Tool CaT
stdout:
MAYBE
Problem:
f(h(x,x)) -> f(i(x))
f(i(x)) -> a()
i(x) -> h(x,x)
Proof:
OpenTool IRC1
stdout:
MAYBE
Warning when parsing problem:
Unsupported strategy 'OUTERMOST'Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(h(x, x)) -> f(i(x))
, f(i(x)) -> a()
, i(x) -> h(x, 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^#(h(x, x)) -> c_0(f^#(i(x)))
, 2: f^#(i(x)) -> c_1()
, 3: i^#(x) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{i(x) -> h(x, x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h) = {}, Uargs(i) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(i^#) = {}
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]
h(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [1]
[0 0 0] [0 0 0] [1]
[0 0 0] [0 0 0] [1]
i(x1) = [1 0 0] x1 + [3]
[3 0 0] [3]
[3 0 0] [3]
a() = [0]
[0]
[0]
f^#(x1) = [1 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_1() = [0]
[0]
[0]
i^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^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: {f^#(h(x, x)) -> c_0(f^#(i(x)))}
Weak Rules: {i(x) -> h(x, x)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{2}: NA
-----------------
The usable rules for this path are:
{i(x) -> h(x, x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h) = {}, Uargs(i) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(i^#) = {}
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]
h(x1, x2) = [1 2 1] x1 + [1 1 1] x2 + [0]
[0 1 0] [0 1 3] [1]
[0 0 1] [0 0 1] [1]
i(x1) = [3 3 3] x1 + [3]
[0 3 3] [3]
[0 0 3] [3]
a() = [0]
[0]
[0]
f^#(x1) = [3 0 0] 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]
c_1() = [0]
[0]
[0]
i^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* 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(i) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(i^#) = {}
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]
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]
i(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
f^#(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]
c_1() = [0]
[0]
[0]
i^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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: {i^#(x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(i^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
i^#(x1) = [0 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
c_2() = [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: f^#(h(x, x)) -> c_0(f^#(i(x)))
, 2: f^#(i(x)) -> c_1()
, 3: i^#(x) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{i(x) -> h(x, x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h) = {}, Uargs(i) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(i^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
h(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
i(x1) = [1 0] x1 + [3]
[3 0] [3]
a() = [0]
[0]
f^#(x1) = [1 0] x1 + [0]
[3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1() = [0]
[0]
i^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^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: {f^#(h(x, x)) -> c_0(f^#(i(x)))}
Weak Rules: {i(x) -> h(x, x)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{2}: NA
-----------------
The usable rules for this path are:
{i(x) -> h(x, x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h) = {}, Uargs(i) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(i^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
h(x1, x2) = [1 0] x1 + [1 3] x2 + [0]
[0 1] [0 1] [1]
i(x1) = [3 3] x1 + [3]
[0 3] [3]
a() = [0]
[0]
f^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1() = [0]
[0]
i^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* 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(i) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(i^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
i(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
i^#(x1) = [0 0] x1 + [0]
[0 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: {i^#(x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(i^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
i^#(x1) = [0 0] x1 + [7]
[0 0] [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^#(h(x, x)) -> c_0(f^#(i(x)))
, 2: f^#(i(x)) -> c_1()
, 3: i^#(x) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{i(x) -> h(x, x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h) = {}, Uargs(i) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(i^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
h(x1, x2) = [1] x1 + [1] x2 + [1]
i(x1) = [2] x1 + [3]
a() = [0]
f^#(x1) = [3] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1() = [0]
i^#(x1) = [0] x1 + [0]
c_2() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^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: {f^#(h(x, x)) -> c_0(f^#(i(x)))}
Weak Rules: {i(x) -> h(x, x)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{2}: NA
-----------------
The usable rules for this path are:
{i(x) -> h(x, x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h) = {}, Uargs(i) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(i^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
h(x1, x2) = [1] x1 + [1] x2 + [0]
i(x1) = [3] x1 + [3]
a() = [0]
f^#(x1) = [3] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1() = [0]
i^#(x1) = [0] x1 + [0]
c_2() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* 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(i) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(i^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
h(x1, x2) = [0] x1 + [0] x2 + [0]
i(x1) = [0] x1 + [0]
a() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
i^#(x1) = [0] x1 + [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: {i^#(x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(i^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
i^#(x1) = [0] x1 + [7]
c_2() = [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
Warning when parsing problem:
Unsupported strategy 'OUTERMOST'Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f(h(x, x)) -> f(i(x))
, f(i(x)) -> a()
, i(x) -> h(x, 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^#(h(x, x)) -> c_0(f^#(i(x)))
, 2: f^#(i(x)) -> c_1()
