Tool CaT
stdout:
MAYBE
Problem:
f(0(),1(),x) -> f(h(x),h(x),x)
h(0()) -> 0()
h(g(x,y)) -> y
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(0(), 1(), x) -> f(h(x), h(x), x)
, h(0()) -> 0()
, h(g(x, y)) -> y}
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^#(0(), 1(), x) -> c_0(f^#(h(x), h(x), x))
, 2: h^#(0()) -> c_1()
, 3: h^#(g(x, y)) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ h(0()) -> 0()
, h(g(x, y)) -> y}
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^#(0(), 1(), x) -> c_0(f^#(h(x), h(x), x))
, h(0()) -> 0()
, h(g(x, y)) -> y}
Proof Output:
The input cannot be shown compatible
* 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(h) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
1() = [0]
[0]
[0]
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]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 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]
h^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [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: {h^#(0()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
h^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_1() = [0]
[1]
[1]
* Path {3}: 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(h) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
1() = [0]
[0]
[0]
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]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 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]
h^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [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: {h^#(g(x, y)) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g) = {}, Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 0] [2]
[0 0 0] [0 0 0] [2]
h^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_2() = [0]
[1]
[1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(0(), 1(), x) -> c_0(f^#(h(x), h(x), x))
, 2: h^#(0()) -> c_1()
, 3: h^#(g(x, y)) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ h(0()) -> 0()
, h(g(x, y)) -> y}
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^#(0(), 1(), x) -> c_0(f^#(h(x), h(x), x))
, h(0()) -> 0()
, h(g(x, y)) -> y}
Proof Output:
The input cannot be shown compatible
* 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(h) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
1() = [0]
[0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
h^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2() = [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: {h^#(0()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
h^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_1() = [0]
[1]
* Path {3}: 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(h) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
1() = [0]
[0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
h^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2() = [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: {h^#(g(x, y)) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g) = {}, Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
h^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_2() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(0(), 1(), x) -> c_0(f^#(h(x), h(x), x))
, 2: h^#(0()) -> c_1()
, 3: h^#(g(x, y)) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ h(0()) -> 0()
, h(g(x, y)) -> y}
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^#(0(), 1(), x) -> c_0(f^#(h(x), h(x), x))
, h(0()) -> 0()
, h(g(x, y)) -> y}
Proof Output:
The input cannot be shown compatible
* 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(h) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
1() = [0]
h(x1) = [0] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [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: {h^#(0()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
h^#(x1) = [1] x1 + [7]
c_1() = [1]
* Path {3}: 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(h) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
1() = [0]
h(x1) = [0] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [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: {h^#(g(x, y)) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g) = {}, Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2) = [0] x1 + [0] x2 + [7]
h^#(x1) = [1] x1 + [7]
c_2() = [1]
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:
{ f(0(), 1(), x) -> f(h(x), h(x), x)
, h(0()) -> 0()
, h(g(x, y)) -> y}
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^#(0(), 1(), x) -> c_0(f^#(h(x), h(x), x))
, 2: h^#(0()) -> c_1()
, 3: h^#(g(x, y)) -> c_2(y)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(n^3)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ h(0()) -> 0()
, h(g(x, y)) -> y}
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^#(0(), 1(), x) -> c_0(f^#(h(x), h(x), x))
, h(0()) -> 0()
, h(g(x, y)) -> y}
Proof Output:
The input cannot be shown compatible
* 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(h) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
1() = [0]
[0]
[0]
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]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 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]
h^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[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]
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: {h^#(0()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
h^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_1() = [0]
[1]
[1]
* Path {3}: YES(?,O(n^3))
-----------------------
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(h) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
1() = [0]
[0]
[0]
h(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(x1, x2) = [1 3 3] x1 + [0 0 0] x2 + [0]
[0 1 1] [0 0 0] [0]
[0 0 1] [0 0 0] [0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 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]
h^#(x1) = [1 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 1] 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(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {h^#(g(x, y)) -> c_2(y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g) = {}, Uargs(h^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
[0 0 0] [0 0 2] [2]
[0 0 0] [0 0 0] [2]
h^#(x1) = [2 2 2] x1 + [3]
[2 2 2] [3]
[2 2 2] [3]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(0(), 1(), x) -> c_0(f^#(h(x), h(x), x))
, 2: h^#(0()) -> c_1()
, 3: h^#(g(x, y)) -> c_2(y)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(n^2)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ h(0()) -> 0()
, h(g(x, y)) -> y}
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^#(0(), 1(), x) -> c_0(f^#(h(x), h(x), x))
, h(0()) -> 0()
, h(g(x, y)) -> y}
Proof Output:
The input cannot be shown compatible
* 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(h) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
1() = [0]
[0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
h^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(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: {h^#(0()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
h^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_1() = [0]
[1]
* Path {3}: YES(?,O(n^2))
-----------------------
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(h) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
1() = [0]
[0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
h^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {h^#(g(x, y)) -> c_2(y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g) = {}, Uargs(h^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
[0 0] [0 0] [2]
h^#(x1) = [2 2] x1 + [7]
[2 0] [7]
c_2(x1) = [0 0] x1 + [0]
[0 0] [1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(0(), 1(), x) -> c_0(f^#(h(x), h(x), x))
, 2: h^#(0()) -> c_1()
, 3: h^#(g(x, y)) -> c_2(y)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ h(0()) -> 0()
, h(g(x, y)) -> y}
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^#(0(), 1(), x) -> c_0(f^#(h(x), h(x), x))
, h(0()) -> 0()
, h(g(x, y)) -> y}
Proof Output:
The input cannot be shown compatible
* 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(h) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
1() = [0]
h(x1) = [0] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(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: {h^#(0()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
h^#(x1) = [1] x1 + [7]
c_1() = [1]
* Path {3}: YES(?,O(n^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(h) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
1() = [0]
h(x1) = [0] x1 + [0]
g(x1, x2) = [1] x1 + [0] x2 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
h^#(x1) = [3] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {h^#(g(x, y)) -> c_2(y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g) = {}, Uargs(h^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2) = [0] x1 + [1] x2 + [7]
h^#(x1) = [1] x1 + [7]
c_2(x1) = [1] x1 + [1]
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.