Tool CaT
stdout:
MAYBE
Problem:
+(X,0()) -> X
+(X,s(Y)) -> s(+(X,Y))
f(0(),s(0()),X) -> f(X,+(X,X),X)
g(X,Y) -> X
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:
{ +(X, 0()) -> X
, +(X, s(Y)) -> s(+(X, Y))
, f(0(), s(0()), X) -> f(X, +(X, X), X)
, g(X, Y) -> X
, 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: +^#(X, 0()) -> c_0()
, 2: +^#(X, s(Y)) -> c_1(+^#(X, Y))
, 3: f^#(0(), s(0()), X) -> c_2(f^#(X, +(X, X), X))
, 4: g^#(X, Y) -> c_3()
, 5: g^#(X, Y) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ MAYBE ]
->{2} [ YES(?,O(n^3)) ]
|
`->{1} [ YES(?,O(n^3)) ]
Sub-problems:
-------------
* Path {2}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_1) = {1}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(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]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 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]
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]
+^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(X, s(Y)) -> c_1(+^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(+^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 4] x1 + [2]
[0 1 2] [2]
[0 0 1] [2]
+^#(x1, x2) = [0 0 0] x1 + [0 1 2] x2 + [2]
[4 4 4] [2 2 2] [0]
[4 4 4] [2 0 2] [0]
c_1(x1) = [1 0 0] x1 + [3]
[0 0 0] [6]
[2 0 0] [2]
* Path {2}->{1}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_1) = {1}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [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]
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]
+^#(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() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(X, 0()) -> c_0()}
Weak Rules: {+^#(X, s(Y)) -> c_1(+^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(+^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 4 0] x1 + [0]
[0 1 3] [2]
[0 0 1] [2]
+^#(x1, x2) = [0 0 0] x1 + [0 3 2] x2 + [0]
[4 4 0] [2 2 2] [0]
[4 4 0] [2 2 2] [0]
c_0() = [1]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {3}: MAYBE
---------------
The usable rules for this path are:
{ +(X, 0()) -> X
, +(X, s(Y)) -> s(+(X, 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(), s(0()), X) -> c_2(f^#(X, +(X, X), X))
, +(X, 0()) -> X
, +(X, s(Y)) -> s(+(X, Y))}
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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_1) = {}, Uargs(f^#) = {}, Uargs(c_2) = {},
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [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]
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]
+^#(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() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(X, Y) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [7]
[0 0 0] [0 0 0] [7]
[0 0 0] [0 0 0] [7]
c_3() = [0]
[3]
[3]
* Path {5}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_1) = {}, Uargs(f^#) = {}, Uargs(c_2) = {},
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [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]
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]
+^#(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() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(X, Y) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [7]
[0 0 0] [0 0 0] [7]
[0 0 0] [0 0 0] [7]
c_4() = [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: +^#(X, 0()) -> c_0()
, 2: +^#(X, s(Y)) -> c_1(+^#(X, Y))
, 3: f^#(0(), s(0()), X) -> c_2(f^#(X, +(X, X), X))
, 4: g^#(X, Y) -> c_3()
, 5: g^#(X, Y) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ MAYBE ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_1) = {1}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [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]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[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: {+^#(X, s(Y)) -> c_1(+^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(+^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [2]
+^#(x1, x2) = [0 0] x1 + [0 1] x2 + [0]
[4 4] [4 5] [0]
c_1(x1) = [1 0] x1 + [1]
[0 0] [7]
* Path {2}->{1}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_1) = {1}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [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]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[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: {+^#(X, 0()) -> c_0()}
Weak Rules: {+^#(X, s(Y)) -> c_1(+^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(+^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 3] x1 + [2]
[0 0] [2]
+^#(x1, x2) = [0 0] x1 + [3 0] x2 + [4]
[0 4] [2 2] [0]
c_0() = [1]
[0]
c_1(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {3}: MAYBE
---------------
The usable rules for this path are:
{ +(X, 0()) -> X
, +(X, s(Y)) -> s(+(X, 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(), s(0()), X) -> c_2(f^#(X, +(X, X), X))
, +(X, 0()) -> X
, +(X, s(Y)) -> s(+(X, Y))}
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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_1) = {}, Uargs(f^#) = {}, Uargs(c_2) = {},
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [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]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [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_2(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(X, Y) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [7]
