Tool CaT
stdout:
MAYBE
Problem:
g(tt(),x) -> g(h(f(x,x)),s(x))
h(f(0(),y)) -> tt()
f(s(x),y) -> f(x,s(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:
{ g(tt(), x) -> g(h(f(x, x)), s(x))
, h(f(0(), y)) -> tt()
, f(s(x), y) -> f(x, s(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: g^#(tt(), x) -> c_0(g^#(h(f(x, x)), s(x)))
, 2: h^#(f(0(), y)) -> c_1()
, 3: f^#(s(x), y) -> c_2(f^#(x, s(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} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ h(f(0(), y)) -> tt()
, f(s(x), y) -> f(x, s(y))}
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.
* 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(g) = {}, Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {},
Uargs(g^#) = {}, Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
tt() = [0]
[0]
[0]
h(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [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_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]
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]
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^#(f(0(), y)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [2 1 0] x1 + [0 0 0] x2 + [2]
[2 0 2] [0 0 0] [0]
[2 0 0] [0 0 0] [2]
0() = [0]
[2]
[0]
h^#(x1) = [2 0 0] x1 + [7]
[0 0 0] [7]
[0 2 2] [7]
c_1() = [0]
[1]
[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(g) = {}, Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {},
Uargs(g^#) = {}, Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
tt() = [0]
[0]
[0]
h(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]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [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_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]
f^#(x1, x2) = [0 0 0] x1 + [0 3 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(s(x), y) -> c_2(f^#(x, s(y)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 0 2] [2]
[0 0 0] [2]
f^#(x1, x2) = [2 2 1] x1 + [0 0 2] x2 + [0]
[1 2 0] [0 0 2] [2]
[2 2 2] [4 0 4] [0]
c_2(x1) = [1 0 0] x1 + [5]
[0 0 0] [7]
[0 2 0] [0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: g^#(tt(), x) -> c_0(g^#(h(f(x, x)), s(x)))
, 2: h^#(f(0(), y)) -> c_1()
, 3: f^#(s(x), y) -> c_2(f^#(x, s(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(f(0(), y)) -> tt()
, f(s(x), y) -> f(x, s(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:
{ g^#(tt(), x) -> c_0(g^#(h(f(x, x)), s(x)))
, h(f(0(), y)) -> tt()
, f(s(x), y) -> f(x, s(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(g) = {}, Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {},
Uargs(g^#) = {}, Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
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]
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^#(f(0(), y)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[3 0] [0 0] [0]
0() = [2]
[0]
h^#(x1) = [0 2] x1 + [3]
[0 0] [7]
c_1() = [0]
[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(g) = {}, Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {},
Uargs(g^#) = {}, Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
f^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(s(x), y) -> c_2(f^#(x, s(y)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0] x1 + [0]
[0 1] [2]
f^#(x1, x2) = [0 1] x1 + [6 0] x2 + [0]
[0 2] [0 4] [4]
c_2(x1) = [1 0] x1 + [1]
[2 0] [7]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: g^#(tt(), x) -> c_0(g^#(h(f(x, x)), s(x)))
, 2: h^#(f(0(), y)) -> c_1()
, 3: f^#(s(x), y) -> c_2(f^#(x, s(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(f(0(), y)) -> tt()
, f(s(x), y) -> f(x, s(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:
{ g^#(tt(), x) -> c_0(g^#(h(f(x, x)), s(x)))
, h(f(0(), y)) -> tt()
, f(s(x), y) -> f(x, s(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(g) = {}, Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {},
Uargs(g^#) = {}, Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
h(x1) = [0] x1 + [0]
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [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: innermost DP runtime-complexity with respect to
Strict Rules: {h^#(f(0(), y)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [2] x1 + [0] x2 + [0]
0() = [0]
h^#(x1) = [2] x1 + [7]
c_1() = [0]
* 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(g) = {}, Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {},
Uargs(g^#) = {}, Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
h(x1) = [0] x1 + [0]
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
f^#(x1, x2) = [3] x1 + [3] x2 + [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: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(s(x), y) -> c_2(f^#(x, s(y)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
f^#(x1, x2) = [4] x1 + [0] x2 + [0]
c_2(x1) = [1] x1 + [7]
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:
{ g(tt(), x) -> g(h(f(x, x)), s(x))
, h(f(0(), y)) -> tt()
, f(s(x), y) -> f(x, s(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: g^#(tt(), x) -> c_0(g^#(h(f(x, x)), s(x)))
, 2: h^#(f(0(), y)) -> c_1()
, 3: f^#(s(x), y) -> c_2(f^#(x, s(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} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ h(f(0(), y)) -> tt()
, f(s(x), y) -> f(x, s(y))}
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.
