(ignored inputs)COMMENT submitted by: Johannes Waldmann Rewrite Rules: [ a(a(?x)) -> c(c(?x)), c(a(?x)) -> a(a(?x)), c(b(?x)) -> a(c(?x)), c(c(?x)) -> c(a(?x)), a(b(?x)) -> a(b(?x)) ] Apply Direct Methods... Inner CPs: [ a(a(b(?x_4))) = c(c(b(?x_4))), c(c(c(?x))) = a(a(a(?x))), c(a(b(?x_4))) = a(a(b(?x_4))), c(a(a(?x_1))) = c(a(a(?x_1))), c(a(c(?x_2))) = c(a(b(?x_2))), a(c(c(?x))) = c(c(a(?x))), c(c(a(?x))) = c(a(c(?x))) ] Outer CPs: [ ] not Overlay, check Termination... unknown/not Terminating unknown Knuth & Bendix Linear unknown Development Closed unknown Strongly Closed unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow inner CP cond (upside-parallel) innter CP Cond (outside) unknown Upside-Parallel-Closed/Outside-Closed (inner) Parallel CPs: (not computed) unknown Toyama (Parallel CPs) Simultaneous CPs: [ a(c(c(?x_1))) = c(c(a(?x_1))), a(a(b(?x_5))) = c(c(b(?x_5))), c(c(c(c(?x_1)))) = a(c(c(a(?x_1)))), c(c(a(b(?x_5)))) = a(c(c(b(?x_5)))), a(a(c(c(?x_1)))) = c(c(c(a(?x_1)))), a(a(a(b(?x_5)))) = c(c(c(b(?x_5)))), c(c(a(?x))) = a(c(c(?x))), a(a(a(?x))) = c(c(c(?x))), c(c(c(?x_2))) = a(a(a(?x_2))), c(a(b(?x_5))) = a(a(b(?x_5))), c(a(c(c(?x_2)))) = c(a(a(a(?x_2)))), c(a(a(b(?x_5)))) = c(a(a(b(?x_5)))), c(a(a(?x))) = c(a(a(?x))), c(a(b(?x))) = c(a(c(?x))), c(c(a(?x_1))) = c(a(c(?x_1))), c(a(c(?x_4))) = c(a(b(?x_4))), c(a(c(a(?x_1)))) = c(c(a(c(?x_1)))), c(a(a(a(?x_3)))) = c(c(a(a(?x_3)))), c(a(a(c(?x_4)))) = c(c(a(b(?x_4)))), c(a(c(?x))) = c(c(a(?x))), c(c(b(?x))) = a(a(b(?x))), a(a(b(?x))) = c(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 <[([],0)], []> Critical Pair by Rules <0, 1> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],3)], [([],0)]> Critical Pair by Rules <4, 1> preceded by [(c,1)] joinable by a reduction of rules <[([],1)], []> Critical Pair by Rules <1, 3> preceded by [(c,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <2, 3> preceded by [(c,1)] joinable by a reduction of rules <[], [([],1),([],0),([(c,1)],2)]> Critical Pair by Rules <0, 0> preceded by [(a,1)] joinable by a reduction of rules <[([(a,1)],3),([(a,1)],1)], [([(c,1)],1),([],1)]> joinable by a reduction of rules <[([(a,1)],3),([(a,1)],1)], [([],3),([],1)]> Critical Pair by Rules <3, 3> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],1),([(c,1)],0)], [([],1),([],0)]> joinable by a reduction of rules <[([],3),([(c,1)],0)], [([],1),([],0)]> unknown Diagram Decreasing check Non-Confluence... obtain 10 rules by 3 steps unfolding obtain 89 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(a(?x)) -> c(c(?x)), c(a(?x)) -> a(a(?x)), c(b(?x)) -> a(c(?x)), c(c(?x)) -> c(a(?x)) ] P: [ a(b(?x)) -> a(b(?x)) ] S: unknown termination failure(Step 1) STEP: 2 (linear) S: [ a(a(?x)) -> c(c(?x)), c(a(?x)) -> a(a(?x)), c(b(?x)) -> a(c(?x)), c(c(?x)) -> c(a(?x)) ] P: [ a(b(?x)) -> a(b(?x)) ] S: unknown termination failure(Step 2) STEP: 3 (relative) S: [ a(a(?x)) -> c(c(?x)), c(a(?x)) -> a(a(?x)), c(b(?x)) -> a(c(?x)), c(c(?x)) -> c(a(?x)) ] P: [ a(b(?x)) -> a(b(?x)) ] Check relative termination: [ a(a(?x)) -> c(c(?x)), c(a(?x)) -> a(a(?x)), c(b(?x)) -> a(c(?x)), c(c(?x)) -> c(a(?x)) ] [ a(b(?x)) -> a(b(?x)) ]