(ignored inputs)COMMENT submitted by: Miguel Vitores and Salvador Lucas secret problem 2022 category: TRS
Rewrite Rules:
   [ g(g(g(c,d),g(b,e)),b) -> a,
     g(a,?y) -> g(b,b),
     g(?x,?y) -> g(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [ g(g(g(b,e),g(c,d)),b) = a,
     g(g(g(d,c),g(b,e)),b) = a,
     g(g(g(c,d),g(e,b)),b) = a ]
Outer CPs:
   [ a = g(b,g(g(c,d),g(b,e))),
     g(b,b) = g(?y,a) ]
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:
   [ g(b,g(g(e,b),g(d,c))) = a,
     g(b,g(g(b,e),g(d,c))) = a,
     g(b,g(g(e,b),g(c,d))) = a,
     g(b,g(g(d,c),g(e,b))) = a,
     g(g(g(e,b),g(d,c)),b) = a,
     g(b,g(g(b,e),g(c,d))) = a,
     g(b,g(g(d,c),g(b,e))) = a,
     g(b,g(g(c,d),g(e,b))) = a,
     g(g(g(b,e),g(d,c)),b) = a,
     g(g(g(e,b),g(c,d)),b) = a,
     g(g(g(d,c),g(e,b)),b) = a,
     g(b,g(g(c,d),g(b,e))) = a,
     g(g(g(b,e),g(c,d)),b) = a,
     g(g(g(d,c),g(b,e)),b) = a,
     g(g(g(c,d),g(e,b)),b) = a,
     g(?y,a) = g(b,b),
     a = g(b,g(g(c,d),g(b,e))),
     g(b,b) = g(?y,a),
     a = g(g(g(b,e),g(c,d)),b),
     a = g(g(g(d,c),g(b,e)),b),
     a = g(g(g(c,d),g(e,b)),b) ]
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 <g(g(g(b,e),g(c,d)),b), a> by Rules <2, 0> preceded by [(g,1)]
 joinable by a reduction of rules <[([(g,1)],2),([],0)], []>
Critical Pair <g(g(g(d,c),g(b,e)),b), a> by Rules <2, 0> preceded by [(g,1),(g,1)]
 joinable by a reduction of rules <[([(g,1),(g,1)],2),([],0)], []>
Critical Pair <g(g(g(c,d),g(e,b)),b), a> by Rules <2, 0> preceded by [(g,1),(g,2)]
 joinable by a reduction of rules <[([(g,1),(g,2)],2),([],0)], []>
Critical Pair <g(b,g(g(c,d),g(b,e))), a> by Rules <2, 0> preceded by []
 joinable by a reduction of rules <[([],2),([],0)], []>
Critical Pair <g(?y_1,a), g(b,b)> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[([],2),([],1)], []>
unknown Diagram Decreasing
check Non-Confluence...
obtain 5 rules by 3 steps unfolding
obtain 14 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
Check relative termination:
   [ g(g(g(c,d),g(b,e)),b) -> a,
     g(a,?y) -> g(b,b) ]
   [ g(?x,?y) -> g(?y,?x) ]
Polynomial Interpretation:
  a:= 0
  b:= (1)
  c:= 0
  d:= 0
  e:= 0
  g:= (1)*x1*x1+(1)*x2*x2
retract g(g(g(c,d),g(b,e)),b) -> a
Polynomial Interpretation:
  a:= (1)
  b:= 0
  c:= 0
  d:= 0
  e:= 0
  g:= (8)*x1+(8)*x2
relatively terminating
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ g(g(g(c,d),g(b,e)),b) -> a,
     g(a,?y) -> g(b,b) ]
P:
   [ g(?x,?y) -> g(?y,?x) ]
S: terminating
CP(S,S):
PCP_in(symP,S):
<g(g(g(c,d),g(?x_1,?y_1)),b), a> --> <g(g(g(c,d),g(?x_1,?y_1)),b), a> => no
<g(g(g(?x_2,?y_2),g(b,e)),b), a> --> <g(g(g(?x_2,?y_2),g(b,e)),b), a> => no