* 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(g) = {}, Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {},
Uargs(g^#) = {}, Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
tt() = [0]
[0]
[0]
h(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [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_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]
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]
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^#(f(0(), y)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [2 1 0] x1 + [0 0 0] x2 + [2]
[2 0 2] [0 0 0] [0]
[2 0 0] [0 0 0] [2]
0() = [0]
[2]
[0]
h^#(x1) = [2 0 0] x1 + [7]
[0 0 0] [7]
[0 2 2] [7]
c_1() = [0]
[1]
[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(g) = {}, Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {},
Uargs(g^#) = {}, Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
tt() = [0]
[0]
[0]
h(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]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [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_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]
f^#(x1, x2) = [0 0 0] x1 + [0 3 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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: {f^#(s(x), y) -> c_2(f^#(x, s(y)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 0 2] [2]
[0 0 0] [2]
f^#(x1, x2) = [2 2 1] x1 + [0 0 2] x2 + [0]
[1 2 0] [0 0 2] [2]
[2 2 2] [4 0 4] [0]
c_2(x1) = [1 0 0] x1 + [5]
[0 0 0] [7]
[0 2 0] [0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: g^#(tt(), x) -> c_0(g^#(h(f(x, x)), s(x)))
, 2: h^#(f(0(), y)) -> c_1()
, 3: f^#(s(x), y) -> c_2(f^#(x, s(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(f(0(), y)) -> tt()
, f(s(x), y) -> f(x, s(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:
{ g^#(tt(), x) -> c_0(g^#(h(f(x, x)), s(x)))
, h(f(0(), y)) -> tt()
, f(s(x), y) -> f(x, s(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(g) = {}, Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {},
Uargs(g^#) = {}, Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
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]
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^#(f(0(), y)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[3 0] [0 0] [0]
0() = [2]
[0]
h^#(x1) = [0 2] x1 + [3]
[0 0] [7]
c_1() = [0]
[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(g) = {}, Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {},
Uargs(g^#) = {}, Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
f^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [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: {f^#(s(x), y) -> c_2(f^#(x, s(y)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0] x1 + [0]
[0 1] [2]
f^#(x1, x2) = [0 1] x1 + [6 0] x2 + [0]
[0 2] [0 4] [4]
c_2(x1) = [1 0] x1 + [1]
[2 0] [7]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: g^#(tt(), x) -> c_0(g^#(h(f(x, x)), s(x)))
, 2: h^#(f(0(), y)) -> c_1()
, 3: f^#(s(x), y) -> c_2(f^#(x, s(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(f(0(), y)) -> tt()
, f(s(x), y) -> f(x, s(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:
{ g^#(tt(), x) -> c_0(g^#(h(f(x, x)), s(x)))
, h(f(0(), y)) -> tt()
, f(s(x), y) -> f(x, s(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(g) = {}, Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {},
Uargs(g^#) = {}, Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
h(x1) = [0] x1 + [0]
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [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^#(f(0(), y)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [2] x1 + [0] x2 + [0]
0() = [0]
h^#(x1) = [2] x1 + [7]
c_1() = [0]
* 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(g) = {}, Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {},
Uargs(g^#) = {}, Uargs(c_0) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
h(x1) = [0] x1 + [0]
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
f^#(x1, x2) = [3] x1 + [3] x2 + [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: {f^#(s(x), y) -> c_2(f^#(x, s(y)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
f^#(x1, x2) = [4] x1 + [0] x2 + [0]
c_2(x1) = [1] x1 + [7]
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.