, 3: i^#(x) -> c_2(x, x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{i(x) -> h(x, x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h) = {}, Uargs(i) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(i^#) = {}, Uargs(c_2) = {}
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]
h(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [1]
[0 0 0] [0 0 0] [1]
[0 0 0] [0 0 0] [1]
i(x1) = [1 0 0] x1 + [3]
[3 0 0] [3]
[3 0 0] [3]
a() = [0]
[0]
[0]
f^#(x1) = [1 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_1() = [0]
[0]
[0]
i^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(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]
Complexity induced by the adequate RMI: YES(?,O(n^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: {f^#(h(x, x)) -> c_0(f^#(i(x)))}
Weak Rules: {i(x) -> h(x, x)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{2}: NA
-----------------
The usable rules for this path are:
{i(x) -> h(x, x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h) = {}, Uargs(i) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(i^#) = {}, Uargs(c_2) = {}
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]
h(x1, x2) = [1 2 1] x1 + [1 1 1] x2 + [0]
[0 1 0] [0 1 3] [1]
[0 0 1] [0 0 1] [1]
i(x1) = [3 3 3] x1 + [3]
[0 3 3] [3]
[0 0 3] [3]
a() = [0]
[0]
[0]
f^#(x1) = [3 0 0] 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]
c_1() = [0]
[0]
[0]
i^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(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]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* 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(i) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(i^#) = {}, Uargs(c_2) = {}
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]
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]
i(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
f^#(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]
c_1() = [0]
[0]
[0]
i^#(x1) = [3 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1, x2) = [1 2 2] x1 + [1 1 1] x2 + [0]
[0 0 0] [0 0 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: {i^#(x) -> c_2(x, x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(i^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
i^#(x1) = [7 7 7] x1 + [7]
[7 7 7] [7]
[7 7 7] [7]
c_2(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [1]
[3 3 3] [3 3 3] [1]
[3 3 3] [3 3 3] [1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(h(x, x)) -> c_0(f^#(i(x)))
, 2: f^#(i(x)) -> c_1()
, 3: i^#(x) -> c_2(x, x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{i(x) -> h(x, x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h) = {}, Uargs(i) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(i^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
h(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
i(x1) = [1 0] x1 + [3]
[3 0] [3]
a() = [0]
[0]
f^#(x1) = [1 0] x1 + [0]
[3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1() = [0]
[0]
i^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^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: {f^#(h(x, x)) -> c_0(f^#(i(x)))}
Weak Rules: {i(x) -> h(x, x)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{2}: NA
-----------------
The usable rules for this path are:
{i(x) -> h(x, x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h) = {}, Uargs(i) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(i^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
h(x1, x2) = [1 0] x1 + [1 3] x2 + [0]
[0 1] [0 1] [1]
i(x1) = [3 3] x1 + [3]
[0 3] [3]
a() = [0]
[0]
f^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1() = [0]
[0]
i^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* 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(i) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(i^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
i(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
i^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_2(x1, x2) = [0 2] x1 + [3 1] x2 + [0]
[0 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: {i^#(x) -> c_2(x, x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(i^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
i^#(x1) = [7 7] x1 + [7]
[7 7] [7]
c_2(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(h(x, x)) -> c_0(f^#(i(x)))
, 2: f^#(i(x)) -> c_1()
, 3: i^#(x) -> c_2(x, x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{i(x) -> h(x, x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h) = {}, Uargs(i) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(i^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
h(x1, x2) = [1] x1 + [1] x2 + [1]
i(x1) = [2] x1 + [3]
a() = [0]
f^#(x1) = [3] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1() = [0]
i^#(x1) = [0] x1 + [0]
c_2(x1, x2) = [0] x1 + [0] x2 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^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: {f^#(h(x, x)) -> c_0(f^#(i(x)))}
Weak Rules: {i(x) -> h(x, x)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{2}: NA
-----------------
The usable rules for this path are:
{i(x) -> h(x, x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h) = {}, Uargs(i) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(i^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
h(x1, x2) = [1] x1 + [1] x2 + [0]
i(x1) = [3] x1 + [3]
a() = [0]
f^#(x1) = [3] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1() = [0]
i^#(x1) = [0] x1 + [0]
c_2(x1, x2) = [0] x1 + [0] x2 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* 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(i) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(i^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
h(x1, x2) = [0] x1 + [0] x2 + [0]
i(x1) = [0] x1 + [0]
a() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
i^#(x1) = [3] x1 + [0]
c_2(x1, x2) = [0] x1 + [3] x2 + [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: {i^#(x) -> c_2(x, x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(i^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
i^#(x1) = [7] x1 + [7]
c_2(x1, x2) = [3] x1 + [3] x2 + [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.