from rhs term... <{5}, a(c(?x_5))> <{2}, c(a(?x_5))> <{}, c(a(?x_5))> Outside Critical Pair: by Rules <7, 6> develop reducts from lhs term... <{8}, a(a(?x_7))> <{}, b(b(?x_7))> develop reducts from rhs term... <{7}, b(b(?x_7))> <{6}, a(a(?x_7))> <{}, a(a(?x_7))> Inside Critical Pair: by Rules <6, 0> develop reducts from lhs term... <{3}, c(a(a(?x_6)))> <{0}, b(a(a(?x_6)))> <{7}, b(b(b(?x_6)))> <{6}, b(a(a(?x_6)))> <{}, b(a(a(?x_6)))> develop reducts from rhs term... <{3}, c(a(a(?x_6)))> <{0}, b(a(a(?x_6)))> <{7}, b(b(b(?x_6)))> <{6}, b(a(a(?x_6)))> <{}, b(a(a(?x_6)))> Inside Critical Pair: by Rules <7, 0> develop reducts from lhs term... <{8}, a(a(b(?x_7)))> <{8}, b(a(a(?x_7)))> <{}, b(b(b(?x_7)))> develop reducts from rhs term... <{3}, c(a(a(?x_7)))> <{0}, b(a(a(?x_7)))> <{7}, b(b(b(?x_7)))> <{6}, b(a(a(?x_7)))> <{}, b(a(a(?x_7)))> Inside Critical Pair: by Rules <2, 1> develop reducts from lhs term... <{1}, a(a(a(?x_2)))> <{5}, c(a(c(?x_2)))> <{2}, c(c(a(?x_2)))> <{}, c(c(a(?x_2)))> develop reducts from rhs term... <{7}, b(b(a(?x_2)))> <{7}, a(b(b(?x_2)))> <{6}, a(a(a(?x_2)))> <{}, a(a(a(?x_2)))> Inside Critical Pair: by Rules <4, 1> develop reducts from lhs term... <{1}, a(a(b(?x_4)))> <{4}, c(c(b(?x_4)))> <{}, c(c(b(?x_4)))> develop reducts from rhs term... <{7}, b(b(b(?x_4)))> <{6}, a(a(b(?x_4)))> <{}, a(a(b(?x_4)))> Inside Critical Pair: by Rules <5, 1> develop reducts from lhs term... <{5}, a(c(c(?x_5)))> <{2}, c(a(c(?x_5)))> <{}, c(a(c(?x_5)))> develop reducts from rhs term... <{7}, b(b(a(?x_5)))> <{7}, a(b(b(?x_5)))> <{6}, a(a(a(?x_5)))> <{}, a(a(a(?x_5)))> Inside Critical Pair: by Rules <6, 2> develop reducts from lhs term... <{5}, a(c(a(?x_6)))> <{2}, c(a(a(?x_6)))> <{7}, c(b(b(?x_6)))> <{6}, c(a(a(?x_6)))> <{}, c(a(a(?x_6)))> develop reducts from rhs term... <{5}, a(c(a(?x_6)))> <{2}, c(a(a(?x_6)))> <{7}, c(b(b(?x_6)))> <{6}, c(a(a(?x_6)))> <{}, c(a(a(?x_6)))> Inside Critical Pair: by Rules <7, 2> develop reducts from lhs term... <{4}, c(b(b(?x_7)))> <{8}, c(a(a(?x_7)))> <{}, c(b(b(?x_7)))> develop reducts from rhs term... <{5}, a(c(a(?x_7)))> <{2}, c(a(a(?x_7)))> <{7}, c(b(b(?x_7)))> <{6}, c(a(a(?x_7)))> <{}, c(a(a(?x_7)))> Inside Critical Pair: by Rules <6, 3> develop reducts from lhs term... <{3}, c(a(a(?x_6)))> <{0}, b(a(a(?x_6)))> <{7}, b(b(b(?x_6)))> <{6}, b(a(a(?x_6)))> <{}, b(a(a(?x_6)))> develop reducts from rhs term... <{5}, a(c(a(?x_6)))> <{2}, c(a(a(?x_6)))> <{7}, c(b(b(?x_6)))> <{6}, c(a(a(?x_6)))> <{}, c(a(a(?x_6)))> Inside Critical Pair: by Rules <7, 3> develop reducts from lhs term... <{8}, a(a(b(?x_7)))> <{8}, b(a(a(?x_7)))> <{}, b(b(b(?x_7)))> develop reducts from rhs term... <{5}, a(c(a(?x_7)))> <{2}, c(a(a(?x_7)))> <{7}, c(b(b(?x_7)))> <{6}, c(a(a(?x_7)))> <{}, c(a(a(?x_7)))> Inside Critical Pair: by Rules <0, 4> develop reducts from lhs term... <{4}, c(b(a(?x)))> <{3}, c(c(a(?x)))> <{0}, c(b(a(?x)))> <{}, c(b(a(?x)))> develop reducts from rhs term... <{4}, c(b(a(?x)))> <{3}, c(c(a(?x)))> <{0}, c(b(a(?x)))> <{}, c(b(a(?x)))> Inside Critical Pair: by Rules <3, 4> develop reducts from lhs term... <{1}, a(a(a(?x_3)))> <{5}, c(a(c(?x_3)))> <{2}, c(c(a(?x_3)))> <{}, c(c(a(?x_3)))> develop reducts from rhs term... <{4}, c(b(a(?x_3)))> <{3}, c(c(a(?x_3)))> <{0}, c(b(a(?x_3)))> <{}, c(b(a(?x_3)))> Inside Critical Pair: by Rules <8, 4> develop reducts from lhs term... <{5}, a(c(a(?x_8)))> <{2}, c(a(a(?x_8)))> <{7}, c(b(b(?x_8)))> <{6}, c(a(a(?x_8)))> <{}, c(a(a(?x_8)))> develop reducts from rhs term... <{4}, c(b(b(?x_8)))> <{8}, c(a(a(?x_8)))> <{}, c(b(b(?x_8)))> Inside Critical Pair: by Rules <6, 5> develop reducts from lhs term... <{5}, a(c(a(?x_6)))> <{2}, c(a(a(?x_6)))> <{7}, c(b(b(?x_6)))> <{6}, c(a(a(?x_6)))> <{}, c(a(a(?x_6)))> develop reducts from rhs term... <{5}, a(a(c(?x_6)))> <{2}, a(c(a(?x_6)))> <{}, a(c(a(?x_6)))> Inside Critical Pair: by Rules <7, 5> develop reducts from lhs term... <{4}, c(b(b(?x_7)))> <{8}, c(a(a(?x_7)))> <{}, c(b(b(?x_7)))> develop reducts from rhs term... <{5}, a(a(c(?x_7)))> <{2}, a(c(a(?x_7)))> <{}, a(c(a(?x_7)))> Inside Critical Pair: by Rules <7, 6> develop reducts from lhs term... <{8}, a(a(a(?x_7)))> <{}, a(b(b(?x_7)))> develop reducts from rhs term... <{7}, b(b(a(?x_7)))> <{7}, a(b(b(?x_7)))> <{6}, a(a(a(?x_7)))> <{}, a(a(a(?x_7)))> Inside Critical Pair: by Rules <6, 7> develop reducts from lhs term... <{7}, b(b(a(?x_6)))> <{7}, a(b(b(?x_6)))> <{6}, a(a(a(?x_6)))> <{}, a(a(a(?x_6)))> develop reducts from rhs term... <{8}, a(a(a(?x_6)))> <{3}, b(c(a(?x_6)))> <{0}, b(b(a(?x_6)))> <{}, b(b(a(?x_6)))> Inside Critical Pair: by Rules <0, 8> develop reducts from lhs term... <{8}, a(a(a(?x)))> <{3}, b(c(a(?x)))> <{0}, b(b(a(?x)))> <{}, b(b(a(?x)))> develop reducts from rhs term... <{7}, b(b(a(?x)))> <{7}, a(b(b(?x)))> <{6}, a(a(a(?x)))> <{}, a(a(a(?x)))> Inside Critical Pair: by Rules <3, 8> develop reducts from lhs term... <{5}, b(a(c(?x_3)))> <{2}, b(c(a(?x_3)))> <{}, b(c(a(?x_3)))> develop reducts from rhs term... <{7}, b(b(a(?x_3)))> <{7}, a(b(b(?x_3)))> <{6}, a(a(a(?x_3)))> <{}, a(a(a(?x_3)))> Try A Minimal Decomposition {1,5,8,4,3,2,0,7}{6} {1,5,8,4,3,2,0,7} (cm)Rewrite Rules: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(b(?x)) -> a(a(?x)), c(b(?x)) -> c(b(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Apply Direct Methods... Inner CPs: [ c(a(c(?x_1))) = a(a(a(?x_1))), c(c(b(?x_3))) = a(a(b(?x_3))), c(c(a(?x_5))) = a(a(a(?x_5))), c(b(b(?x_7))) = a(c(a(?x_7))), b(c(a(?x_4))) = a(a(a(?x_4))), b(b(a(?x_6))) = a(a(a(?x_6))), c(a(a(?x_2))) = c(b(b(?x_2))), c(c(a(?x_4))) = c(b(a(?x_4))), c(b(a(?x_6))) = c(b(a(?x_6))), b(b(b(?x_7))) = c(a(a(?x_7))), c(b(b(?x_7))) = c(a(a(?x_7))), b(b(b(?x_7))) = b(a(a(?x_7))), c(a(a(?x))) = a(a(c(?x))), b(a(a(?x))) = a(a(b(?x))), a(b(b(?x))) = b(b(a(?x))) ] Outer CPs: [ a(c(?x_1)) = c(a(?x_1)), c(a(?x_4)) = b(a(?