(ignored inputs)COMMENT submitted by: Raul Gutierrez secret problem 2021 category: SRS
Rewrite Rules:
   [ a(b(?x)) -> b(c(?x)),
     a(c(?x)) -> c(a(?x)),
     b(b(?x)) -> a(c(?x)),
     c(b(?x)) -> b(c(?x)),
     c(b(?x)) -> c(c(?x)),
     c(c(?x)) -> c(b(?x)),
     0(1(2(?x))) -> 2(0(1(?x))),
     2(2(2(2(2(2(2(1(1(1(1(2(?x)))))))))))) -> 2(1(2(2(0(1(2(1(1(0(1(0(?x)))))))))))) ]
Apply Direct Methods...
Inner CPs:
   [ a(a(c(?x_2))) = b(c(b(?x_2))),
     a(b(c(?x_3))) = c(a(b(?x_3))),
     a(c(c(?x_4))) = c(a(b(?x_4))),
     a(c(b(?x_5))) = c(a(c(?x_5))),
     c(a(c(?x_2))) = b(c(b(?x_2))),
     c(a(c(?x_2))) = c(c(b(?x_2))),
     c(b(c(?x_3))) = c(b(b(?x_3))),
     c(c(c(?x_4))) = c(b(b(?x_4))),
     0(1(2(1(2(2(0(1(2(1(1(0(1(0(?x_7)))))))))))))) = 2(0(1(2(2(2(2(2(2(1(1(1(1(2(?x_7)))))))))))))),
     b(a(c(?x))) = a(c(b(?x))),
     c(c(b(?x))) = c(b(c(?x))),
     2(2(2(2(2(2(2(1(1(1(1(2(1(2(2(0(1(2(1(1(0(1(0(?x))))))))))))))))))))))) = 2(1(2(2(0(1(2(1(1(0(1(0(2(2(2(2(2(2(1(1(1(1(2(?x))))))))))))))))))))))) ]
Outer CPs:
   [ b(c(?x_3)) = c(c(?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
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ a(a(c(?x_3))) = b(c(b(?x_3))),
     a(b(c(?x_4))) = c(a(b(?x_4))),
     a(c(c(?x_5))) = c(a(b(?x_5))),
     a(c(b(?x_6))) = c(a(c(?x_6))),
     b(a(c(?x_1))) = a(c(b(?x_1))),
     a(c(a(c(?x_1)))) = b(a(c(b(?x_1)))),
     b(c(a(c(?x_1)))) = a(a(c(b(?x_1)))),
     b(c(a(c(?x_1)))) = c(a(c(b(?x_1)))),
     c(c(a(c(?x_1)))) = c(a(c(b(?x_1)))),
     a(c(b(?x))) = b(a(c(?x))),
     b(c(b(?x))) = a(a(c(?x))),
     b(c(b(?x))) = c(a(c(?x))),
     c(c(b(?x))) = c(a(c(?x))),
     c(c(?x)) = b(c(?x)),
     c(a(c(?x_4))) = b(c(b(?x_4))),
     c(a(a(c(?x_4)))) = a(b(c(b(?x_4)))),
     c(b(a(c(?x_4)))) = c(b(c(b(?x_4)))),
     c(a(b(?x))) = a(b(c(?x))),
     c(b(b(?x))) = c(b(c(?x))),
     b(c(?x)) = c(c(?x)),
     c(a(c(?x_4))) = c(c(b(?x_4))),
     c(a(a(c(?x_4)))) = a(c(c(b(?x_4)))),
     c(b(a(c(?x_4)))) = c(c(c(b(?x_4)))),
     c(a(b(?x))) = a(c(c(?x))),
     c(b(b(?x))) = c(c(c(?x))),
     c(c(b(?x_1))) = c(b(c(?x_1))),
     c(b(c(?x_5))) = c(b(b(?x_5))),
     c(c(c(?x_6))) = c(b(b(?x_6))),
     c(b(c(b(?x_1)))) = c(c(b(c(?x_1)))),
     c(b(b(c(?x_5)))) = c(c(b(b(?x_5)))),
     c(b(c(c(?x_6)))) = c(c(b(b(?x_6)))),
