Tool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(X) -> cons(X, f(g(X)))
, g(0()) -> s(0())
, g(s(X)) -> s(s(g(X)))
, sel(0(), cons(X, Y)) -> X
, sel(s(X), cons(Y, Z)) -> sel(X, Z)}
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^#(X) -> c_0(f^#(g(X)))
, 2: g^#(0()) -> c_1()
, 3: g^#(s(X)) -> c_2(g^#(X))
, 4: sel^#(0(), cons(X, Y)) -> c_3()
, 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^3)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{3} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^2)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ g(0()) -> s(0())
, g(s(X)) -> s(s(g(X)))}
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^#(X) -> c_0(f^#(g(X)))
, g(0()) -> s(0())
, g(s(X)) -> s(s(g(X)))}
Proof Output:
The input cannot be shown compatible
* Path {3}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_4) = {}
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]
cons(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]
g(x1) = [0 0 0] x1 + [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]
sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
sel^#(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(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: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(s(X)) -> c_2(g^#(X))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [2]
[0 0 0] [0]
g^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_2(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {3}->{2}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_4) = {}
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]
cons(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]
g(x1) = [0 0 0] x1 + [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]
sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
sel^#(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(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: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(0()) -> c_1()}
Weak Rules: {g^#(s(X)) -> c_2(g^#(X))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 1 0] x1 + [0]
[0 1 1] [1]
[0 0 0] [0]
g^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_1() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
* Path {5}: YES(?,O(n^3))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {1}
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]
cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
g(x1) = [0 0 0] x1 + [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]
sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_3() = [0]
[0]
[0]
c_4(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^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 4 4] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 1] [2]
s(x1) = [1 4 4] x1 + [2]
[0 1 2] [2]
[0 0 0] [2]
sel^#(x1, x2) = [0 2 0] x1 + [0 0 2] x2 + [0]
[2 0 2] [2 0 0] [0]
[0 2 2] [1 0 0] [0]
c_4(x1) = [1 0 0] x1 + [5]
[2 0 2] [3]
[0 0 0] [7]
* Path {5}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {1}
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]
cons(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]
g(x1) = [0 0 0] x1 + [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]
sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sel^#(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(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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {sel^#(0(), cons(X, Y)) -> c_3()}
Weak Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 3 0] x2 + [2]
[0 0 0] [0 1 4] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[2]
[0]
s(x1) = [1 1 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [2]
sel^#(x1, x2) = [2 2 0] x1 + [2 0 0] x2 + [0]
[1 0 2] [4 2 0] [0]
[4 0 2] [1 1 0] [0]
c_3() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [4]
[0 0 0] [7]
[0 0 0] [3]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(X) -> c_0(f^#(g(X)))
, 2: g^#(0()) -> c_1()
, 3: g^#(s(X)) -> c_2(g^#(X))
, 4: sel^#(0(), cons(X, Y)) -> c_3()
, 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{3} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ g(0()) -> s(0())
, g(s(X)) -> s(s(g(X)))}
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^#(X) -> c_0(f^#(g(X)))
, g(0()) -> s(0())
, g(s(X)) -> s(s(g(X)))}
Proof Output:
The input cannot be shown compatible
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(s(X)) -> c_2(g^#(X))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
g^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {3}->{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(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(0()) -> c_1()}
Weak Rules: {g^#(s(X)) -> c_2(g^#(X))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [3]
g^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [5]
[2 0] [3]
* Path {5}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_3() = [0]
[0]
c_4(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: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [4]
s(x1) = [1 2] x1 + [2]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 2] x2 + [0]
[4 0] [4 0] [0]
c_4(x1) = [1 0] x1 + [5]
[0 0] [7]
* Path {5}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {sel^#(0(), cons(X, Y)) -> c_3()}
Weak Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 3] x2 + [2]
[0 0] [0 1] [2]
0() = [2]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
sel^#(x1, x2) = [2 0] x1 + [2 3] x2 + [0]
[2 1] [2 2] [0]
c_3() = [1]
[0]
c_4(x1) = [1 0] x1 + [7]
[0 0] [6]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(X) -> c_0(f^#(g(X)))
, 2: g^#(0()) -> c_1()
, 3: g^#(s(X)) -> c_2(g^#(X))
, 4: sel^#(0(), cons(X, Y)) -> c_3()
, 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^1)) ]
|
`->{4} [ NA ]
->{3} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ g(0()) -> s(0())
, g(s(X)) -> s(s(g(X)))}
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^#(X) -> c_0(f^#(g(X)))
, g(0()) -> s(0())
, g(s(X)) -> s(s(g(X)))}
Proof Output:
The input cannot be shown compatible
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [3] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(s(X)) -> c_2(g^#(X))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
g^#(x1) = [2] x1 + [0]
c_2(x1) = [1] x1 + [7]
* Path {3}->{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(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(0()) -> c_1()}
Weak Rules: {g^#(s(X)) -> c_2(g^#(X))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
g^#(x1) = [2] x1 + [0]
c_1() = [1]
c_2(x1) = [1] x1 + [0]