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: [ c(a(a(?x_1))) = a(a(c(?x_1))), c(a(c(?x_2))) = a(a(a(?x_2))), c(c(b(?x_4))) = a(a(b(?x_4))), c(c(a(?x_6))) = a(a(a(?x_6))), a(a(a(a(?x_1)))) = c(a(a(c(?x_1)))), a(a(a(c(?x_2)))) = c(a(a(a(?x_2)))), a(a(c(b(?x_4)))) = c(a(a(b(?x_4)))), a(a(c(a(?x_6)))) = c(a(a(a(?x_6)))), a(a(c(?x))) = c(a(a(?x))), c(a(?x)) = a(c(?x)), c(b(b(?x_8))) = a(c(a(?x_8))), a(a(b(b(?x_8)))) = c(a(c(a(?x_8)))), a(a(a(?x))) = c(a(c(?x))), b(a(a(?x_1))) = a(a(b(?x_1))), b(c(a(?x_5))) = a(a(a(?x_5))), b(b(a(?x_7))) = a(a(a(?x_7))), a(a(a(a(?x_1)))) = b(a(a(b(?x_1)))), a(a(c(a(?x_5)))) = b(a(a(a(?x_5)))), a(a(b(a(?x_7)))) = b(a(a(a(?x_7)))), c(b(a(a(?x_1)))) = c(a(a(b(?x_1)))), c(b(c(a(?x_5)))) = c(a(a(a(?x_5)))), c(b(b(a(?x_7)))) = c(a(a(a(?x_7)))), a(a(b(?x))) = b(a(a(?x))), c(b(b(?x))) = c(a(a(?x))), c(a(a(?x_4))) = c(b(b(?x_4))), c(c(a(?x_5))) = c(b(a(?x_5))), c(b(a(?x_7))) = c(b(a(?x_7))), a(a(a(a(?x_4)))) = c(c(b(b(?x_4)))), a(a(c(a(?x_5)))) = c(c(b(a(?x_5)))), a(a(b(a(?x_7)))) = c(c(b(a(?x_7)))), a(a(b(?x))) = c(c(b(?x))), b(a(?x)) = c(a(?x)), b(b(b(?x_8))) = c(a(a(?x_8))), a(a(b(b(?x_8)))) = b(c(a(a(?x_8)))), c(b(b(b(?x_8)))) = c(c(a(a(?x_8)))), a(a(a(?x))) = b(c(a(?x))), c(b(a(?x))) = c(c(a(?x))), a(c(?x)) = c(a(?x)), a(a(b(b(?x_8)))) = c(c(a(a(?x_8)))), a(a(a(?x))) = c(c(a(?x))), c(a(?x)) = b(a(?x)), b(b(b(?x_8))) = b(a(a(?x_8))), a(a(b(b(?x_8)))) = b(b(a(a(?x_8)))), c(b(b(b(?x_8)))) = c(b(a(a(?x_8)))), a(a(a(?x))) = b(b(a(?x))), a(b(b(?x_1))) = b(b(a(?x_1))), b(b(b(b(?x_1)))) = a(b(b(a(?x_1)))), a(c(b(b(?x_1)))) = c(b(b(a(?x_1)))), c(a(b(b(?x_1)))) = b(b(b(a(?x_1)))), c(a(b(b(?x_1)))) = c(b(b(a(?x_1)))), b(a(b(b(?x_1)))) = b(b(b(a(?x_1)))), b(b(a(?x))) = a(b(b(?x))), a(c(a(?x))) = c(b(b(?x))), c(a(a(?x))) = b(b(b(?x))), b(a(a(?x))) = b(b(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 <1, 0> preceded by [(c,1)] joinable by a reduction of rules <[([],1),([(a,1)],0)], []> Critical Pair by Rules <3, 0> preceded by [(c,1)] joinable by a reduction of rules <[([],0)], []> Critical Pair by Rules <5, 0> preceded by [(c,1)] joinable by a reduction of rules <[([],0)], []> Critical Pair by Rules <7, 1> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],2),([],1)], []> Critical Pair by Rules <4, 2> preceded by [(b,1)] joinable by a reduction of rules <[], [([],7),([(b,1)],4)]> Critical Pair by Rules <6, 2> preceded by [(b,1)] joinable by a reduction of rules <[([],2)], []> joinable by a reduction of rules <[], [([],7)]> Critical Pair by Rules <2, 3> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],7)], []> joinable by a reduction of rules <[], [([(c,1)],2)]> Critical Pair by Rules <4, 3> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],4)]> Critical Pair by Rules <6, 3> preceded by [(c,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <7, 4> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],2),([],4)], []> Critical Pair by Rules <7, 5> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],2)], []> joinable by a reduction of rules <[], [([(c,1)],7)]> Critical Pair by Rules <7, 6> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],2)], []> joinable by a reduction of rules <[], [([(b,1)],7)]> Critical Pair by Rules <0, 0> preceded by [(c,1)] joinable by a reduction of rules <[([],1),([(a,1)],1)], []> Critical Pair by Rules <2, 2> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],7)], [([],7)]> Critical Pair by Rules <7, 7> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],2)], [([],2)]> Critical Pair by Rules <5, 1> preceded by [] joinable by a reduction of rules <[([],1)], []> Critical Pair by Rules <6, 4> preceded by [] joinable by a reduction of rules <[([],4)], []> unknown Diagram Decreasing check Non-Confluence... obtain 11 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: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] P: [ b(b(?x)) -> a(a(?x)), c(b(?x)) -> c(b(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] S: terminating CP(S,S): --> => yes --> => yes PCP_in(symP,S): --> => yes --> => yes --> => no --> => no CP(S,symP): --> => yes --> => yes --> => yes --> => yes check joinability condition: check modulo reachablity from a(a(c(?x_1))) to c(b(b(?x_1))): maybe not reachable check modulo reachablity from a(a(c(?x_1))) to b(b(b(?x_1))): maybe not reachable failed failure(Step 1) [ ] Added S-Rules: [ ] Added P-Rules: [ ] STEP: 2 (linear) S: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] P: [ b(b(?x)) -> a(a(?x)), c(b(?x)) -> c(b(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] S: terminating CP(S,S): --> => yes --> => yes CP_in(symP,S): --> => no --> => no --> => yes --> => yes CP(S,symP): --> => yes --> => yes --> => yes --> => yes check joinability condition: check modulo reachablity from a(a(c(?x))) to c(b(b(?x))): maybe not reachable check modulo reachablity from a(a(c(?x))) to b(b(b(?x))): maybe not reachable failed failure(Step 2) [ ] Added S-Rules: [ ] Added P-Rules: [ ] STEP: 3 (relative) S: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] P: [ b(b(?x)) -> a(a(?x)), c(b(?x)) -> c(b(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Check relative termination: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] [ b(b(?x)) -> a(a(?x)), c(b(?x)) -> c(b(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Polynomial Interpretation: a:= (2)*x1 b:= (1)+(2)*x1 c:= (2)*x1 retract b(a(?x)) -> c(a(?x)) retract b(b(?x)) -> a(a(?x)) Polynomial Interpretation: a:= (1)+(1)*x1 b:= (1)*x1 c:= (5)*x1+(3)*x1*x1 retract c(a(?x)) -> a(c(?x)) retract b(a(?x)) -> c(a(?x)) retract b(b(?x)) -> a(a(?x)) retract a(a(?x)) -> b(b(?x)) Polynomial Interpretation: a:= (2)*x1*x1 b:= (1)*x1 c:= (1)+(2)*x1*x1 relatively terminating S/P: relatively terminating check CP condition: failed failure(Step 3) failure(no possibility remains) unknown Reduction-Preserving Completion Direct Methods: Can't judge Try Persistent Decomposition for... [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(b(?x)) -> a(a(?x)), c(b(?x)) -> c(b(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Sort Assignment: a : 15=>15 b : 15=>15 c : 15=>15 maximal types: {15} Persistent Decomposition failed: Can't judge Try Layer Preserving Decomposition for... [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(b(?x)) -> a(a(?x)), c(b(?x)) -> c(b(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Layer Preserving Decomposition failed: Can't judge No further decomposition possible {6} (cm)Rewrite Rules: [ a(a(?x)) -> a(a(?x)) ] Apply Direct Methods... Inner CPs: [ a(a(a(?x))) = a(a(a(?x))) ] Outer