Problem SK90 2.56

Tool Bounds

Execution Time	5.36232e-2ms
Answer	YES(?,O(n^1))
Input	SK90 2.56

stdout:

YES(?,O(n^1))

We consider the following Problem:

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

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

Proof:
  The problem is match-bounded by 2.
  The enriched problem is compatible with the following automaton:
  {  a_0() -> 1
   , a_1() -> 3
   , f_0(1, 1) -> 1
   , f_1(2, 1) -> 1
   , f_2(4, 1) -> 1
   , g_0(1, 1) -> 1
   , g_1(3, 1) -> 1
   , b_0() -> 1
   , b_1() -> 2
   , b_2() -> 4}

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

Tool CDI

Execution Time	0.17292905ms
Answer	YES(?,O(n^2))
Input	SK90 2.56

stdout:

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

This TRS is terminating using the deltarestricted interpretation
b(delta) =  + 0 + 0*delta
f(delta, X1, X0) =  + 0*X0 + 0*X1 + 0 + 2*X0*delta + 2*X1*delta + 2*delta
a(delta) =  + 0 + 3*delta
g(delta, X1, X0) =  + 0*X0 + 0*X1 + 0 + 2*X0*delta + 2*X1*delta + 0*delta
f_tau_1(delta) = delta/(0 + 2 * delta)
f_tau_2(delta) = delta/(0 + 2 * delta)
g_tau_1(delta) = delta/(0 + 2 * delta)
g_tau_2(delta) = delta/(0 + 2 * delta)

Time: 0.135726 seconds
Statistics:
Number of monomials: 136
Last formula building started for bound 3
Last SAT solving started for bound 3

Tool EDA

Execution Time	0.11097598ms
Answer	YES(?,O(n^1))
Input	SK90 2.56

stdout:

YES(?,O(n^1))

We consider the following Problem:

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

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

Proof:
  We have the following EDA-non-satisfying matrix interpretation:
  Interpretation Functions:
   a() = [3]
   f(x1, x2) = [1] x1 + [1] x2 + [3]
   g(x1, x2) = [1] x1 + [1] x2 + [2]
   b() = [1]

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

Tool IDA

Execution Time	0.310189ms
Answer	YES(?,O(n^1))
Input	SK90 2.56

stdout:

YES(?,O(n^1))

We consider the following Problem:

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

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

Proof:
  We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
  Interpretation Functions:
   a() = [3]
   f(x1, x2) = [1] x1 + [1] x2 + [3]
   g(x1, x2) = [1] x1 + [1] x2 + [2]
   b() = [1]

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

Tool TRI

Execution Time	7.109189e-2ms
Answer	YES(?,O(n^1))
Input	SK90 2.56

stdout:

YES(?,O(n^1))

We consider the following Problem:

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

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

Proof:
  We have the following triangular matrix interpretation:
  Interpretation Functions:
   a() = [3]
   f(x1, x2) = [1] x1 + [1] x2 + [3]
   g(x1, x2) = [1] x1 + [1] x2 + [2]
   b() = [1]

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

Tool TRI2

Execution Time	5.7843924e-2ms
Answer	YES(?,O(n^2))
Input	SK90 2.56

stdout:

YES(?,O(n^2))

We consider the following Problem:

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

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

Proof:
  We have the following triangular matrix interpretation:
  Interpretation Functions:
   a() = [2]
         [0]
   f(x1, x2) = [1 0] x1 + [1 3] x2 + [2]
               [0 0]      [0 1]      [0]
   g(x1, x2) = [1 3] x1 + [1 3] x2 + [1]
               [0 1]      [0 1]      [0]
   b() = [0]
         [3]

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