Tool CaT
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 4.53 |
|---|
stdout:
MAYBE
Problem:
f(a()) -> b()
f(c()) -> d()
f(g(x,y)) -> g(f(x),f(y))
f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
g(x,x) -> h(e(),x)
Proof:
OpenTool IRC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 4.53 |
|---|
stdout:
MAYBE
Tool IRC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 4.53 |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(a()) -> c_0()
, 2: f^#(c()) -> c_1()
, 3: f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
, 4: f^#(h(x, y)) -> c_3(g^#(h(y, f(x)), h(x, f(y))))
, 5: g^#(x, x) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
`->{5} [ NA ]
->{3} [ inherited ]
|
`->{5} [ MAYBE ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
d() = [0]
[0]
[0]
g(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
h(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
e() = [0]
[0]
[0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(a()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a() = [2]
[2]
[2]
f^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_0() = [0]
[1]
[1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
d() = [0]
[0]
[0]
g(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
h(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
e() = [0]
[0]
[0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(c()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
c() = [2]
[2]
[2]
f^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_1() = [0]
[1]
[1]
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{5}.
* Path {3}->{5}: MAYBE
--------------------
The usable rules for this path are:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
, g^#(x, x) -> c_4()
, f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
Proof Output:
The input cannot be shown compatible
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{5}.
* Path {4}->{5}: NA
-----------------
The usable rules for this path are:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(a()) -> c_0()
, 2: f^#(c()) -> c_1()
, 3: f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
, 4: f^#(h(x, y)) -> c_3(g^#(h(y, f(x)), h(x, f(y))))
, 5: g^#(x, x) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
`->{5} [ NA ]
->{3} [ inherited ]
|
`->{5} [ MAYBE ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
d() = [0]
[0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
e() = [0]
[0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(a()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a() = [2]
[2]
f^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_0() = [0]
[1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
d() = [0]
[0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
e() = [0]
[0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(c()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
c() = [2]
[2]
f^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_1() = [0]
[1]
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{5}.
* Path {3}->{5}: MAYBE
--------------------
The usable rules for this path are:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
, g^#(x, x) -> c_4()
, f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
Proof Output:
The input cannot be shown compatible
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{5}.
* Path {4}->{5}: NA
-----------------
The usable rules for this path are:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(a()) -> c_0()
, 2: f^#(c()) -> c_1()
, 3: f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
, 4: f^#(h(x, y)) -> c_3(g^#(h(y, f(x)), h(x, f(y))))
, 5: g^#(x, x) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
`->{5} [ NA ]
->{3} [ inherited ]
|
`->{5} [ MAYBE ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
a() = [0]
b() = [0]
c() = [0]
d() = [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
h(x1, x2) = [0] x1 + [0] x2 + [0]
e() = [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(a()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a() = [7]
f^#(x1) = [1] x1 + [7]
c_0() = [1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
a() = [0]
b() = [0]
c() = [0]
d() = [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
h(x1, x2) = [0] x1 + [0] x2 + [0]
e() = [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(c()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
c() = [7]
f^#(x1) = [1] x1 + [7]
c_1() = [1]
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{5}.
* Path {3}->{5}: MAYBE
--------------------
The usable rules for this path are:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
, g^#(x, x) -> c_4()
, f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
Proof Output:
The input cannot be shown compatible
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{5}.
* Path {4}->{5}: NA
-----------------
The usable rules for this path are:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 4.53 |
|---|
stdout:
MAYBE
Tool RC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 4.53 |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(a()) -> c_0()
, 2: f^#(c()) -> c_1()
, 3: f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
, 4: f^#(h(x, y)) -> c_3(g^#(h(y, f(x)), h(x, f(y))))
, 5: g^#(x, x) -> c_4(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
`->{5} [ NA ]
->{3} [ inherited ]
|
`->{5} [ MAYBE ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
d() = [0]
[0]
[0]
g(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
h(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
e() = [0]
[0]
[0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(a()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a() = [2]
[2]
[2]
f^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_0() = [0]
[1]
[1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
d() = [0]
[0]
[0]
g(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
h(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
e() = [0]
[0]
[0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(c()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
c() = [2]
[2]
[2]
f^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_1() = [0]
[1]
[1]
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{5}.
* Path {3}->{5}: MAYBE
--------------------
The usable rules for this path are:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
, g^#(x, x) -> c_4(x)
, f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
Proof Output:
The input cannot be shown compatible
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{5}.
* Path {4}->{5}: NA
-----------------
The usable rules for this path are:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(a()) -> c_0()
, 2: f^#(c()) -> c_1()
, 3: f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
, 4: f^#(h(x, y)) -> c_3(g^#(h(y, f(x)), h(x, f(y))))
, 5: g^#(x, x) -> c_4(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
`->{5} [ NA ]
->{3} [ inherited ]
|
`->{5} [ MAYBE ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
d() = [0]
[0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
e() = [0]
[0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(a()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a() = [2]
[2]
f^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_0() = [0]
[1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
d() = [0]
[0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
e() = [0]
[0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(c()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
c() = [2]
[2]
f^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_1() = [0]
[1]
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{5}.
* Path {3}->{5}: MAYBE
--------------------
The usable rules for this path are:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
, g^#(x, x) -> c_4(x)
, f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
Proof Output:
The input cannot be shown compatible
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{5}.
* Path {4}->{5}: NA
-----------------
The usable rules for this path are:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(a()) -> c_0()
, 2: f^#(c()) -> c_1()
, 3: f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
, 4: f^#(h(x, y)) -> c_3(g^#(h(y, f(x)), h(x, f(y))))
, 5: g^#(x, x) -> c_4(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
`->{5} [ NA ]
->{3} [ inherited ]
|
`->{5} [ MAYBE ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
a() = [0]
b() = [0]
c() = [0]
d() = [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
h(x1, x2) = [0] x1 + [0] x2 + [0]
e() = [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(a()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a() = [7]
f^#(x1) = [1] x1 + [7]
c_0() = [1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
a() = [0]
b() = [0]
c() = [0]
d() = [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
h(x1, x2) = [0] x1 + [0] x2 + [0]
e() = [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(c()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
c() = [7]
f^#(x1) = [1] x1 + [7]
c_1() = [1]
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{5}.
* Path {3}->{5}: MAYBE
--------------------
The usable rules for this path are:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
, g^#(x, x) -> c_4(x)
, f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
Proof Output:
The input cannot be shown compatible
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{5}.
* Path {4}->{5}: NA
-----------------
The usable rules for this path are:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool pair1rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 4.53 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair1 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair2rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 4.53 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3irc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 4.53 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 4.53 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool tup3irc
| Execution Time | 60.825024ms |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 4.53 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x)}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'tup3 (timeout of 60.0 seconds)':
------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..