Problem Transformed CSR 04 Ex25 Luc06 C

Tool CaT

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex25 Luc06 C

stdout:

YES(?,O(n^1))

Problem:
active(f(f(X))) -> mark(c(f(g(f(X)))))
active(c(X)) -> mark(d(X))
active(h(X)) -> mark(c(d(X)))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
h(mark(X)) -> mark(h(X))
proper(f(X)) -> f(proper(X))
proper(c(X)) -> c(proper(X))
proper(g(X)) -> g(proper(X))
proper(d(X)) -> d(proper(X))
proper(h(X)) -> h(proper(X))
f(ok(X)) -> ok(f(X))
c(ok(X)) -> ok(c(X))
g(ok(X)) -> ok(g(X))
d(ok(X)) -> ok(d(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Proof:
Bounds Processor:
  bound: 1
  enrichment: match
  automaton:
   final states: {10,9,8,7,6,5,4,3}
   transitions:
    top1(70) -> 71*
    active1(85) -> 86*
    active1(79) -> 80*
    proper1(77) -> 78*
    proper1(69) -> 70*
    ok1(67) -> 68*
    ok1(59) -> 60*
    ok1(49) -> 50*
    ok1(39) -> 40*
    ok1(31) -> 32*
    h1(29) -> 30*
    h1(21) -> 22*
    d1(61) -> 62*
    d1(58) -> 59*
    g1(51) -> 52*
    g1(48) -> 49*
    c1(41) -> 42*
    c1(38) -> 39*
    f1(19) -> 20*
    f1(11) -> 12*
    mark1(22) -> 23*
    mark1(12) -> 13*
    active0(2) -> 3*
    active0(1) -> 3*
    f0(2) -> 4*
    f0(1) -> 4*
    mark0(2) -> 1*
    mark0(1) -> 1*
    c0(2) -> 7*
    c0(1) -> 7*
    g0(2) -> 8*
    g0(1) -> 8*
    d0(2) -> 9*
    d0(1) -> 9*
    h0(2) -> 5*
    h0(1) -> 5*
    proper0(2) -> 6*
    proper0(1) -> 6*
    ok0(2) -> 2*
    ok0(1) -> 2*
    top0(2) -> 10*
    top0(1) -> 10*
    1 -> 85,77,61,51,41,29,19
    2 -> 79,69,58,48,38,21,11
    12 -> 31*
    13 -> 20,12,31,4
    20 -> 12*
    22 -> 67*
    23 -> 30,22,67,5
    30 -> 22*
    32 -> 12,31,4
    40 -> 39,7
    42 -> 39*
    50 -> 49,8
    52 -> 49*
    60 -> 59,9
    62 -> 59*
    68 -> 22,67,5
    71 -> 10*
    78 -> 70*
    80 -> 70*
    86 -> 70*
  problem:

  Qed

Tool IRC1

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex25 Luc06 C

stdout:

YES(?,O(n^1))

Tool IRC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex25 Luc06 C

stdout:

YES(?,O(n^1))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  active(f(f(X))) -> mark(c(f(g(f(X)))))
     , active(c(X)) -> mark(d(X))
     , active(h(X)) -> mark(c(d(X)))
     , active(f(X)) -> f(active(X))
     , active(h(X)) -> h(active(X))
     , f(mark(X)) -> mark(f(X))
     , h(mark(X)) -> mark(h(X))
     , proper(f(X)) -> f(proper(X))
     , proper(c(X)) -> c(proper(X))
     , proper(g(X)) -> g(proper(X))
     , proper(d(X)) -> d(proper(X))
     , proper(h(X)) -> h(proper(X))
     , f(ok(X)) -> ok(f(X))
     , c(ok(X)) -> ok(c(X))
     , g(ok(X)) -> ok(g(X))
     , d(ok(X)) -> ok(d(X))
     , h(ok(X)) -> ok(h(X))
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X))}

Proof Output:
  'Bounds with minimal-enrichment and initial automaton 'match'' proved the best result:

  Details:
  --------
    'Bounds with minimal-enrichment and initial automaton 'match'' succeeded with the following output:
     'Bounds with minimal-enrichment and initial automaton 'match''
     --------------------------------------------------------------
     Answer:           YES(?,O(n^1))
     Input Problem:    innermost runtime-complexity with respect to
       Rules:
         {  active(f(f(X))) -> mark(c(f(g(f(X)))))
          , active(c(X)) -> mark(d(X))
          , active(h(X)) -> mark(c(d(X)))
          , active(f(X)) -> f(active(X))
          , active(h(X)) -> h(active(X))
          , f(mark(X)) -> mark(f(X))
          , h(mark(X)) -> mark(h(X))
          , proper(f(X)) -> f(proper(X))
          , proper(c(X)) -> c(proper(X))
          , proper(g(X)) -> g(proper(X))
          , proper(d(X)) -> d(proper(X))
          , proper(h(X)) -> h(proper(X))
          , f(ok(X)) -> ok(f(X))
          , c(ok(X)) -> ok(c(X))
          , g(ok(X)) -> ok(g(X))
          , d(ok(X)) -> ok(d(X))
          , h(ok(X)) -> ok(h(X))
          , top(mark(X)) -> top(proper(X))
          , top(ok(X)) -> top(active(X))}