     c(a(c(b(?x_1)))) = a(c(b(c(?x_1)))),
     c(a(b(c(?x_5)))) = a(c(b(b(?x_5)))),
     c(a(c(c(?x_6)))) = a(c(b(b(?x_6)))),
     c(b(c(?x))) = c(c(b(?x))),
     c(a(c(?x))) = a(c(b(?x))),
     0(1(2(1(2(2(0(1(2(1(1(0(1(0(?x_8)))))))))))))) = 2(0(1(2(2(2(2(2(2(1(1(1(1(2(?x_8)))))))))))))),
     2(2(2(2(2(2(2(1(1(1(1(2(1(2(2(0(1(2(1(1(0(1(0(?x_1))))))))))))))))))))))) = 2(1(2(2(0(1(2(1(1(0(1(0(2(2(2(2(2(2(1(1(1(1(2(?x_1))))))))))))))))))))))),
     2(1(2(2(0(1(2(1(1(0(1(0(2(2(2(2(2(2(1(1(1(1(2(1(2(2(0(1(2(1(1(0(1(0(?x_1)))))))))))))))))))))))))))))))))) = 2(2(2(2(2(2(2(1(1(1(1(2(1(2(2(0(1(2(1(1(0(1(0(2(2(2(2(2(2(1(1(1(1(2(?x_1)))))))))))))))))))))))))))))))))),
     2(0(1(2(2(2(2(2(2(1(1(1(1(2(1(2(2(0(1(2(1(1(0(1(0(?x_1))))))))))))))))))))))))) = 0(1(2(1(2(2(0(1(2(1(1(0(1(0(2(2(2(2(2(2(1(1(1(1(2(?x_1))))))))))))))))))))))))),
     2(1(2(2(0(1(2(1(1(0(1(0(2(2(2(2(2(2(1(1(1(1(2(?x))))))))))))))))))))))) = 2(2(2(2(2(2(2(1(1(1(1(2(1(2(2(0(1(2(1(1(0(1(0(?x))))))))))))))))))))))),
     2(0(1(2(2(2(2(2(2(1(1(1(1(2(?x)))))))))))))) = 0(1(2(1(2(2(0(1(2(1(1(0(1(0(?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 <a(a(c(?x_2))), b(c(b(?x_2)))> by Rules <2, 0> preceded by [(a,1)]
unknown Diagram Decreasing
check Non-Confluence...
obtain 12 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(b(?x)) -> b(c(?x)),
     a(c(?x)) -> c(a(?x)),
     b(b(?x)) -> a(c(?x)),
     c(b(?x)) -> b(c(?x)),
     0(1(2(?x))) -> 2(0(1(?x))),
     2(2(2(2(2(2(2(1(1(1(1(2(?x)))))))))))) -> 2(1(2(2(0(1(2(1(1(0(1(0(?x)))))))))))) ]
P:
   [ c(b(?x)) -> c(c(?x)),
     c(c(?x)) -> c(b(?x)) ]
S: terminating
CP(S,S):
<a(a(c(?x_2))), b(c(b(?x_2)))> --> <c(a(a(?x_2))), c(c(a(?x_2)))> => no
<a(b(c(?x_3))), c(a(b(?x_3)))> --> <b(c(c(?x_3))), b(c(c(?x_3)))> => yes
<b(a(c(?x))), a(c(b(?x)))> --> <b(c(a(?x))), b(c(c(?x)))> => no
<c(a(c(?x_2))), b(c(b(?x_2)))> --> <c(c(a(?x_2))), c(c(a(?x_2)))> => yes
<0(1(2(1(2(2(0(1(2(1(1(0(1(0(?x_5)))))))))))))), 2(0(1(2(2(2(2(2(2(1(1(1(1(2(?x_5))))))))))))))> --> <2(0(1(1(2(2(2(0(1(1(1(0(1(0(?x_5)))))))))))))), 2(2(2(2(2(2(2(0(1(1(1(1(1(2(?x_5))))))))))))))> => no
<2(2(2(2(2(2(2(1(1(1(1(2(1(2(2(0(1(2(1(1(0(1(0(?x))))))))))))))))))))))), 2(1(2(2(0(1(2(1(1(0(1(0(2(2(2(2(2(2(1(1(1(1(2(?x)))))))))))))))))))))))> --> <2(1(2(2(2(0(1(1(1(2(2(2(0(1(0(1(0(1(1(1(0(1(0(?x))))))))))))))))))))))), 2(1(2(2(2(0(1(1(1(0(1(0(2(2(2(2(2(2(1(1(1(1(2(?x)))))))))))))))))))))))> => no