[0 0] [0 0] [7]
c_3() = [0]
[1]
* Path {5}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_1) = {}, Uargs(f^#) = {}, Uargs(c_2) = {},
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [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]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [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_2(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(X, Y) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [7]
[0 0] [0 0] [7]
c_4() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: +^#(X, 0()) -> c_0()
, 2: +^#(X, s(Y)) -> c_1(+^#(X, Y))
, 3: f^#(0(), s(0()), X) -> c_2(f^#(X, +(X, X), X))
, 4: g^#(X, Y) -> c_3()
, 5: g^#(X, Y) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ MAYBE ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_1) = {1}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
+^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [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: {+^#(X, s(Y)) -> c_1(+^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(+^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
+^#(x1, x2) = [7] x1 + [2] x2 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{1}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_1) = {1}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [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: {+^#(X, 0()) -> c_0()}
Weak Rules: {+^#(X, s(Y)) -> c_1(+^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(+^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
+^#(x1, x2) = [7] x1 + [2] x2 + [4]
c_0() = [1]
c_1(x1) = [1] x1 + [4]
* Path {3}: MAYBE
---------------
The usable rules for this path are:
{ +(X, 0()) -> X
, +(X, s(Y)) -> s(+(X, 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(), s(0()), X) -> c_2(f^#(X, +(X, X), X))
, +(X, 0()) -> X
, +(X, s(Y)) -> s(+(X, Y))}
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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_1) = {}, Uargs(f^#) = {}, Uargs(c_2) = {},
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(X, Y) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0] x1 + [0] x2 + [7]
c_3() = [0]
* Path {5}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_1) = {}, Uargs(f^#) = {}, Uargs(c_2) = {},
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(X, Y) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0] x1 + [0] x2 + [7]
c_4() = [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:
{ +(X, 0()) -> X
, +(X, s(Y)) -> s(+(X, Y))
, f(0(), s(0()), X) -> f(X, +(X, X), X)
, g(X, Y) -> X
, 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: +^#(X, 0()) -> c_0(X)
, 2: +^#(X, s(Y)) -> c_1(+^#(X, Y))
, 3: f^#(0(), s(0()), X) -> c_2(f^#(X, +(X, X), X))
, 4: g^#(X, Y) -> c_3(X)
, 5: g^#(X, Y) -> c_4(Y)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ MAYBE ]
->{2} [ YES(?,O(n^3)) ]
|
`->{1} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {2}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1},
Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(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]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 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]
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]
+^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {+^#(X, s(Y)) -> c_1(+^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(+^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 4] x1 + [2]
[0 1 2] [2]
[0 0 1] [2]
+^#(x1, x2) = [0 0 0] x1 + [0 1 2] x2 + [2]
[4 4 4] [2 2 2] [0]
[4 4 4] [2 0 2] [0]
c_1(x1) = [1 0 0] x1 + [3]
[0 0 0] [6]
[2 0 0] [2]
* Path {2}->{1}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1},
Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [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]
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]
+^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {+^#(X, 0()) -> c_0(X)}
Weak Rules: {+^#(X, s(Y)) -> c_1(+^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(+^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 2 3] x1 + [0]
[0 0 3] [2]
[0 0 1] [2]
+^#(x1, x2) = [0 0 0] x1 + [2 2 2] x2 + [2]
[0 0 0] [0 2 2] [0]
[4 0 4] [4 0 0] [4]
c_0(x1) = [0 0 0] x1 + [1]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [5]
[0 0 0] [7]
[2 2 0] [0]
* Path {3}: MAYBE
---------------
The usable rules for this path are:
{ +(X, 0()) -> X
, +(X, s(Y)) -> s(+(X, 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(), s(0()), X) -> c_2(f^#(X, +(X, X), X))
, +(X, 0()) -> X
, +(X, s(Y)) -> s(+(X, Y))}
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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [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]
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]
+^#(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]
c_1(x1) = [0 0 0] x1 + [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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(X, Y) -> c_3(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [7 7 7] x1 + [0 0 0] x2 + [7]
[7 7 7] [0 0 0] [7]
[7 7 7] [0 0 0] [7]
c_3(x1) = [3 3 3] x1 + [0]
[3 1 3] [1]
[1 1 1] [1]