<g(g(g(?x_2,?y_2),g(?x_1,?y_1)),b), a> --> <g(g(g(?x_2,?y_2),g(?x_1,?y_1)),b), a> => no
<g(g(g(b,e),g(c,d)),b), a> --> <g(g(g(b,e),g(c,d)),b), a> => no
CP(S,symP):
<a, g(b,g(g(c,d),g(b,e)))> --> <a, g(b,g(g(c,d),g(b,e)))> => no
<g(b,b), g(?y,a)> --> <g(b,b), g(?y,a)> => no
check joinability condition:
check modulo reachablity from a to g(g(g(c,d),g(?x_1,?y_1)),b): maybe not reachable
check modulo reachablity from a to g(g(g(?x_2,?y_2),g(b,e)),b): maybe not reachable
check modulo reachablity from a to g(g(g(?x_2,?y_2),g(?x_1,?y_1)),b): maybe not reachable
check modulo reachablity from a to g(g(g(b,e),g(c,d)),b): maybe not reachable
check modulo reachablity from a to g(b,g(g(c,d),g(b,e))): maybe not reachable
check modulo reachablity from g(b,b) to g(?y,a): maybe not reachable
failed
failure(Step 1)
   [ g(g(g(c,d),g(?x_1,?y_1)),b) -> a,
     g(g(g(?x_2,?y_2),g(b,e)),b) -> a,
     g(g(g(?x_2,?y_2),g(?x_1,?y_1)),b) -> a,
     g(g(g(b,e),g(c,d)),b) -> a,
     g(b,g(g(c,d),g(b,e))) -> a ]
Added S-Rules:
   [ g(g(g(c,d),g(?x_1,?y_1)),b) -> a,
     g(g(g(?x_2,?y_2),g(b,e)),b) -> a,
     g(g(g(?x_2,?y_2),g(?x_1,?y_1)),b) -> a,
     g(g(g(b,e),g(c,d)),b) -> a,
     g(b,g(g(c,d),g(b,e))) -> a ]
Added P-Rules:
   [ ]
replace: g(a,?y) -> g(b,b) => g(a,?y) -> g(b,b)
STEP: 2 (linear)
S:
   [ g(g(g(c,d),g(b,e)),b) -> a,
     g(a,?y) -> g(b,b) ]
P:
   [ g(?x,?y) -> g(?y,?x) ]
S: terminating
CP(S,S):
CP_in(symP,S):
<g(g(g(b,e),g(c,d)),b), a> --> <g(g(g(b,e),g(c,d)),b), a> => no
<g(g(g(d,c),g(b,e)),b), a> --> <g(g(g(d,c),g(b,e)),b), a> => no
<g(g(g(c,d),g(e,b)),b), a> --> <g(g(g(c,d),g(e,b)),b), a> => no
CP(S,symP):
<a, g(b,g(g(c,d),g(b,e)))> --> <a, g(b,g(g(c,d),g(b,e)))> => no
<g(b,b), g(?y,a)> --> <g(b,b), g(?y,a)> => no
check joinability condition:
check modulo reachablity from a to g(g(g(b,e),g(c,d)),b): maybe not reachable
check modulo reachablity from a to g(g(g(d,c),g(b,e)),b): maybe not reachable
check modulo reachablity from a to g(g(g(c,d),g(e,b)),b): maybe not reachable
check modulo reachablity from a to g(b,g(g(c,d),g(b,e))): maybe not reachable
check modulo reachablity from g(b,b) to g(?y,a): maybe not reachable
failed
failure(Step 2)
   [ g(g(g(b,e),g(c,d)),b) -> a,
     g(g(g(d,c),g(b,e)),b) -> a,
     g(g(g(c,d),g(e,b)),b) -> a,
     g(b,g(g(c,d),g(b,e))) -> a ]
Added S-Rules:
   [ g(g(g(b,e),g(c,d)),b) -> a,
     g(g(g(d,c),g(b,e)),b) -> a,
     g(g(g(c,d),g(e,b)),b) -> a,
     g(b,g(g(c,d),g(b,e))) -> a ]
Added P-Rules:
   [ ]
replace: g(a,?y) -> g(b,b) => g(a,?y) -> g(b,b)
STEP: 3 (relative)
S:
   [ g(g(g(c,d),g(b,e)),b) -> a,
     g(a,?y) -> g(b,b) ]
P:
   [ g(?x,?y) -> g(?y,?x) ]
Check relative termination:
   [ g(g(g(c,d),g(b,e)),b) -> a,
     g(a,?y) -> g(b,b) ]
   [ g(?x,?y) -> g(?y,?x) ]