PCP_in(symP,S):
<a(c(c(?x_2))), c(a(b(?x_2)))> --> <c(c(a(?x_2))), b(c(c(?x_2)))> => no
<a(c(b(?x_1))), c(a(c(?x_1)))> --> <b(c(c(?x_1))), c(c(a(?x_1)))> => no
CP(S,symP):
<c(a(c(?x))), c(c(b(?x)))> --> <c(c(a(?x))), b(c(c(?x)))> => no
<c(b(c(?x))), c(b(b(?x)))> --> <b(c(c(?x))), c(c(a(?x)))> => no
<b(c(?x)), c(c(?x))> --> <b(c(?x)), c(c(?x))> => no
check joinability condition:
check modulo joinability of c(a(a(?x_2))) and c(c(a(?x_2))): maybe not joinable
check modulo joinability of b(c(a(?x))) and b(c(c(?x))): joinable by {0}
check modulo joinability of 2(0(1(1(2(2(2(0(1(1(1(0(1(0(?x_5)))))))))))))) and 2(2(2(2(2(2(2(0(1(1(1(1(1(2(?x_5)))))))))))))): maybe not joinable
check modulo joinability of 2(1(2(2(2(0(1(1(1(2(2(2(0(1(0(1(0(1(1(1(0(1(0(?x))))))))))))))))))))))) and 2(1(2(2(2(0(1(1(1(0(1(0(2(2(2(2(2(2(1(1(1(1(2(?x))))))))))))))))))))))): maybe not joinable
check modulo joinability of c(c(a(?x_2))) and b(c(c(?x_2))): joinable by {0}
check modulo joinability of b(c(c(?x_1))) and c(c(a(?x_1))): joinable by {0}
check modulo joinability of c(c(a(?x))) and b(c(c(?x))): joinable by {0}
check modulo joinability of b(c(c(?x))) and c(c(a(?x))): joinable by {0}
check modulo reachablity from b(c(?x)) to c(c(?x)): maybe not reachable
failed
failure(Step 1)
   [ c(c(?x)) -> b(c(?x)) ]
Added S-Rules:
   [ c(c(?x)) -> b(c(?x)) ]
Added P-Rules:
   [ ]
STEP: 2 (linear)
S:
   [ a(b(?x)) -> b(c(?x)),
     a(c(?x)) -> c(a(?x)),
     b(b(?x)) -> a(c(?x)),
     c(b(?x)) -> b(c(?x)),
     0(1(2(?x))) -> 2(0(1(?x))),
     2(2(2(2(2(2(2(1(1(1(1(2(?x)))))))))))) -> 2(1(2(2(0(1(2(1(1(0(1(0(?x)))))))))))) ]
P:
   [ c(b(?x)) -> c(c(?x)),
     c(c(?x)) -> c(b(?x)) ]
S: terminating
CP(S,S):
<a(a(c(?x_2))), b(c(b(?x_2)))> --> <c(a(a(?x_2))), c(c(a(?x_2)))> => no
<a(b(c(?x_3))), c(a(b(?x_3)))> --> <b(c(c(?x_3))), b(c(c(?x_3)))> => yes
<b(a(c(?x))), a(c(b(?x)))> --> <b(c(a(?x))), b(c(c(?x)))> => no
<c(a(c(?x_2))), b(c(b(?x_2)))> --> <c(c(a(?x_2))), c(c(a(?x_2)))> => yes
<0(1(2(1(2(2(0(1(2(1(1(0(1(0(?x_5)))))))))))))), 2(0(1(2(2(2(2(2(2(1(1(1(1(2(?x_5))))))))))))))> --> <2(0(1(1(2(2(2(0(1(1(1(0(1(0(?x_5)))))))))))))), 2(2(2(2(2(2(2(0(1(1(1(1(1(2(?x_5))))))))))))))> => no