CPs: [ ] not Overlay, check Termination... unknown/not Terminating unknown Knuth & Bendix Linear Development Closed Direct Methods: CR Try A Minimal Decomposition {8,6,1,5,3,2,0,7}{4} {8,6,1,5,3,2,0,7} (cm)Rewrite Rules: [ b(b(?x)) -> a(a(?x)), a(a(?x)) -> a(a(?x)), c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Apply Direct Methods... Inner CPs: [ b(c(a(?x_4))) = a(a(a(?x_4))), b(b(a(?x_6))) = a(a(a(?x_6))), a(b(b(?x_7))) = a(a(a(?x_7))), c(a(c(?x_3))) = a(a(a(?x_3))), c(c(a(?x_5))) = a(a(a(?x_5))), c(a(a(?x_1))) = a(c(a(?x_1))), c(b(b(?x_7))) = a(c(a(?x_7))), b(a(a(?x_1))) = c(a(a(?x_1))), b(b(b(?x_7))) = c(a(a(?x_7))), c(a(a(?x_1))) = c(a(a(?x_1))), c(b(b(?x_7))) = c(a(a(?x_7))), b(a(a(?x_1))) = b(a(a(?x_1))), b(b(b(?x_7))) = b(a(a(?x_7))), a(a(a(?x_1))) = b(b(a(?x_1))), b(a(a(?x))) = a(a(b(?x))), a(a(a(?x))) = a(a(a(?x))), c(a(a(?x))) = a(a(c(?x))), a(b(b(?x))) = b(b(a(?x))) ] Outer CPs: [ a(a(?x_1)) = b(b(?x_1)), a(c(?x_3)) = c(a(?x_3)), c(a(?x_4)) = b(a(?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(a(?x_1))) = a(a(b(?x_1))), b(c(a(?x_5))) = a(a(a(?x_5))), b(b(a(?x_7))) = a(a(a(?x_7))), a(a(a(a(?x_1)))) = b(a(a(b(?x_1)))), a(a(c(a(?x_5)))) = b(a(a(a(?x_5)))), a(a(b(a(?x_7)))) = b(a(a(a(?x_7)))), a(a(b(?x))) = b(a(a(?x))), b(b(?x)) = a(a(?x)), a(a(a(?x_1))) = a(a(a(?x_1))), a(b(b(?x_8))) = a(a(a(?x_8))), a(a(a(a(?x_1)))) = a(a(a(a(?x_1)))), a(a(b(b(?x_8)))) = a(a(a(a(?x_8)))), a(c(a(a(?x_1)))) = c(a(a(a(?x_1)))), a(c(b(b(?x_8)))) = c(a(a(a(?x_8)))), c(a(a(a(?x_1)))) = b(a(a(a(?x_1)))), c(a(b(b(?x_8)))) = b(a(a(a(?x_8)))), c(a(a(a(?x_1)))) = c(a(a(a(?x_1)))), c(a(b(b(?x_8)))) = c(a(a(a(?x_8)))), b(a(a(a(?x_1)))) = b(a(a(a(?x_1)))), b(a(b(b(?x_8)))) = b(a(a(a(?x_8)))), b(b(a(a(?x_1)))) = a(a(a(a(?x_1)))), b(b(b(b(?x_8)))) = a(a(a(a(?x_8)))), a(c(a(?x))) = c(a(a(?x))), c(a(a(?x))) = b(a(a(?x))), c(a(a(?x))) = c(a(a(?x))), b(a(a(?x))) = b(a(a(?x))), c(a(a(?x_1))) = a(a(c(?x_1))), c(a(c(?x_4))) = a(a(a(?x_4))), c(c(a(?x_6))) = a(a(a(?x_6))), a(a(a(a(?x_1)))) = c(a(a(c(?x_1)))), a(a(a(c(?x_4)))) = c(a(a(a(?x_4)))), a(a(c(a(?x_6)))) = c(a(a(a(?x_6)))), a(a(c(?x))) = c(a(a(?x))), c(a(?x)) = a(c(?x)), c(a(a(?x_3))) = a(c(a(?x_3))), c(b(b(?x_8))) = a(c(a(?x_8))), a(a(a(a(?x_3)))) = c(a(c(a(?x_3)))), a(a(b(b(?x_8)))) = c(a(c(a(?x_8)))), a(a(a(?x))) = c(a(c(?x))), b(a(?x)) = c(a(?x)), b(a(a(?x_3))) = c(a(a(?x_3))), b(b(b(?x_8))) = c(a(a(?x_8))), a(a(a(a(?x_3)))) = b(c(a(a(?x_3)))), a(a(b(b(?x_8)))) = b(c(a(a(?x_8)))), a(a(a(?x))) = b(c(a(?x))), a(c(?x)) = c(a(?x)), c(b(b(?x_8))) = c(a(a(?x_8))), a(a(a(a(?x_3)))) = c(c(a(a(?x_3)))), a(a(b(b(?x_8)))) = c(c(a(a(?x_8)))), a(a(a(?x))) = c(c(a(?x))), c(a(?x)) = b(a(?x)), b(b(b(?x_8))) = b(a(a(?x_8))), a(a(a(a(?x_3)))) = b(b(a(a(?x_3)))), a(a(b(b(?x_8)))) = b(b(a(a(?x_8)))), a(a(a(?x))) = b(b(a(?x))), a(a(?x)) = b(b(?x)), a(b(b(?x_1))) = b(b(a(?x_1))), b(b(b(b(?x_1)))) = a(b(b(a(?x_1)))), b(b(a(a(?x_3)))) = a(b(b(a(?x_3)))), a(a(b(b(?x_1)))) = a(b(b(a(?x_1)))), a(a(a(a(?x_3)))) = a(b(b(a(?x_3)))), a(c(b(b(?x_1)))) = c(b(b(a(?x_1)))), a(c(a(a(?x_3)))) = c(b(b(a(?x_3)))), c(a(b(b(?x_1)))) = b(b(b(a(?x_1)))), c(a(a(a(?x_3)))) = b(b(b(a(?x_3)))), c(a(b(b(?x_1)))) = c(b(b(a(?x_1)))), c(a(a(a(?x_3)))) = c(b(b(a(?x_3)))), b(a(b(b(?x_1)))) = b(b(b(a(?x_1)))), b(a(a(a(?x_3)))) = b(b(b(a(?x_3)))), b(b(a(?x))) = a(b(b(?x))), a(a(a(?x))) = a(b(b(?x))), a(c(a(?x))) = c(b(b(?x))), c(a(a(?x))) = b(b(b(?x))), c(a(a(?x))) = c(b(b(?x))), b(a(a(?x))) = b(b(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 [(b,1)] joinable by a reduction of rules <[], [([],7),([(b,1)],4)]> Critical Pair by Rules <6, 0> preceded by [(b,1)] joinable by a reduction of rules <[([],0)], []> joinable by a reduction of rules <[], [([],7)]> Critical Pair by Rules <7, 1> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],0)], []> joinable by a reduction of rules <[], [([(a,1)],7)]> Critical Pair by Rules <3, 2> preceded by [(c,1)] joinable by a reduction of rules <[([],3),([(a,1)],2)], []> Critical Pair by Rules <5, 2> preceded by [(c,1)] joinable by a reduction of rules <[([],2)], []> Critical Pair by Rules <1, 3> preceded by [(c,1)] joinable by a reduction of rules <[([],3)], []> Critical Pair by Rules <7, 3> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],0),([],3)], []> Critical Pair by Rules <1, 4> preceded by [(b,1)] joinable by a reduction of rules <[([],4)], []> Critical Pair by Rules <7, 4> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],0),([],4)], []> Critical Pair by Rules <1, 5> preceded by [(c,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <7, 5> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],0)], []> joinable by a reduction of rules <[], [([(c,1)],7)]> Critical Pair by Rules <1, 6> preceded by [(b,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <7, 6> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],0)], []> joinable by a reduction of rules <[], [([(b,1)],7)]> Critical Pair by Rules <1, 7> preceded by [(a,1)] joinable by a reduction of rules <[([],7)], []> joinable by a reduction of rules <[], [([],0)]> Critical Pair by Rules <0, 0> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],7)], [([],7)]> Critical Pair by Rules <1, 1> preceded by [(a,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <2, 2> preceded by [(c,1)] joinable by a reduction of rules <[([],3),([(a,1)],3)], []> Critical Pair by Rules <7, 7> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],0)], [([],0)]> Critical Pair by Rules <7, 1> preceded by [] joinable by a reduction of rules <[([],0)], []> joinable by a reduction of rules <[], [([],7)]> Critical Pair by Rules <5, 3> preceded by [] joinable by a reduction of rules <[([],3)], []> Critical Pair by Rules <6, 4> preceded by [] joinable by a reduction of rules <[([],4)], []> unknown Diagram Decreasing check Non-Confluence... obtain 10 rules by 3 steps unfolding obtain 99 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: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] P: [ b(b(?x)) -> a(a(?x)), a(a(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] S: terminating CP(S,S): --> => yes --> => yes PCP_in(symP,S): --> => yes --> => yes --> => no --> => yes --> => no