Problem Rubio 04 p266

Tool Bounds

Execution Time	4.2328835e-2ms
Answer	YES(?,O(n^1))
Input	Rubio 04 p266

stdout:

YES(?,O(n^1))

We consider the following Problem:

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

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

Proof:
  The problem is match-bounded by 3.
  The enriched problem is compatible with the following automaton:
  {  f_0(1) -> 1
   , f_1(1) -> 4
   , f_1(2) -> 1
   , f_1(2) -> 4
   , f_2(2) -> 7
   , f_2(5) -> 4
   , b_0(1) -> 1
   , b_1(1) -> 1
   , b_1(1) -> 4
   , b_1(4) -> 3
   , b_2(7) -> 6
   , a_0(1) -> 1
   , a_1(1) -> 1
   , a_1(3) -> 2
   , a_2(1) -> 1
   , a_2(1) -> 4
   , a_2(4) -> 3
   , a_2(6) -> 5
   , a_3(7) -> 6
   , g_0(1) -> 1}

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

Tool CDI

Execution Time	1.8474228ms
Answer	YES(?,O(n^2))
Input	Rubio 04 p266

stdout:

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

This TRS is terminating using the deltarestricted interpretation
g(delta, X0) =  + 0*X0 + 0 + 1*X0*delta + 2*delta
b(delta, X0) =  + 0*X0 + 0 + 1*X0*delta + 2*delta
a(delta, X0) =  + 0*X0 + 0 + 1*X0*delta + 0*delta
f(delta, X0) =  + 0*X0 + 3 + 2*X0*delta + 2*delta
g_tau_1(delta) = delta/(0 + 1 * delta)
b_tau_1(delta) = delta/(0 + 1 * delta)
a_tau_1(delta) = delta/(0 + 1 * delta)
f_tau_1(delta) = delta/(0 + 2 * delta)

Time: 1.807852 seconds
Statistics:
Number of monomials: 290
Last formula building started for bound 3
Last SAT solving started for bound 3

Tool EDA

Execution Time	0.595181ms
Answer	YES(?,O(n^2))
Input	Rubio 04 p266

stdout:

YES(?,O(n^2))

We consider the following Problem:

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

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

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

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

Tool IDA

Execution Time	0.791898ms
Answer	YES(?,O(n^2))
Input	Rubio 04 p266

stdout:

YES(?,O(n^2))

We consider the following Problem:

  Strict Trs:
    {  b(X) -> a(X)
     , f(a(g(X))) -> b(X)
     , f(f(X)) -> f(a(b(f(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:
   f(x1) = [1 1] x1 + [0]
           [0 1]      [3]
   b(x1) = [1 0] x1 + [1]
           [0 0]      [2]
   a(x1) = [1 0] x1 + [0]
           [0 0]      [1]
   g(x1) = [1 0] x1 + [3]
           [0 0]      [0]

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

Tool TRI

Execution Time	0.18569708ms
Answer	YES(?,O(n^2))
Input	Rubio 04 p266

stdout:

YES(?,O(n^2))

We consider the following Problem:

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

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

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

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

Tool TRI2

Execution Time	0.18313193ms
Answer	YES(?,O(n^2))
Input	Rubio 04 p266

stdout:

YES(?,O(n^2))

We consider the following Problem:

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

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

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

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