<2(2(2(2(2(2(2(1(1(1(1(2(1(2(2(0(1(2(1(1(0(1(0(?x))))))))))))))))))))))), 2(1(2(2(0(1(2(1(1(0(1(0(2(2(2(2(2(2(1(1(1(1(2(?x)))))))))))))))))))))))> --> <2(1(2(2(2(0(1(1(1(2(2(2(0(1(0(1(0(1(1(1(0(1(0(?x))))))))))))))))))))))), 2(1(2(2(2(0(1(1(1(0(1(0(2(2(2(2(2(2(1(1(1(1(2(?x)))))))))))))))))))))))> => no
CP_in(symP,S):
<a(c(b(?x))), c(a(c(?x)))> --> <b(c(c(?x))), c(c(a(?x)))> => no
<a(c(c(?x))), c(a(b(?x)))> --> <c(c(a(?x))), b(c(c(?x)))> => no
CP(S,symP):
<c(a(c(?x))), c(c(b(?x)))> --> <c(c(a(?x))), b(c(c(?x)))> => no
<c(b(c(?x))), c(b(b(?x)))> --> <b(c(c(?x))), c(c(a(?x)))> => no
<b(c(?x)), c(c(?x))> --> <b(c(?x)), c(c(?x))> => no
check joinability condition:
check modulo joinability of c(a(a(?x_2))) and c(c(a(?x_2))): maybe not joinable
check modulo joinability of b(c(a(?x))) and b(c(c(?x))): maybe not joinable
check modulo joinability of 2(0(1(1(2(2(2(0(1(1(1(0(1(0(?x_5)))))))))))))) and 2(2(2(2(2(2(2(0(1(1(1(1(1(2(?x_5)))))))))))))): maybe not joinable
check modulo joinability of 2(1(2(2(2(0(1(1(1(2(2(2(0(1(0(1(0(1(1(1(0(1(0(?x))))))))))))))))))))))) and 2(1(2(2(2(0(1(1(1(0(1(0(2(2(2(2(2(2(1(1(1(1(2(?x))))))))))))))))))))))): maybe not joinable
check modulo joinability of b(c(c(?x))) and c(c(a(?x))): joinable by {0}
check modulo joinability of c(c(a(?x))) and b(c(c(?x))): joinable by {0}
check modulo joinability of c(c(a(?x))) and b(c(c(?x))): joinable by {0}
check modulo joinability of b(c(c(?x))) and c(c(a(?x))): joinable by {0}
check modulo reachablity from b(c(?x)) to c(c(?x)): maybe not reachable
failed
failure(Step 2)
   [ c(c(?x)) -> b(c(?x)) ]
Added S-Rules:
   [ c(c(?x)) -> b(c(?x)) ]
Added P-Rules:
   [ ]
STEP: 3 (relative)
S:
   [ a(b(?x)) -> b(c(?x)),
     a(c(?x)) -> c(a(?x)),
     b(b(?x)) -> a(c(?x)),
     c(b(?x)) -> b(c(?x)),
     0(1(2(?x))) -> 2(0(1(?x))),
     2(2(2(2(2(2(2(1(1(1(1(2(?x)))))))))))) -> 2(1(2(2(0(1(2(1(1(0(1(0(?x)))))))))))) ]
P:
   [ c(b(?x)) -> c(c(?x)),
     c(c(?x)) -> c(b(?x)) ]
Check relative termination:
   [ a(b(?x)) -> b(c(?x)),
     a(c(?x)) -> c(a(?x)),
     b(b(?x)) -> a(c(?x)),
     c(b(?x)) -> b(c(?x)),
     0(1(2(?x))) -> 2(0(1(?x))),
     2(2(2(2(2(2(2(1(1(1(1(2(?x)))))))))))) -> 2(1(2(2(0(1(2(1(1(0(1(0(?x)))))))))))) ]
   [ c(b(?x)) -> c(c(?x)),
     c(c(?x)) -> c(b(?x)) ]