Problem SK90 4.44

Tool Bounds

Execution Time	4.9726963e-2ms
Answer	YES(?,O(n^1))
Input	SK90 4.44

stdout:

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  i(a()) -> b()
     , h(a()) -> b()
     , g(i(x)) -> g(h(x))
     , f(h(x)) -> f(i(x))}
  StartTerms: all
  Strategy: none

Certificate: YES(?,O(n^1))

Proof:
  The problem is match-bounded by 1.
  The enriched problem is compatible with the following automaton:
  {  h_0(1) -> 1
   , h_1(1) -> 2
   , f_0(1) -> 1
   , f_1(3) -> 1
   , i_0(1) -> 1
   , i_1(1) -> 3
   , g_0(1) -> 1
   , g_1(2) -> 1
   , a_0() -> 1
   , b_0() -> 1
   , b_1() -> 1
   , b_1() -> 2
   , b_1() -> 3}

Hurray, we answered YES(?,O(n^1))

Tool CDI

Execution Time	9.32951e-2ms
Answer	YES(?,O(n^2))
Input	SK90 4.44

stdout:

YES(?,O(n^2))
QUADRATIC upper bound on the derivational complexity

This TRS is terminating using the deltarestricted interpretation
a(delta) =  + 0 + 1*delta
b(delta) =  + 0 + 0*delta
g(delta, X0) =  + 0*X0 + 0 + 1*X0*delta + 0*delta
h(delta, X0) =  + 0*X0 + 1 + 2*X0*delta + 0*delta
i(delta, X0) =  + 0*X0 + 0 + 2*X0*delta + 2*delta
f(delta, X0) =  + 1*X0 + 0 + 3*X0*delta + 0*delta
g_tau_1(delta) = delta/(0 + 1 * delta)
h_tau_1(delta) = delta/(0 + 2 * delta)
i_tau_1(delta) = delta/(0 + 2 * delta)
f_tau_1(delta) = delta/(1 + 3 * delta)

Time: 0.054248 seconds
Statistics:
Number of monomials: 98
Last formula building started for bound 3
Last SAT solving started for bound 3

Tool EDA

Execution Time	0.3283038ms
Answer	YES(?,O(n^2))
Input	SK90 4.44

stdout:

YES(?,O(n^2))

We consider the following Problem:

  Strict Trs:
    {  i(a()) -> b()
     , h(a()) -> b()
     , g(i(x)) -> g(h(x))
     , f(h(x)) -> f(i(x))}
  StartTerms: all
  Strategy: none

Certificate: YES(?,O(n^2))

Proof:
  We have the following EDA-non-satisfying matrix interpretation:
  Interpretation Functions:
   h(x1) = [1 0] x1 + [3]
           [0 0]      [1]
   f(x1) = [1 0] x1 + [2]
           [0 0]      [1]
   i(x1) = [1 0] x1 + [2]
           [0 0]      [3]
   g(x1) = [1 2] x1 + [0]
           [0 0]      [1]
   a() = [2]
         [0]
   b() = [0]
         [1]

Hurray, we answered YES(?,O(n^2))

Tool IDA

Execution Time	0.53002ms
Answer	YES(?,O(n^2))
Input	SK90 4.44

stdout:

YES(?,O(n^2))

We consider the following Problem:

  Strict Trs:
    {  i(a()) -> b()
     , h(a()) -> b()
     , g(i(x)) -> g(h(x))
     , f(h(x)) -> f(i(x))}
  StartTerms: all
  Strategy: none

Certificate: YES(?,O(n^2))

Proof:
  We have the following EDA-non-satisfying and IDA(2)-non-satisfying matrix interpretation:
  Interpretation Functions:
   h(x1) = [1 0] x1 + [3]
           [0 0]      [1]
   f(x1) = [1 0] x1 + [2]
           [0 0]      [1]
   i(x1) = [1 0] x1 + [2]
           [0 0]      [3]
   g(x1) = [1 1] x1 + [0]
           [0 0]      [1]
   a() = [2]
         [0]
   b() = [0]
         [1]

Hurray, we answered YES(?,O(n^2))

Tool TRI

Execution Time	0.14532804ms
Answer	YES(?,O(n^2))
Input	SK90 4.44

stdout:

YES(?,O(n^2))

We consider the following Problem:

  Strict Trs:
    {  i(a()) -> b()
     , h(a()) -> b()
     , g(i(x)) -> g(h(x))
     , f(h(x)) -> f(i(x))}
  StartTerms: all
  Strategy: none

Certificate: YES(?,O(n^2))

Proof:
  We have the following triangular matrix interpretation:
  Interpretation Functions:
   h(x1) = [1 1] x1 + [2]
           [0 1]      [2]
   f(x1) = [1 3] x1 + [0]
           [0 1]      [3]
   i(x1) = [1 1] x1 + [3]
           [0 0]      [0]
   g(x1) = [1 0] x1 + [2]
           [0 0]      [3]
   a() = [2]
         [0]
   b() = [0]
         [0]

Hurray, we answered YES(?,O(n^2))

Tool TRI2

Execution Time	9.0991974e-2ms
Answer	YES(?,O(n^2))
Input	SK90 4.44

stdout:

YES(?,O(n^2))

We consider the following Problem:

  Strict Trs:
    {  i(a()) -> b()
     , h(a()) -> b()
     , g(i(x)) -> g(h(x))
     , f(h(x)) -> f(i(x))}
  StartTerms: all
  Strategy: none

Certificate: YES(?,O(n^2))

Proof:
  We have the following triangular matrix interpretation:
  Interpretation Functions:
   h(x1) = [1 1] x1 + [2]
           [0 1]      [2]
   f(x1) = [1 3] x1 + [0]
           [0 1]      [3]
   i(x1) = [1 1] x1 + [3]
           [0 0]      [0]
   g(x1) = [1 0] x1 + [2]
           [0 0]      [3]
   a() = [2]
         [0]
   b() = [0]
         [0]

Hurray, we answered YES(?,O(n^2))