Tool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ h(X) -> g(X, X)
, g(a(), X) -> f(b(), X)
, f(X, X) -> h(a())
, a() -> b()}
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: h^#(X) -> c_0(g^#(X, X))
, 2: g^#(a(), X) -> c_1(f^#(b(), X))
, 3: f^#(X, X) -> c_2(h^#(a()))
, 4: a^#() -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{a() -> b()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(h^#) = {1},
Uargs(c_0) = {1}, Uargs(g^#) = {}, Uargs(c_1) = {1},
Uargs(f^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(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]
a() = [3]
[3]
[3]
f(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]
b() = [1]
[1]
[1]
h^#(x1) = [3 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
g^#(x1, x2) = [1 1 0] x1 + [1 1 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
f^#(x1, x2) = [0 0 0] x1 + [1 1 1] x2 + [0]
[0 0 0] [3 3 3] [0]
[0 0 0] [3 3 3] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
a^#() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ h^#(X) -> c_0(g^#(X, X))
, f^#(X, X) -> c_2(h^#(a()))
, g^#(a(), X) -> c_1(f^#(b(), X))}
Weak Rules: {a() -> b()}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(h^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(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]
a() = [0]
[0]
[0]
f(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]
b() = [0]
[0]
[0]
h^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a^#() = [0]
[0]
[0]
c_3() = [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: {a^#() -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
[7]
[7]
c_3() = [0]
[3]
[3]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: h^#(X) -> c_0(g^#(X, X))
, 2: g^#(a(), X) -> c_1(f^#(b(), X))
, 3: f^#(X, X) -> c_2(h^#(a()))
, 4: a^#() -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{a() -> b()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(h^#) = {1},
Uargs(c_0) = {1}, Uargs(g^#) = {}, Uargs(c_1) = {1},
Uargs(f^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
a() = [3]
[3]
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
b() = [1]
[1]
h^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
g^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
f^#(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [3 3] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
a^#() = [0]
[0]
c_3() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ h^#(X) -> c_0(g^#(X, X))
, f^#(X, X) -> c_2(h^#(a()))
, g^#(a(), X) -> c_1(f^#(b(), X))}
Weak Rules: {a() -> b()}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(h^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
a() = [0]
[0]
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
b() = [0]
[0]
h^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
a^#() = [0]
[0]
c_3() = [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: {a^#() -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
[7]
c_3() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: h^#(X) -> c_0(g^#(X, X))
, 2: g^#(a(), X) -> c_1(f^#(b(), X))
, 3: f^#(X, X) -> c_2(h^#(a()))
, 4: a^#() -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{a() -> b()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(h^#) = {1},
Uargs(c_0) = {1}, Uargs(g^#) = {}, Uargs(c_1) = {1},
Uargs(f^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1) = [0] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
a() = [3]
f(x1, x2) = [0] x1 + [0] x2 + [0]
b() = [0]
h^#(x1) = [3] x1 + [0]
c_0(x1) = [1] x1 + [0]
g^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [1] x1 + [0]
f^#(x1, x2) = [0] x1 + [1] x2 + [0]
c_2(x1) = [1] x1 + [0]
a^#() = [0]
c_3() = [0]
Complexity induced by the adequate RMI: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ h^#(X) -> c_0(g^#(X, X))
, f^#(X, X) -> c_2(h^#(a()))
, g^#(a(), X) -> c_1(f^#(b(), X))}
Weak Rules: {a() -> b()}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(h^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1) = [0] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
a() = [0]
f(x1, x2) = [0] x1 + [0] x2 + [0]
b() = [0]
h^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
a^#() = [0]
c_3() = [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: {a^#() -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
c_3() = [0]
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
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ h(X) -> g(X, X)
, g(a(), X) -> f(b(), X)
, f(X, X) -> h(a())
, a() -> b()}
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: h^#(X) -> c_0(g^#(X, X))
, 2: g^#(a(), X) -> c_1(f^#(b(), X))
, 3: f^#(X, X) -> c_2(h^#(a()))
, 4: a^#() -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{a() -> b()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(h^#) = {1},
Uargs(c_0) = {1}, Uargs(g^#) = {1, 2}, Uargs(c_1) = {1},
Uargs(f^#) = {2}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(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]
a() = [3]
[3]
[3]
f(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]
b() = [1]
[1]
[1]
h^#(x1) = [3 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
g^#(x1, x2) = [1 1 0] x1 + [1 1 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
f^#(x1, x2) = [0 0 0] x1 + [1 1 1] x2 + [0]
[0 0 0] [3 3 3] [0]
[0 0 0] [3 3 3] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
a^#() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ h^#(X) -> c_0(g^#(X, X))
, f^#(X, X) -> c_2(h^#(a()))
, g^#(a(), X) -> c_1(f^#(b(), X))}
Weak Rules: {a() -> b()}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(h^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(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]
a() = [0]
[0]
[0]
f(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]
b() = [0]
[0]
[0]
h^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a^#() = [0]
[0]
[0]
c_3() = [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: {a^#() -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
[7]
[7]
c_3() = [0]
[3]
[3]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: h^#(X) -> c_0(g^#(X, X))
, 2: g^#(a(), X) -> c_1(f^#(b(), X))
, 3: f^#(X, X) -> c_2(h^#(a()))
, 4: a^#() -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{a() -> b()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(h^#) = {1},
Uargs(c_0) = {1}, Uargs(g^#) = {1, 2}, Uargs(c_1) = {1},
Uargs(f^#) = {2}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
a() = [3]
[3]
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
b() = [1]
[1]
h^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
g^#(x1, x2) = [1 0] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
f^#(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [3 3] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
a^#() = [0]
[0]
c_3() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ h^#(X) -> c_0(g^#(X, X))
, f^#(X, X) -> c_2(h^#(a()))
, g^#(a(), X) -> c_1(f^#(b(), X))}
Weak Rules: {a() -> b()}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(h^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
a() = [0]
[0]
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
b() = [0]
[0]
h^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
a^#() = [0]
[0]
c_3() = [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: {a^#() -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
[7]
c_3() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: h^#(X) -> c_0(g^#(X, X))
, 2: g^#(a(), X) -> c_1(f^#(b(), X))
, 3: f^#(X, X) -> c_2(h^#(a()))
, 4: a^#() -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{a() -> b()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(h^#) = {1},
Uargs(c_0) = {1}, Uargs(g^#) = {1, 2}, Uargs(c_1) = {1},
Uargs(f^#) = {2}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1) = [0] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
a() = [3]
f(x1, x2) = [0] x1 + [0] x2 + [0]
b() = [2]
h^#(x1) = [3] x1 + [0]
c_0(x1) = [1] x1 + [0]
g^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [1] x1 + [0]
f^#(x1, x2) = [0] x1 + [1] x2 + [0]
c_2(x1) = [1] x1 + [0]
a^#() = [0]
c_3() = [0]
Complexity induced by the adequate RMI: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ h^#(X) -> c_0(g^#(X, X))
, f^#(X, X) -> c_2(h^#(a()))
, g^#(a(), X) -> c_1(f^#(b(), X))}
Weak Rules: {a() -> b()}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(h^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1) = [0] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
a() = [0]
f(x1, x2) = [0] x1 + [0] x2 + [0]
b() = [0]
h^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
a^#() = [0]
c_3() = [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: {a^#() -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
c_3() = [0]
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.