     Proof Output:
       The problem is match-bounded by 1.
       The enriched problem is compatible with the following automaton:
       {  active_0(2) -> 1
        , active_1(2) -> 4
        , f_0(2) -> 1
        , f_1(2) -> 3
        , mark_0(2) -> 2
        , mark_1(3) -> 1
        , mark_1(3) -> 3
        , c_0(2) -> 1
        , c_1(2) -> 3
        , g_0(2) -> 1
        , g_1(2) -> 3
        , d_0(2) -> 1
        , d_1(2) -> 3
        , h_0(2) -> 1
        , h_1(2) -> 3
        , proper_0(2) -> 1
        , proper_1(2) -> 4
        , ok_0(2) -> 2
        , ok_1(3) -> 1
        , ok_1(3) -> 3
        , top_0(2) -> 1
        , top_1(4) -> 1}

Tool RC1

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex25 Luc06 C

stdout:

YES(?,O(n^1))

Tool RC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex25 Luc06 C

stdout:

YES(?,O(n^1))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    runtime-complexity with respect to
  Rules:
    {  active(f(f(X))) -> mark(c(f(g(f(X)))))
     , active(c(X)) -> mark(d(X))
     , active(h(X)) -> mark(c(d(X)))
     , active(f(X)) -> f(active(X))
     , active(h(X)) -> h(active(X))
     , f(mark(X)) -> mark(f(X))
     , h(mark(X)) -> mark(h(X))
     , proper(f(X)) -> f(proper(X))
     , proper(c(X)) -> c(proper(X))
     , proper(g(X)) -> g(proper(X))
     , proper(d(X)) -> d(proper(X))
     , proper(h(X)) -> h(proper(X))
     , f(ok(X)) -> ok(f(X))
     , c(ok(X)) -> ok(c(X))
     , g(ok(X)) -> ok(g(X))
     , d(ok(X)) -> ok(d(X))
     , h(ok(X)) -> ok(h(X))
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X))}

Proof Output:
  'Bounds with minimal-enrichment and initial automaton 'match'' proved the best result:

  Details:
  --------
    'Bounds with minimal-enrichment and initial automaton 'match'' succeeded with the following output:
     'Bounds with minimal-enrichment and initial automaton 'match''
     --------------------------------------------------------------
     Answer:           YES(?,O(n^1))
     Input Problem:    runtime-complexity with respect to
       Rules:
         {  active(f(f(X))) -> mark(c(f(g(f(X)))))
          , active(c(X)) -> mark(d(X))
          , active(h(X)) -> mark(c(d(X)))
          , active(f(X)) -> f(active(X))
          , active(h(X)) -> h(active(X))
          , f(mark(X)) -> mark(f(X))
          , h(mark(X)) -> mark(h(X))
          , proper(f(X)) -> f(proper(X))
          , proper(c(X)) -> c(proper(X))
          , proper(g(X)) -> g(proper(X))
          , proper(d(X)) -> d(proper(X))
          , proper(h(X)) -> h(proper(X))
          , f(ok(X)) -> ok(f(X))
          , c(ok(X)) -> ok(c(X))
          , g(ok(X)) -> ok(g(X))
          , d(ok(X)) -> ok(d(X))
          , h(ok(X)) -> ok(h(X))
          , top(mark(X)) -> top(proper(X))
          , top(ok(X)) -> top(active(X))}

     Proof Output:
       The problem is match-bounded by 1.
       The enriched problem is compatible with the following automaton:
       {  active_0(2) -> 1
        , active_1(2) -> 4
        , f_0(2) -> 1
        , f_1(2) -> 3
        , mark_0(2) -> 2
        , mark_1(3) -> 1
        , mark_1(3) -> 3
        , c_0(2) -> 1
        , c_1(2) -> 3
        , g_0(2) -> 1
        , g_1(2) -> 3
        , d_0(2) -> 1
        , d_1(2) -> 3
        , h_0(2) -> 1
        , h_1(2) -> 3
        , proper_0(2) -> 1
        , proper_1(2) -> 4
        , ok_0(2) -> 2
        , ok_1(3) -> 1
        , ok_1(3) -> 3
        , top_0(2) -> 1
        , top_1(4) -> 1}