* Path {5}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [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]
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]
+^#(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]
c_1(x1) = [0 0 0] x1 + [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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [1 1 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(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(X, Y) -> c_4(Y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0 0 0] x1 + [7 7 7] x2 + [7]
[0 0 0] [7 7 7] [7]
[0 0 0] [7 7 7] [7]
c_4(x1) = [3 3 3] x1 + [0]
[3 1 3] [1]
[1 1 1] [1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: +^#(X, 0()) -> c_0(X)
, 2: +^#(X, s(Y)) -> c_1(+^#(X, Y))
, 3: f^#(0(), s(0()), X) -> c_2(f^#(X, +(X, X), X))
, 4: g^#(X, Y) -> c_3(X)
, 5: g^#(X, Y) -> c_4(Y)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ MAYBE ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1},
Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [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]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {+^#(X, s(Y)) -> c_1(+^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(+^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [2]
+^#(x1, x2) = [0 0] x1 + [0 1] x2 + [0]
[4 4] [4 5] [0]
c_1(x1) = [1 0] x1 + [1]
[0 0] [7]
* Path {2}->{1}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1},
Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [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]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 1] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {+^#(X, 0()) -> c_0(X)}
Weak Rules: {+^#(X, s(Y)) -> c_1(+^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(+^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [2 0] x2 + [4]
[4 4] [2 2] [0]
c_0(x1) = [0 0] x1 + [1]
[0 0] [0]
c_1(x1) = [1 0] x1 + [3]
[0 0] [4]
* Path {3}: MAYBE
---------------
The usable rules for this path are:
{ +(X, 0()) -> X
, +(X, s(Y)) -> s(+(X, 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(), s(0()), X) -> c_2(f^#(X, +(X, X), X))
, +(X, 0()) -> X
, +(X, s(Y)) -> s(+(X, Y))}
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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [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]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [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_2(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [1 1] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(X, Y) -> c_3(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [7 7] x1 + [0 0] x2 + [7]
[7 7] [0 0] [7]
c_3(x1) = [1 3] x1 + [0]
[3 1] [3]
* Path {5}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [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]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [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_2(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [1 1] 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: {g^#(X, Y) -> c_4(Y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0 0] x1 + [7 7] x2 + [7]
[0 0] [7 7] [7]
c_4(x1) = [1 3] x1 + [0]
[3 1] [3]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: +^#(X, 0()) -> c_0(X)
, 2: +^#(X, s(Y)) -> c_1(+^#(X, Y))
, 3: f^#(0(), s(0()), X) -> c_2(f^#(X, +(X, X), X))
, 4: g^#(X, Y) -> c_3(X)
, 5: g^#(X, Y) -> c_4(Y)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ MAYBE ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1},
Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
+^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {+^#(X, s(Y)) -> c_1(+^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(+^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
+^#(x1, x2) = [7] x1 + [2] x2 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{1}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1},
Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
+^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {+^#(X, 0()) -> c_0(X)}
Weak Rules: {+^#(X, s(Y)) -> c_1(+^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(+^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
+^#(x1, x2) = [7] x1 + [3] x2 + [2]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [5]
* Path {3}: MAYBE
---------------
The usable rules for this path are:
{ +(X, 0()) -> X
, +(X, s(Y)) -> s(+(X, 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(), s(0()), X) -> c_2(f^#(X, +(X, X), X))
, +(X, 0()) -> X
, +(X, s(Y)) -> s(+(X, Y))}
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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(X, Y) -> c_3(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [7] x1 + [0] x2 + [7]
c_3(x1) = [1] x1 + [0]
* Path {5}: 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(+) = {}, Uargs(s) = {}, Uargs(f) = {}, Uargs(g) = {},
Uargs(+^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(f^#) = {},
Uargs(c_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
+(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] 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: {g^#(X, Y) -> c_4(Y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0] x1 + [7] x2 + [7]
c_4(x1) = [1] x1 + [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.