CP(S,symP): --> => yes --> => yes --> => yes check joinability condition: check modulo reachablity from a(a(c(?x_1))) to c(b(b(?x_1))): maybe not reachable check modulo reachablity from a(a(c(?x_1))) to b(b(b(?x_1))): maybe not reachable failed failure(Step 1) [ ] Added S-Rules: [ ] Added P-Rules: [ ] STEP: 2 (linear) S: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] P: [ b(b(?x)) -> a(a(?x)), a(a(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] S: terminating CP(S,S): --> => yes --> => yes CP_in(symP,S): --> => no --> => no --> => yes --> => yes --> => yes CP(S,symP): --> => yes --> => yes --> => yes check joinability condition: check modulo reachablity from a(a(c(?x))) to c(b(b(?x))): maybe not reachable check modulo reachablity from a(a(c(?x))) to b(b(b(?x))): maybe not reachable failed failure(Step 2) [ ] Added S-Rules: [ ] Added P-Rules: [ ] STEP: 3 (relative) S: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] P: [ b(b(?x)) -> a(a(?x)), a(a(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Check relative termination: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] [ b(b(?x)) -> a(a(?x)), a(a(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Polynomial Interpretation: a:= (2)*x1 b:= (1)+(2)*x1 c:= (2)*x1 retract b(a(?x)) -> c(a(?x)) retract b(b(?x)) -> a(a(?x)) Polynomial Interpretation: a:= (1)+(1)*x1 b:= (1)*x1 c:= (5)*x1+(3)*x1*x1 retract c(a(?x)) -> a(c(?x)) retract b(a(?x)) -> c(a(?x)) retract b(b(?x)) -> a(a(?x)) retract a(a(?x)) -> b(b(?x)) Polynomial Interpretation: a:= (2)*x1*x1 b:= (1)*x1 c:= (1)+(2)*x1*x1 relatively terminating S/P: relatively terminating check CP condition: failed failure(Step 3) failure(no possibility remains) unknown Reduction-Preserving Completion Direct Methods: Can't judge Try Persistent Decomposition for... [ b(b(?x)) -> a(a(?x)), a(a(?x)) -> a(a(?x)), c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Sort Assignment: a : 15=>15 b : 15=>15 c : 15=>15 maximal types: {15} Persistent Decomposition failed: Can't judge Try Layer Preserving Decomposition for... [ b(b(?x)) -> a(a(?x)), a(a(?x)) -> a(a(?x)), c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Layer Preserving Decomposition failed: Can't judge No further decomposition possible {4} (cm)Rewrite Rules: [ c(b(?x)) -> c(b(?x)) ] Apply Direct Methods... Inner CPs: [ ] Outer CPs: [ ] Overlay, check Innermost Termination... unknown Innermost Terminating unknown Knuth & Bendix Linear Development Closed Direct Methods: CR Try A Minimal Decomposition {8,1,5,3,2,0,7}{4}{6} {8,1,5,3,2,0,7} (cm)Rewrite Rules: [ b(b(?x)) -> a(a(?x)), c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Apply Direct Methods... Inner CPs: [ b(c(a(?x_3))) = a(a(a(?x_3))), b(b(a(?x_5))) = a(a(a(?x_5))), c(a(c(?x_2))) = a(a(a(?x_2))), c(c(a(?x_4))) = a(a(a(?x_4))), c(b(b(?x_6))) = a(c(a(?x_6))), b(b(b(?x_6))) = c(a(a(?x_6))), c(b(b(?x_6))) = c(a(a(?x_6))), b(b(b(?x_6))) = b(a(a(?x_6))), b(a(a(?x))) = a(a(b(?x))), c(a(a(?x))) = a(a(c(?x))), a(b(b(?x))) = b(b(a(?x))) ] Outer CPs: [ a(c(?x_2)) = c(a(?x_2)), c(a(?x_3)) = b(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: [ b(a(a(?x_1))) = a(a(b(?x_1))), b(c(a(?x_4))) = a(a(a(?x_4))), b(b(a(?x_6))) = a(a(a(?x_6))), a(a(a(a(?x_1)))) = b(a(a(b(?x_1)))), a(a(c(a(?x_4)))) = b(a(a(a(?x_4)))), a(a(b(a(?x_6)))) = b(a(a(a(?x_6)))), a(a(b(?x))) = b(a(a(?x))), c(a(a(?x_1))) = a(a(c(?x_1))), c(a(c(?x_3))) = a(a(a(?x_3))), c(c(a(?x_5))) = a(a(a(?x_5))), a(a(a(a(?x_1)))) = c(a(a(c(?x_1)))), a(a(a(c(?x_3)))) = c(a(a(a(?x_3)))), a(a(c(a(?x_5)))) = c(a(a(a(?x_5)))), a(a(c(?x))) = c(a(a(?x))), c(a(?x)) = a(c(?x)), c(b(b(?x_7))) = a(c(a(?x_7))), a(a(b(b(?x_7)))) = c(a(c(a(?x_7)))), a(a(a(?x))) = c(a(c(?x))), b(a(?x)) = c(a(?x)), b(b(b(?x_7))) = c(a(a(?x_7))), a(a(b(b(?x_7)))) = b(c(a(a(?x_7)))), a(a(a(?x))) = b(c(a(?x))), a(c(?x)) = c(a(?x)), c(b(b(?x_7))) = c(a(a(?x_7))), a(a(b(b(?x_7)))) = c(c(a(a(?x_7)))), a(a(a(?x))) = c(c(a(?x))), c(a(?x)) = b(a(?x)), b(b(b(?x_7))) = b(a(a(?x_7))), a(a(b(b(?x_7)))) = b(b(a(a(?x_7)))), a(a(a(?x))) = b(b(a(?x))), a(b(b(?x_1))) = b(b(a(?x_1))), b(b(b(b(?x_1)))) = a(b(b(a(?x_1)))), a(c(b(b(?x_1)))) = c(b(b(a(?x_1)))), c(a(b(b(?x_1)))) = b(b(b(a(?x_1)))), c(a(b(b(?x_1)))) = c(b(b(a(?x_1)))), b(a(b(b(?x_1)))) = b(b(b(a(?x_1)))), b(b(a(?x))) = a(b(b(?x))), a(c(a(?x))) = c(b(b(?x))), c(a(a(?x))) = b(b(b(?x))), c(a(a(?x))) = c(b(b(?x))), b(a(a(?x))) = b(b(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 <3, 0> preceded by [(b,1)] joinable by a reduction of rules <[], [([],6),([(b,1)],3)]> Critical Pair by Rules <5, 0> preceded by [(b,1)] joinable by a reduction of rules <[([],0)], []> joinable by a reduction of rules <[], [([],6)]> Critical Pair by Rules <2, 1> preceded by [(c,1)] joinable by a reduction of rules <[([],2),([(a,1)],1)], []> Critical Pair by Rules <4, 1> preceded by [(c,1)] joinable by a reduction of rules <[([],1)], []> Critical Pair by Rules <6, 2> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],0),([],2)], []> Critical Pair by Rules <6, 3> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],0),([],3)], []> Critical Pair by Rules <6, 4> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],0)], []> joinable by a reduction of rules <[], [([(c,1)],6)]> Critical Pair by Rules <6, 5> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],0)], []> joinable by a reduction of rules <[], [([(b,1)],6)]> Critical Pair by Rules <0, 0> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],6)], [([],6)]> Critical Pair by Rules <1, 1> preceded by [(c,1)] joinable by a reduction of rules <[([],2),([(a,1)],2)], []> Critical Pair by Rules <6, 6> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],0)], [([],0)]> Critical Pair by Rules <4, 2> preceded by [] joinable by a reduction of rules <[([],2)], []> Critical Pair by Rules <5, 3> preceded by [] joinable by a reduction of rules <[([],3)], []> unknown Diagram Decreasing check Non-Confluence... obtain 10 rules by 3 steps unfolding obtain 99 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: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] P: [ b(b(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] S: terminating CP(S,S): --> => yes --> => yes PCP_in(symP,S): --> => yes --> => no --> => no CP(S,symP): --> => yes --> => yes --> => yes check joinability condition: check modulo reachablity from a(a(c(?x_1))) to c(b(b(?x_1))): maybe not reachable check modulo reachablity from a(a(c(?x_1))) to b(b(b(?x_1))): maybe not reachable failed failure(Step 1) [ ] Added S-Rules: [ ] Added P-Rules: [ ] STEP: 2 (linear) S: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] P: [ b(b(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] S: terminating CP(S,S): --> => yes --> => yes CP_in(symP,S): --> => no --> => no --> => yes CP(S,symP): --> => yes --> => yes --> => yes check joinability condition: check modulo reachablity from a(a(c(?x))) to c(b(b(?x))): maybe not reachable check modulo reachablity from a(a(c(?x))) to b(b(b(?x))): maybe not reachable failed failure(Step 2) [ ] Added S-Rules: [ ] Added P-Rules: [ ] STEP: 3 (relative) S: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] P: [ b(b(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Check relative termination: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] [ b(b(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Polynomial Interpretation: a:= (2)*x1 b:= (1)+(2)*x1 c:= (2)*x1 retract b(a(?x)) -> c(a(?x)) retract b(b(?x)) -> a(a(?x)) Polynomial Interpretation: a:= (3)*x1 b:= (2)*x1 c:= (1)+(3)*x1 retract c(c(?x)) -> a(a(?x)) retract b(a(?x)) -> c(a(?x)) retract b(b(?x)) -> a(a(?x)) Polynomial Interpretation: a:= (1)+(1)*x1 b:= (2)+(1)*x1 c:= (2)+(1)*x1+(2)*x1*x1 relatively terminating S/P: relatively terminating check CP condition: failed failure(Step 3) failure(no possibility remains) unknown Reduction-Preserving Completion Direct Methods: Can't judge Try Persistent Decomposition for... [ b(b(?x)) -> a(a(?x)), c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Sort Assignment: a : 14=>14 b : 14=>14 c : 14=>14 maximal types: {14} Persistent Decomposition failed: Can't judge Try Layer Preserving Decomposition for... [ b(b(?x)) -> a(a(?x)), c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Layer Preserving Decomposition failed: Can't judge No further decomposition possible {4} (cm)Rewrite Rules: [ c(b(?x)) -> c(b(?x)) ] Apply Direct Methods... Inner CPs: [ ] Outer CPs: [ ] Overlay, check Innermost Termination... unknown Innermost Terminating unknown Knuth & Bendix Linear Development Closed Direct Methods: CR {6} (cm)Rewrite Rules: [ a(a(?x)) -> a(a(?x)) ] Apply Direct Methods... Inner CPs: [ a(a(a(?x))) = a(a(a(?x))) ] Outer CPs: [ ] not Overlay, check Termination... unknown/not Terminating unknown Knuth & Bendix Linear Development Closed Direct Methods: CR Try A Minimal Decomposition {1,5,8,4,3,2,0,7}{6} {1,5,8,4,3,2,0,7} (cm)Rewrite Rules: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(b(?x)) -> a(a(?x)), c(b(?x)) -> c(b(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Apply Direct Methods... Inner CPs: [ c(a(c(?x_1))) = a(a(a(?x_1))), c(c(b(?x_3))) = a(a(b(?x_3))), c(c(a(?x_5))) = a(a(a(?x_5))), c(b(b(?x_7))) = a(c(a(?x_7))), b(c(a(?x_4))) = a(a(a(?x_4))), b(b(a(?x_6))) = a(a(a(?x_6))), c(a(a(?x_2))) = c(b(b(?x_2))), c(c(a(?x_4))) = c(b(a(?x_4))), c(b(a(?x_6))) = c(b(a(?x_6))), b(b(b(?x_7))) = c(a(a(?x_7))), c(b(b(?x_7))) = c(a(a(?x_7))), b(b(b(?x_7))) = b(a(a(?x_7))), c(a(a(?x))) = a(a(c(?x))), b(a(a(?x))) = a(a(b(?x))), a(b(b(?x))) = b(b(a(?x))) ] Outer CPs: [ a(c(?x_1)) = c(a(?x_1)), c(a(?x_4)) = b(a(?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: [ c(a(a(?x_1))) = a(a(c(?x_1))), c(a(c(?x_2))) = a(a(a(?x_2))), c(c(b(?x_4))) = a(a(b(?x_4))), c(c(a(?x_6))) = a(a(a(?x_6))), a(a(a(a(?x_1)))) = c(a(a(c(?x_1)))), a(a(a(c(?x_2)))) = c(a(a(a(?x_2)))), a(a(c(b(?x_4)))) = c(a(a(b(?x_4)))), a(a(c(a(?x_6)))) = c(a(a(a(?x_6)))), a(a(c(?x))) = c(a(a(?x))), c(a(?x)) = a(c(?x)), c(b(b(?x_8))) = a(c(a(?x_8))), a(a(b(b(?x_8)))) = c(a(c(a(?x_8)))), a(a(a(?x))) = c(a(c(?x))), b(a(a(?x_1))) = a(a(b(?x_1))), b(c(a(?x_5))) = a(a(a(?x_5))), b(b(a(?x_7))) = a(a(a(?x_7))), a(a(a(a(?x_1)))) = b(a(a(b(?x_1)))), a(a(c(a(?x_5)))) = b(a(a(a(?x_5)))), a(a(b(a(?x_7)))) = b(a(a(a(?x_7)))), c(b(a(a(?x_1)))) = c(a(a(b(?x_1)))), c(b(c(a(?x_5)))) = c(a(a(a(?x_5)))), c(b(b(a(?x_7)))) = c(a(a(a(?x_7)))), a(a(b(?x))) = b(a(a(?x))), c(b(b(?x))) = c(a(a(?x))), c(a(a(?x_4))) = c(b(b(?x_4))), c(c(a(?x_5))) = c(b(a(?x_5))), c(b(a(?x_7))) = c(b(a(?x_7))), a(a(a(a(?x_4)))) = c(c(b(b(?x_4)))), a(a(c(a(?x_5)))) = c(c(b(a(?x_5)))), a(a(b(a(?x_7)))) = c(c(b(a(?x_7)))), a(a(b(?x))) = c(c(b(?x))), b(a(?x)) = c(a(?x)), b(b(b(?x_8))) = c(a(a(?x_8))), a(a(b(b(?x_8)))) = b(c(a(a(?x_8)))), c(b(b(b(?x_8)))) = c(c(a(a(?x_8)))), a(a(a(?x))) = b(c(a(?x))), c(b(a(?x))) = c(c(a(?x))), a(c(?x)) = c(a(?x)), a(a(b(b(?x_8)))) = c(c(a(a(?x_8)))), a(a(a(?x))) = c(c(a(?x))), c(a(?x)) = b(a(?x)), b(b(b(?x_8))) = b(a(a(?x_8))), a(a(b(b(?x_8)))) = b(b(a(a(?x_8)))), c(b(b(b(?x_8)))) = c(b(a(a(?x_8)))), a(a(a(?x))) = b(b(a(?x))), a(b(b(?x_1))) = b(b(a(?x_1))), b(b(b(b(?x_1)))) = a(b(b(a(?x_1)))), a(c(b(b(?x_1)))) = c(b(b(a(?x_1)))), c(a(b(b(?x_1)))) = b(b(b(a(?x_1)))), c(a(b(b(?x_1)))) = c(b(b(a(?x_1)))), b(a(b(b(?x_1)))) = b(b(b(a(?x_1)))), b(b(a(?x))) = a(b(b(?x))), a(c(a(?x))) = c(b(b(?x))), c(a(a(?x))) = b(b(b(?x))), b(a(a(?x))) = b(b(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 <1, 0> preceded by [(c,1)] joinable by a reduction of rules <[([],1),([(a,1)],0)], []> Critical Pair by Rules <3, 0> preceded by [(c,1)] joinable by a reduction of rules <[([],0)], []> Critical Pair by Rules <5, 0> preceded by [(c,1)] joinable by a reduction of rules <[([],0)], []> Critical Pair by Rules <7, 1> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],2),([],1)], []> Critical Pair by Rules <4, 2> preceded by [(b,1)] joinable by a reduction of rules <[], [([],7),([(b,1)],4)]> Critical Pair by Rules <6, 2> preceded by [(b,1)] joinable by a reduction of rules <[([],2)], []> joinable by a reduction of rules <[], [([],7)]> Critical Pair by Rules <2, 3> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],7)], []> joinable by a reduction of rules <[], [([(c,1)],2)]> Critical Pair by Rules <4, 3> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],4)]> Critical Pair by Rules <6, 3> preceded by [(c,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <7, 4> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],2),([],4)], []> Critical Pair by Rules <7, 5> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],2)], []> joinable by a reduction of rules <[], [([(c,1)],7)]> Critical Pair by Rules <7, 6> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],2)], []> joinable by a reduction of rules <[], [([(b,1)],7)]> Critical Pair by Rules <0, 0> preceded by [(c,1)] joinable by a reduction of rules <[([],1),([(a,1)],1)], []> Critical Pair by Rules <2, 2> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],7)], [([],7)]> Critical Pair by Rules <7, 7> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],2)], [([],2)]> Critical Pair by Rules <5, 1> preceded by [] joinable by a reduction of rules <[([],1)], []> Critical Pair by Rules <6, 4> preceded by [] joinable by a reduction of rules <[([],4)], []> unknown Diagram Decreasing check Non-Confluence... obtain 11 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: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] P: [ b(b(?x)) -> a(a(?x)), c(b(?x)) -> c(b(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] S: terminating CP(S,S): --> => yes --> => yes PCP_in(symP,S): --> => yes --> => yes --> => no --> => no CP(S,symP): --> => yes --> => yes --> => yes --> => yes check joinability condition: check modulo reachablity from a(a(c(?x_1))) to c(b(b(?x_1))): maybe not reachable check modulo reachablity from a(a(c(?x_1))) to b(b(b(?x_1))): maybe not reachable failed failure(Step 1) [ ] Added S-Rules: [ ] Added P-Rules: [ ] STEP: 2 (linear) S: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] P: [ b(b(?x)) -> a(a(?x)), c(b(?x)) -> c(b(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] S: terminating CP(S,S): --> => yes --> => yes CP_in(symP,S): --> => no --> => no --> => yes --> => yes CP(S,symP): --> => yes --> => yes --> => yes --> => yes check joinability condition: check modulo reachablity from a(a(c(?x))) to c(b(b(?x))): maybe not reachable check modulo reachablity from a(a(c(?x))) to b(b(b(?x))): maybe not reachable failed failure(Step 2) [ ] Added S-Rules: [ ] Added P-Rules: [ ] STEP: 3 (relative) S: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] P: [ b(b(?x)) -> a(a(?x)), c(b(?x)) -> c(b(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Check relative termination: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] [ b(b(?x)) -> a(a(?x)), c(b(?x)) -> c(b(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Polynomial Interpretation: a:= (2)*x1 b:= (1)+(2)*x1 c:= (2)*x1 retract b(a(?x)) -> c(a(?x)) retract b(b(?x)) -> a(a(?x)) Polynomial Interpretation: a:= (1)+(1)*x1 b:= (1)*x1 c:= (5)*x1+(3)*x1*x1 retract c(a(?x)) -> a(c(?x)) retract b(a(?x)) -> c(a(?x)) retract b(b(?x)) -> a(a(?x)) retract a(a(?x)) -> b(b(?x)) Polynomial Interpretation: a:= (2)*x1*x1 b:= (1)*x1 c:= (1)+(2)*x1*x1 relatively terminating S/P: relatively terminating check CP condition: failed failure(Step 3) failure(no possibility remains) unknown Reduction-Preserving Completion Direct Methods: Can't judge Try Persistent Decomposition for... [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(b(?x)) -> a(a(?x)), c(b(?x)) -> c(b(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Sort Assignment: a : 15=>15 b : 15=>15 c : 15=>15 maximal types: {15} Persistent Decomposition failed: Can't judge Try Layer Preserving Decomposition for... [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(b(?x)) -> a(a(?x)), c(b(?x)) -> c(b(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Layer Preserving Decomposition failed: Can't judge No further decomposition possible {6} (cm)Rewrite Rules: [ a(a(?x)) -> a(a(?x)) ] Apply Direct Methods... Inner CPs: [ a(a(a(?x))) = a(a(a(?x))) ] Outer CPs: [ ] not Overlay, check Termination... unknown/not Terminating unknown Knuth & Bendix Linear Development Closed Direct Methods: CR Try A Minimal Decomposition {8,6,1,5,3,2,0,7}{4} {8,6,1,5,3,2,0,7} (cm)Rewrite Rules: [ b(b(?x)) -> a(a(?x)), a(a(?x)) -> a(a(?x)), c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Apply Direct Methods... Inner CPs: [ b(c(a(?x_4))) = a(a(a(?x_4))), b(b(a(?x_6))) = a(a(a(?x_6))), a(b(b(?x_7))) = a(a(a(?x_7))), c(a(c(?x_3))) = a(a(a(?x_3))), c(c(a(?x_5))) = a(a(a(?x_5))), c(a(a(?x_1))) = a(c(a(?x_1))), c(b(b(?x_7))) = a(c(a(?x_7))), b(a(a(?x_1))) = c(a(a(?x_1))), b(b(b(?x_7))) = c(a(a(?x_7))), c(a(a(?x_1))) = c(a(a(?x_1))), c(b(b(?x_7))) = c(a(a(?x_7))), b(a(a(?x_1))) = b(a(a(?x_1))), b(b(b(?x_7))) = b(a(a(?x_7))), a(a(a(?x_1))) = b(b(a(?x_1))), b(a(a(?x))) = a(a(b(?x))), a(a(a(?x))) = a(a(a(?x))), c(a(a(?x))) = a(a(c(?x))), a(b(b(?x))) = b(b(a(?x))) ] Outer CPs: [ a(a(?x_1)) = b(b(?x_1)), a(c(?x_3)) = c(a(?x_3)), c(a(?x_4)) = b(a(?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(a(?x_1))) = a(a(b(?x_1))), b(c(a(?x_5))) = a(a(a(?x_5))), b(b(a(?x_7))) = a(a(a(?x_7))), a(a(a(a(?x_1)))) = b(a(a(b(?x_1)))), a(a(c(a(?x_5)))) = b(a(a(a(?x_5)))), a(a(b(a(?x_7)))) = b(a(a(a(?x_7)))), a(a(b(?x))) = b(a(a(?x))), b(b(?x)) = a(a(?x)), a(a(a(?x_1))) = a(a(a(?x_1))), a(b(b(?x_8))) = a(a(a(?x_8))), a(a(a(a(?x_1)))) = a(a(a(a(?x_1)))), a(a(b(b(?x_8)))) = a(a(a(a(?x_8)))), a(c(a(a(?x_1)))) = c(a(a(a(?x_1)))), a(c(b(b(?x_8)))) = c(a(a(a(?x_8)))), c(a(a(a(?x_1)))) = b(a(a(a(?x_1)))), c(a(b(b(?x_8)))) = b(a(a(a(?x_8)))), c(a(a(a(?x_1)))) = c(a(a(a(?x_1)))), c(a(b(b(?x_8)))) = c(a(a(a(?x_8)))), b(a(a(a(?x_1)))) = b(a(a(a(?x_1)))), b(a(b(b(?x_8)))) = b(a(a(a(?x_8)))), b(b(a(a(?x_1)))) = a(a(a(a(?x_1)))), b(b(b(b(?x_8)))) = a(a(a(a(?x_8)))), a(c(a(?x))) = c(a(a(?x))), c(a(a(?x))) = b(a(a(?x))), c(a(a(?x))) = c(a(a(?x))), b(a(a(?x))) = b(a(a(?x))), c(a(a(?x_1))) = a(a(c(?x_1))), c(a(c(?x_4))) = a(a(a(?x_4))), c(c(a(?x_6))) = a(a(a(?x_6))), a(a(a(a(?x_1)))) = c(a(a(c(?x_1)))), a(a(a(c(?x_4)))) = c(a(a(a(?x_4)))), a(a(c(a(?x_6)))) = c(a(a(a(?x_6)))), a(a(c(?x))) = c(a(a(?x))), c(a(?x)) = a(c(?x)), c(a(a(?x_3))) = a(c(a(?x_3))), c(b(b(?x_8))) = a(c(a(?x_8))), a(a(a(a(?x_3)))) = c(a(c(a(?x_3)))), a(a(b(b(?x_8)))) = c(a(c(a(?x_8)))), a(a(a(?x))) = c(a(c(?x))), b(a(?x)) = c(a(?x)), b(a(a(?x_3))) = c(a(a(?x_3))), b(b(b(?x_8))) = c(a(a(?x_8))), a(a(a(a(?x_3)))) = b(c(a(a(?x_3)))), a(a(b(b(?x_8)))) = b(c(a(a(?x_8)))), a(a(a(?x))) = b(c(a(?x))), a(c(?x)) = c(a(?x)), c(b(b(?x_8))) = c(a(a(?x_8))), a(a(a(a(?x_3)))) = c(c(a(a(?x_3)))), a(a(b(b(?x_8)))) = c(c(a(a(?x_8)))), a(a(a(?x))) = c(c(a(?x))), c(a(?x)) = b(a(?x)), b(b(b(?x_8))) = b(a(a(?x_8))), a(a(a(a(?x_3)))) = b(b(a(a(?x_3)))), a(a(b(b(?x_8)))) = b(b(a(a(?x_8)))), a(a(a(?x))) = b(b(a(?x))), a(a(?x)) = b(b(?x)), a(b(b(?x_1))) = b(b(a(?x_1))), b(b(b(b(?x_1)))) = a(b(b(a(?x_1)))), b(b(a(a(?x_3)))) = a(b(b(a(?x_3)))), a(a(b(b(?x_1)))) = a(b(b(a(?x_1)))), a(a(a(a(?x_3)))) = a(b(b(a(?x_3)))), a(c(b(b(?x_1)))) = c(b(b(a(?x_1)))), a(c(a(a(?x_3)))) = c(b(b(a(?x_3)))), c(a(b(b(?x_1)))) = b(b(b(a(?x_1)))), c(a(a(a(?x_3)))) = b(b(b(a(?x_3)))), c(a(b(b(?x_1)))) = c(b(b(a(?x_1)))), c(a(a(a(?x_3)))) = c(b(b(a(?x_3)))), b(a(b(b(?x_1)))) = b(b(b(a(?x_1)))), b(a(a(a(?x_3)))) = b(b(b(a(?x_3)))), b(b(a(?x))) = a(b(b(?x))), a(a(a(?x))) = a(b(b(?x))), a(c(a(?x))) = c(b(b(?x))), c(a(a(?x))) = b(b(b(?x))), c(a(a(?x))) = c(b(b(?x))), b(a(a(?x))) = b(b(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 [(b,1)] joinable by a reduction of rules <[], [([],7),([(b,1)],4)]> Critical Pair by Rules <6, 0> preceded by [(b,1)] joinable by a reduction of rules <[([],0)], []> joinable by a reduction of rules <[], [([],7)]> Critical Pair by Rules <7, 1> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],0)], []> joinable by a reduction of rules <[], [([(a,1)],7)]> Critical Pair by Rules <3, 2> preceded by [(c,1)] joinable by a reduction of rules <[([],3),([(a,1)],2)], []> Critical Pair by Rules <5, 2> preceded by [(c,1)] joinable by a reduction of rules <[([],2)], []> Critical Pair by Rules <1, 3> preceded by [(c,1)] joinable by a reduction of rules <[([],3)], []> Critical Pair by Rules <7, 3> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],0),([],3)], []> Critical Pair by Rules <1, 4> preceded by [(b,1)] joinable by a reduction of rules <[([],4)], []> Critical Pair by Rules <7, 4> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],0),([],4)], []> Critical Pair by Rules <1, 5> preceded by [(c,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <7, 5> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],0)], []> joinable by a reduction of rules <[], [([(c,1)],7)]> Critical Pair by Rules <1, 6> preceded by [(b,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <7, 6> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],0)], []> joinable by a reduction of rules <[], [([(b,1)],7)]> Critical Pair by Rules <1, 7> preceded by [(a,1)] joinable by a reduction of rules <[([],7)], []> joinable by a reduction of rules <[], [([],0)]> Critical Pair by Rules <0, 0> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],7)], [([],7)]> Critical Pair by Rules <1, 1> preceded by [(a,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <2, 2> preceded by [(c,1)] joinable by a reduction of rules <[([],3),([(a,1)],3)], []> Critical Pair by Rules <7, 7> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],0)], [([],0)]> Critical Pair by Rules <7, 1> preceded by [] joinable by a reduction of rules <[([],0)], []> joinable by a reduction of rules <[], [([],7)]> Critical Pair by Rules <5, 3> preceded by [] joinable by a reduction of rules <[([],3)], []> Critical Pair by Rules <6, 4> preceded by [] joinable by a reduction of rules <[([],4)], []> unknown Diagram Decreasing check Non-Confluence... obtain 10 rules by 3 steps unfolding obtain 99 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: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] P: [ b(b(?x)) -> a(a(?x)), a(a(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] S: terminating CP(S,S): --> => yes --> => yes PCP_in(symP,S): --> => yes --> => yes --> => no --> => yes --> => no CP(S,symP): --> => yes --> => yes --> => yes check joinability condition: check modulo reachablity from a(a(c(?x_1))) to c(b(b(?x_1))): maybe not reachable check modulo reachablity from a(a(c(?x_1))) to b(b(b(?x_1))): maybe not reachable failed failure(Step 1) [ ] Added S-Rules: [ ] Added P-Rules: [ ] STEP: 2 (linear) S: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] P: [ b(b(?x)) -> a(a(?x)), a(a(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] S: terminating CP(S,S): --> => yes --> => yes CP_in(symP,S): --> => no --> => no --> => yes --> => yes --> => yes CP(S,symP): --> => yes --> => yes --> => yes check joinability condition: check modulo reachablity from a(a(c(?x))) to c(b(b(?x))): maybe not reachable check modulo reachablity from a(a(c(?x))) to b(b(b(?x))): maybe not reachable failed failure(Step 2) [ ] Added S-Rules: [ ] Added P-Rules: [ ] STEP: 3 (relative) S: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] P: [ b(b(?x)) -> a(a(?x)), a(a(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Check relative termination: [ c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)) ] [ b(b(?x)) -> a(a(?x)), a(a(?x)) -> a(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Polynomial Interpretation: a:= (2)*x1 b:= (1)+(2)*x1 c:= (2)*x1 retract b(a(?x)) -> c(a(?x)) retract b(b(?x)) -> a(a(?x)) Polynomial Interpretation: a:= (1)+(1)*x1 b:= (1)*x1 c:= (5)*x1+(3)*x1*x1 retract c(a(?x)) -> a(c(?x)) retract b(a(?x)) -> c(a(?x)) retract b(b(?x)) -> a(a(?x)) retract a(a(?x)) -> b(b(?x)) Polynomial Interpretation: a:= (2)*x1*x1 b:= (1)*x1 c:= (1)+(2)*x1*x1 relatively terminating S/P: relatively terminating check CP condition: failed failure(Step 3) failure(no possibility remains) unknown Reduction-Preserving Completion Direct Methods: Can't judge Try Persistent Decomposition for... [ b(b(?x)) -> a(a(?x)), a(a(?x)) -> a(a(?x)), c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Sort Assignment: a : 15=>15 b : 15=>15 c : 15=>15 maximal types: {15} Persistent Decomposition failed: Can't judge Try Layer Preserving Decomposition for... [ b(b(?x)) -> a(a(?x)), a(a(?x)) -> a(a(?x)), c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(a(?x)) -> c(a(?x)), c(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Layer Preserving Decomposition failed: Can't judge No further decomposition possible {4} (cm)Rewrite Rules: [ c(b(?x)) -> c(b(?x)) ] Apply Direct