(ignored inputs)COMMENT submitted by: Johannes Waldmann
Rewrite Rules:
[ a(b(?x)) -> b(c(?x)),
c(b(?x)) -> b(c(?x)),
c(b(?x)) -> c(c(?x)),
b(b(?x)) -> a(c(?x)),
a(b(?x)) -> a(b(?x)),
c(c(?x)) -> c(b(?x)),
a(c(?x)) -> c(a(?x)) ]
Apply Direct Methods...
Inner CPs:
[ a(a(c(?x_3))) = b(c(b(?x_3))),
c(a(c(?x_3))) = b(c(b(?x_3))),
c(a(c(?x_3))) = c(c(b(?x_3))),
a(a(c(?x_3))) = a(b(b(?x_3))),
c(b(c(?x_1))) = c(b(b(?x_1))),
c(c(c(?x_2))) = c(b(b(?x_2))),
a(b(c(?x_1))) = c(a(b(?x_1))),
a(c(c(?x_2))) = c(a(b(?x_2))),
a(c(b(?x_5))) = c(a(c(?x_5))),
b(a(c(?x))) = a(c(b(?x))),
c(c(b(?x))) = c(b(c(?x))) ]
Outer CPs:
[ b(c(?x)) = a(b(?x)),
b(c(?x_1)) = c(c(?x_1)) ]
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)) = b(c(?x)),
a(a(c(?x_4))) = b(c(b(?x_4))),
c(c(?x)) = b(c(?x)),
c(a(c(?x_4))) = b(c(b(?x_4))),
c(b(a(c(?x_4)))) = c(b(c(b(?x_4)))),
c(a(a(c(?x_4)))) = a(b(c(b(?x_4)))),
c(b(b(?x))) = c(b(c(?x))),
c(a(b(?x))) = a(b(c(?x))),
b(c(?x)) = c(c(?x)),
c(a(c(?x_4))) = c(c(b(?x_4))),
c(b(a(c(?x_4)))) = c(c(c(b(?x_4)))),
c(a(a(c(?x_4)))) = a(c(c(b(?x_4)))),
c(b(b(?x))) = c(c(c(?x))),
c(a(b(?x))) = a(c(c(?x))),
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(b(a(c(?x_1)))) = a(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))),
a(b(b(?x))) = a(a(c(?x))),
b(c(?x)) = a(b(?x)),
a(a(c(?x_5))) = a(b(b(?x_5))),
c(c(b(?x_1))) = c(b(c(?x_1))),
c(b(c(?x_3))) = c(b(b(?x_3))),
c(c(c(?x_4))) = c(b(b(?x_4))),
c(b(c(b(?x_1)))) = c(c(b(c(?x_1)))),
c(b(b(c(?x_3)))) = c(c(b(b(?x_3)))),
c(b(c(c(?x_4)))) = c(c(b(b(?x_4)))),
c(a(c(b(?x_1)))) = a(c(b(c(?x_1)))),
c(a(b(c(?x_3)))) = a(c(b(b(?x_3)))),
c(a(c(c(?x_4)))) = a(c(b(b(?x_4)))),
c(b(c(?x))) = c(c(b(?x))),
c(a(c(?x))) = a(c(b(?x))),
a(b(c(?x_3))) = c(a(b(?x_3))),
a(c(c(?x_4))) = c(a(b(?x_4))),
a(c(b(?x_7))) = c(a(c(?x_7))) ]
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 [(a,1)]
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) (failure)
unknown Non-Confluence
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
[ a(b(?x)) -> b(c(?x)),
c(b(?x)) -> b(c(?x)),
b(b(?x)) -> a(c(?x)),
a(c(?x)) -> c(a(?x)) ]
P:
[ c(b(?x)) -> c(c(?x)),
a(b(?x)) -> a(b(?x)),
c(c(?x)) -> c(b(?x)) ]
S: terminating
CP(S,S):
--> => no
--> => yes
--> => no
--> => yes
PCP_in(symP,S):
--> => no
--> => no
CP(S,symP):
--> => yes
--> => no
--> => no
--> => yes
--> => 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 c(c(a(?x_3))) and b(c(c(?x_3))): 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 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
