Problem AG01 innermost 4.31

Tool CaT

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	AG01 innermost 4.31

stdout:

YES(?,O(n^1))

Problem:
a(d(x)) -> d(c(b(a(x))))
b(c(x)) -> c(d(a(b(x))))
a(c(x)) -> x
b(d(x)) -> x

Proof:
Bounds Processor:
  bound: 1
  enrichment: match
  automaton:
   final states: {4,3}
   transitions:
    c1(22) -> 23*
    c1(7) -> 8*
    d1(21) -> 22*
    d1(8) -> 9*
    a1(20) -> 21*
    a1(5) -> 6*
    a1(17) -> 18*
    b1(19) -> 20*
    b1(31) -> 32*
    b1(6) -> 7*
    a0(2) -> 3*
    a0(1) -> 3*
    d0(2) -> 1*
    d0(1) -> 1*
    c0(2) -> 2*
    c0(1) -> 2*
    b0(2) -> 4*
    b0(1) -> 4*
    1 -> 7,21,32,20,6,4,3,31,17
    2 -> 7,21,32,20,6,4,3,19,5
    8 -> 7*
    9 -> 21,18,6,3
    18 -> 6*
    22 -> 21*
    23 -> 7,20,4
    32 -> 20*
  problem:

  Qed

Tool IRC1

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	AG01 innermost 4.31

stdout:

YES(?,O(n^1))

Tool IRC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	AG01 innermost 4.31

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:
    {  a(d(x)) -> d(c(b(a(x))))
     , b(c(x)) -> c(d(a(b(x))))
     , a(c(x)) -> x
     , b(d(x)) -> 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:
         {  a(d(x)) -> d(c(b(a(x))))
          , b(c(x)) -> c(d(a(b(x))))
          , a(c(x)) -> x
          , b(d(x)) -> x}

     Proof Output:
       The problem is match-bounded by 1.
       The enriched problem is compatible with the following automaton:
       {  a_0(2) -> 1
        , a_1(2) -> 5
        , a_1(8) -> 7
        , d_0(2) -> 1
        , d_0(2) -> 2
        , d_0(2) -> 4
        , d_0(2) -> 5
        , d_0(2) -> 7
        , d_0(2) -> 8
        , d_1(3) -> 1
        , d_1(3) -> 5
        , d_1(3) -> 7
        , d_1(7) -> 6
        , d_1(7) -> 7
        , c_0(2) -> 1
        , c_0(2) -> 2
        , c_0(2) -> 4
        , c_0(2) -> 5
        , c_0(2) -> 7
        , c_0(2) -> 8
        , c_1(4) -> 3
        , c_1(4) -> 4
        , c_1(6) -> 1
        , c_1(6) -> 4
        , c_1(6) -> 8
        , b_0(2) -> 1
        , b_1(2) -> 8
        , b_1(5) -> 4}

Tool RC1

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	AG01 innermost 4.31

stdout:

YES(?,O(n^1))

Tool RC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	AG01 innermost 4.31

stdout:

YES(?,O(n^1))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    runtime-complexity with respect to
  Rules:
    {  a(d(x)) -> d(c(b(a(x))))
     , b(c(x)) -> c(d(a(b(x))))
     , a(c(x)) -> x
     , b(d(x)) -> 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:
         {  a(d(x)) -> d(c(b(a(x))))
          , b(c(x)) -> c(d(a(b(x))))
          , a(c(x)) -> x
          , b(d(x)) -> x}

     Proof Output:
       The problem is match-bounded by 1.
       The enriched problem is compatible with the following automaton:
       {  a_0(2) -> 1
        , a_1(2) -> 5
        , a_1(8) -> 7
        , d_0(2) -> 1
        , d_0(2) -> 2
        , d_0(2) -> 4
        , d_0(2) -> 5
        , d_0(2) -> 7
        , d_0(2) -> 8
        , d_1(3) -> 1
        , d_1(3) -> 5
        , d_1(3) -> 7
        , d_1(7) -> 6
        , d_1(7) -> 7
        , c_0(2) -> 1
        , c_0(2) -> 2
        , c_0(2) -> 4
        , c_0(2) -> 5
        , c_0(2) -> 7
        , c_0(2) -> 8
        , c_1(4) -> 3
        , c_1(4) -> 4
        , c_1(6) -> 1
        , c_1(6) -> 4
        , c_1(6) -> 8
        , b_0(2) -> 1
        , b_1(2) -> 8
        , b_1(5) -> 4}