Methods... Inner CPs: [ ] Outer CPs: [ ] Overlay, check Innermost Termination... unknown Innermost Terminating unknown Knuth & Bendix Linear Development Closed Direct Methods: CR Try A Minimal Decomposition {6,1,5,8,4,3,0,7}{2} {6,1,5,8,4,3,0,7} (cm)Rewrite Rules: [ a(a(?x)) -> a(a(?x)), c(c(?x)) -> a(a(?x)), c(a(?x)) -> a(c(?x)), b(b(?x)) -> a(a(?x)), c(b(?x)) -> c(b(?x)), b(a(?x)) -> c(a(?x)), b(a(?x)) -> b(a(?x)), a(a(?x)) -> b(b(?x)) ] Apply Direct Methods... Inner CPs: [ a(b(b(?x_7))) = a(a(a(?x_7))), c(a(c(?x_2))) = a(a(a(?x_2))), c(c(b(?x_4))) = a(a(b(?x_4))), c(a(a(?x))) = a(c(a(?x))), c(b(b(?x_7))) = a(c(a(?x_7))), b(c(a(?x_5))) = a(a(a(?x_5))), b(b(a(?x_6))) = a(a(a(?x_6))), c(a(a(?x_3))) = c(b(b(?x_3))), c(c(a(?x_5))) = c(b(a(?x_5))), c(b(a(?x_6))) = c(b(a(?x_6))), b(a(a(?x))) = c(a(a(?x))), b(b(b(?x_7))) = c(a(a(?x_7))), b(a(a(?x))) = b(a(a(?x))), b(b(b(?x_7))) = b(a(a(?x_7))), a(a(a(?x))) = b(b(a(?x))), a(a(a(?x))) = a(a(a(?x))), c(a(a(?x))) = a(a(c(?x))), b(a(a(?x))) = a(a(b(?x))), a(b(b(?x))) = b(b(a(?x))) ] Outer CPs: [ a(a(?x)) = b(b(?x)), c(a(?x_5)) = b(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: [ b(b(?x)) = a(a(?x)), a(a(a(?x_1))) = a(a(a(?x_1))), a(b(b(?x_8))) = a(a(a(?x_8))), a(a(a(a(?x_1)))) = a(a(a(a(?x_1)))), a(a(b(b(?x_8)))) = a(a(a(a(?x_8)))), a(c(a(a(?x_1)))) = c(a(a(a(?x_1)))), a(c(b(b(?x_8)))) = c(a(a(a(?x_8)))), c(a(a(a(?x_1)))) = b(a(a(a(?x_1)))), c(a(b(b(?x_8)))) = b(a(a(a(?x_8)))), b(a(a(a(?x_1)))) = b(a(a(a(?x_1)))), b(a(b(b(?x_8)))) = b(a(a(a(?x_8)))), b(b(a(a(?x_1)))) = a(a(a(a(?x_1)))), b(b(b(b(?x_8)))) = a(a(a(a(?x_8)))), a(c(a(?x))) = c(a(a(?x))), c(a(a(?x))) = b(a(a(?x))), b(a(a(?x))) = b(a(a(?x))), b(b(a(?x))) = a(a(a(?x))), c(a(a(?x_1))) = a(a(c(?x_1))), c(a(c(?x_3))) = a(a(a(?x_3))), c(c(b(?x_5))) = a(a(b(?x_5))), a(a(a(a(?x_1)))) = c(a(a(c(?x_1)))), a(a(a(c(?x_3)))) = c(a(a(a(?x_3)))), a(a(c(b(?x_5)))) = c(a(a(b(?x_5)))), a(a(c(?x))) = c(a(a(?x))), c(a(a(?x_2))) = a(c(a(?x_2))), c(b(b(?x_8))) = a(c(a(?x_8))), a(a(a(a(?x_2)))) = c(a(c(a(?x_2)))), a(a(b(b(?x_8)))) = c(a(c(a(?x_8)))), a(a(a(?x))) = c(a(c(?x))), b(a(a(?x_1))) = a(a(b(?x_1))), b(c(a(?x_6))) = a(a(a(?x_6))), a(a(a(a(?x_1)))) = b(a(a(b(?x_1)))), a(a(c(a(?x_6)))) = b(a(a(a(?x_6)))), a(a(b(a(?x_7)))) = b(a(a(a(?x_7)))), c(b(a(a(?x_1)))) = c(a(a(b(?x_1)))), c(b(c(a(?x_6)))) = c(a(a(a(?x_6)))), c(b(b(a(?x_7)))) = c(a(a(a(?x_7)))), a(a(b(?x))) = b(a(a(?x))), c(b(b(?x))) = c(a(a(?x))), c(a(a(?x_5))) = c(b(b(?x_5))), c(c(a(?x_6))) = c(b(a(?x_6))), c(b(a(?x_7))) = c(b(a(?x_7))), a(a(a(a(?x_5)))) = c(c(b(b(?x_5)))), a(a(c(a(?x_6)))) = c(c(b(a(?x_6)))), a(a(b(a(?x_7)))) = c(c(b(a(?x_7)))), a(a(b(?x))) = c(c(b(?x))), b(a(?x)) = c(a(?x)), b(a(a(?x_2))) = c(a(a(?x_2))), b(b(b(?x_8))) = c(a(a(?x_8))), a(a(a(a(?x_2)))) = b(c(a(a(?x_2)))), a(a(b(b(?x_8)))) = b(c(a(a(?x_8)))), c(b(a(a(?x_2)))) = c(c(a(a(?x_2)))), c(b(b(b(?x_8)))) = c(c(a(a(?x_8)))), a(a(a(?x))) = b(c(a(?x))), c(b(a(?x))) = c(c(a(?x))), c(a(?x)) = b(a(?x)), b(b(b(?x_8))) = b(a(a(?x_8))), a(a(a(a(?x_2)))) = b(b(a(a(?x_2)))), a(a(b(b(?x_8)))) = b(b(a(a(?x_8)))), c(b(a(a(?x_2)))) = c(b(a(a(?x_2)))), c(b(b(b(?x_8)))) = c(b(a(a(?x_8)))), a(a(a(?x))) = b(b(a(?x))), a(a(?x)) = b(b(?x)), a(b(b(?x_1))) = b(b(a(?x_1))), b(b(b(b(?x_1)))) = a(b(b(a(?x_1)))), b(b(a(a(?x_2)))) = a(b(b(a(?x_2)))), a(a(b(b(?x_1)))) = a(b(b(a(?x_1)))), a(a(a(a(?x_2)))) = a(b(b(a(?x_2)))), a(c(b(b(?x_1)))) = c(b(b(a(?x_1)))), a(c(a(a(?x_2)))) = c(b(b(a(?x_2)))), c(a(b(b(?x_1)))) = b(b(b(a(?x_1)))), c(a(a(a(?x_2)))) = b(b(b(a(?x_2)))), b(a(b(b(?x_1)))) = b(b(b(a(?x_1)))), b(a(a(a(?x_2)))) = b(b(b(a(?x_2)))), b(b(a(?x))) = a(b(b(?x))), a(a(a(?x))) = a(b(b(?x))), a(c(a(?x))) = c(b(b(?x))), c(a(a(?x))) = b(b(b(?x))), b(a(a(?x))) = b(b(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 <7, 0> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],3)], []> joinable by a reduction of rules <[], [([(a,1)],7)]> Critical Pair by Rules <2, 1> preceded by [(c,1)] joinable by a reduction of rules <[([],2),([(a,1)],1)], []> Critical Pair by Rules <4, 1> preceded by [(c,1)] joinable by a reduction of rules <[([],1)], []> Critical Pair by Rules <0, 2> preceded by [(c,1)] joinable by a reduction of rules <[([],2)], []> Critical Pair by Rules <7, 2> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],3),([],2)], []> Critical Pair by Rules <5, 3> preceded by [(b,1)] joinable by a reduction of rules <[], [([],7),([(b,1)],5)]> Critical Pair by Rules <6, 3> preceded by [(b,1)] joinable by a reduction of rules <[([],3)], []> joinable by a reduction of rules <[], [([],7)]> Critical Pair by Rules <3, 4> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],7)], []> joinable by a reduction of rules <[], [([(c,1)],3)]> Critical Pair by Rules <5, 4> preceded by [(c,1)] joinable by a reduction of rules <[], [([(c,1)],5)]> Critical Pair by Rules <6, 4> preceded by [(c,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <0, 5> preceded by [(b,1)] joinable by a reduction of rules <[([],5)], []> Critical Pair by Rules <7, 5> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],3),([],5)], []> Critical Pair by Rules <0, 6> preceded by [(b,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <7, 6> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],3)], []> joinable by a reduction of rules <[], [([(b,1)],7)]> Critical Pair by Rules <0, 7> preceded by [(a,1)] joinable by a reduction of rules <[([],7)], []> joinable by a reduction of rules <[], [([],3)]> Critical Pair by Rules <0, 0> preceded by [(a,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <1, 1> preceded by [(c,1)] joinable by a reduction of rules <[([],2),([(a,1)],2)], []> Critical Pair by Rules <3, 3> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],7)], [([],7)]> Critical Pair by Rules <7, 7> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],3)], [([],3)]> Critical Pair by Rules <7, 0> preceded by [] joinable by a reduction of rules <[([],3)], []> joinable by a reduction of rules <[], [([],7)]> Critical Pair by Rules <6, 5> preceded by [] joinable by a reduction of rules <[([],5)], []> unknown Diagram Decreasing check Non-Confluence... obtain 10 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)