check modulo joinability of c(c(a(?x))) and b(c(c(?x))): joinable by {0}
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)),
c(b(?x)) -> b(c(?x)),
b(b(?x)) -> a(c(?x)),
a(c(?x)) -> c(a(?x)) ]
P:
[ c(b(?x)) -> c(c(?x)),
a(b(?x)) -> a(b(?x)),
c(c(?x)) -> c(b(?x)) ]
S: terminating
CP(S,S):
--> => no
--> => yes
--> => no
--> => yes
CP_in(symP,S):
--> => no
--> => no
CP(S,symP):
--> => yes
--> => no
--> => no
--> => yes
--> => 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 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 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
check modulo joinability of c(c(a(?x))) and b(c(c(?x))): joinable by {0}
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)),
c(b(?x)) -> b(c(?x)),
b(b(?x)) -> a(c(?x)),
a(c(?x)) -> c(a(?x)) ]
P:
[ c(b(?x)) -> c(c(?x)),
a(b(?x)) -> a(b(?x)),
c(c(?x)) -> c(b(?x)) ]
Check relative termination:
[ a(b(?x)) -> b(c(?x)),
c(b(?x)) -> b(c(?x)),
b(b(?x)) -> a(c(?x)),
a(c(?x)) -> c(a(?x)) ]
[ c(b(?x)) -> c(c(?x)),
a(b(?x)) -> a(b(?x)),
c(c(?x)) -> c(b(?x)) ]
Polynomial Interpretation:
a:= (2)+(2)*x1
b:= (2)*x1
c:= (2)*x1
retract a(b(?x)) -> b(c(?x))
Polynomial Interpretation:
a:= (2)*x1
b:= (3)+(2)*x1
c:= (2)*x1
retract a(b(?x)) -> b(c(?x))
retract c(b(?x)) -> b(c(?x))
retract b(b(?x)) -> a(c(?x))
retract c(b(?x)) -> c(c(?x))
Polynomial Interpretation:
a:= (1)*x1*x1
b:= (2)+(2)*x1+(1)*x1*x1
c:= (2)+(2)*x1+(1)*x1*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
[ a(b(?x)) -> b(c(?x)),
c(b(?x)) -> b(c(?x)),
b(b(?x)) -> a(c(?x)),
a(c(?x)) -> c(a(?x)),
c(c(?x)) -> b(c(?x)) ]
P:
[ c(b(?x)) -> c(c(?x)),
a(b(?x)) -> a(b(?x)),
c(c(?x)) -> c(b(?x)) ]
S: unknown termination
failure(Step 4)
STEP: 5 (linear)
S:
[ a(b(?x)) -> b(c(?x)),
c(b(?x)) -> b(c(?x)),
b(b(?x)) -> a(c(?x)),
a(c(?x)) -> c(a(?x)),
c(c(?x)) -> b(c(?x)) ]
P:
[ c(b(?x)) -> c(c(?x)),
a(b(?x)) -> a(b(?x)),
c(c(?x)) -> c(b(?x)) ]
S: unknown termination
failure(Step 5)
STEP: 6 (relative)
S:
[ a(b(?x)) -> b(c(?x)),
c(b(?x)) -> b(c(?x)),
b(b(?x)) -> a(c(?x)),
a(c(?x)) -> c(a(?x)),
c(c(?x)) -> b(c(?x)) ]
P:
[ c(b(?x)) -> c(c(?x)),
a(b(?x)) -> a(b(?x)),
c(c(?x)) -> c(b(?x)) ]
Check relative termination:
[ a(b(?x)) -> b(c(?x)),
c(b(?x)) -> b(c(?x)),
b(b(?x)) -> a(c(?x)),
a(c(?x)) -> c(a(?x)),
c(c(?x)) -> b(c(?x)) ]
[ c(b(?x)) -> c(c(?x)),
a(b(?x)) -> a(b(?x)),
c(c(?x)) -> c(b(?x)) ]