* Path {5}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
sel^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_3() = [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(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [2]
sel^#(x1, x2) = [3] x1 + [0] x2 + [2]
c_4(x1) = [1] x1 + [5]
* Path {5}->{4}: NA
-----------------
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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f(X) -> cons(X, f(g(X)))
, g(0()) -> s(0())
, g(s(X)) -> s(s(g(X)))
, sel(0(), cons(X, Y)) -> X
, sel(s(X), cons(Y, Z)) -> sel(X, Z)}
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^#(X) -> c_0(X, f^#(g(X)))
, 2: g^#(0()) -> c_1()
, 3: g^#(s(X)) -> c_2(g^#(X))
, 4: sel^#(0(), cons(X, Y)) -> c_3(X)
, 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^3)) ]
|
`->{4} [ YES(?,O(n^3)) ]
->{3} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^2)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ g(0()) -> s(0())
, g(s(X)) -> s(s(g(X)))}
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^#(X) -> c_0(X, f^#(g(X)))
, g(0()) -> s(0())
, g(s(X)) -> s(s(g(X)))}
Proof Output:
The input cannot be shown compatible
* Path {3}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}
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]
cons(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]
g(x1) = [0 0 0] x1 + [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]
sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_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]
g^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
sel^#(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: {g^#(s(X)) -> c_2(g^#(X))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [2]
[0 0 0] [0]
g^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_2(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {3}->{2}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}
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]
cons(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]
g(x1) = [0 0 0] x1 + [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]
sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_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]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
sel^#(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: {g^#(0()) -> c_1()}
Weak Rules: {g^#(s(X)) -> c_2(g^#(X))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 1 0] x1 + [0]
[0 1 1] [1]
[0 0 0] [0]
g^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_1() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
* Path {5}: YES(?,O(n^3))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {1}
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]
cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
g(x1) = [0 0 0] x1 + [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]
sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_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]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(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^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 4 4] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 1] [2]
s(x1) = [1 4 4] x1 + [2]
[0 1 2] [2]
[0 0 0] [2]
sel^#(x1, x2) = [0 2 0] x1 + [0 0 2] x2 + [0]
[2 0 2] [2 0 0] [0]
[0 2 2] [1 0 0] [0]
c_4(x1) = [1 0 0] x1 + [5]
[2 0 2] [3]
[0 0 0] [7]
* Path {5}->{4}: YES(?,O(n^3))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {1}
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]
cons(x1, x2) = [1 1 1] x1 + [0 0 0] x2 + [0]
[0 1 3] [0 0 0] [0]
[0 0 1] [0 0 0] [0]
g(x1) = [0 0 0] x1 + [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]
sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_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]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sel^#(x1, x2) = [0 0 0] x1 + [3 1 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(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^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {sel^#(0(), cons(X, Y)) -> c_3(X)}
Weak Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2 2] x1 + [1 2 2] x2 + [0]
[0 0 2] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [2]
[3]
[0]
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [2]
[0 0 0] [2]
sel^#(x1, x2) = [2 2 0] x1 + [2 0 0] x2 + [0]
[2 2 2] [2 2 0] [0]
[3 3 0] [2 2 0] [0]
c_3(x1) = [0 0 0] x1 + [1]
[0 0 0] [0]
[0 0 0] [1]
c_4(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [6]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(X) -> c_0(X, f^#(g(X)))
, 2: g^#(0()) -> c_1()
, 3: g^#(s(X)) -> c_2(g^#(X))
, 4: sel^#(0(), cons(X, Y)) -> c_3(X)
, 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{3} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ g(0()) -> s(0())
, g(s(X)) -> s(s(g(X)))}
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^#(X) -> c_0(X, f^#(g(X)))
, g(0()) -> s(0())
, g(s(X)) -> s(s(g(X)))}
Proof Output:
The input cannot be shown compatible
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
sel^#(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: {g^#(s(X)) -> c_2(g^#(X))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
g^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {3}->{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(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
sel^#(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: {g^#(0()) -> c_1()}
Weak Rules: {g^#(s(X)) -> c_2(g^#(X))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [3]
g^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [5]
[2 0] [3]
* Path {5}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(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: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [4]
s(x1) = [1 2] x1 + [2]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 2] x2 + [0]
[4 0] [4 0] [0]
c_4(x1) = [1 0] x1 + [5]
[0 0] [7]
* Path {5}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [1 3] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [0]
c_4(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^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {sel^#(0(), cons(X, Y)) -> c_3(X)}
Weak Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
[0 0] [0 1] [0]
0() = [2]
[2]
s(x1) = [1 0] x1 + [3]
[0 1] [0]
sel^#(x1, x2) = [2 2] x1 + [3 3] x2 + [0]
[2 2] [2 5] [0]
c_3(x1) = [1 0] x1 + [1]
[0 0] [0]
c_4(x1) = [1 0] x1 + [7]
[0 0] [7]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(X) -> c_0(X, f^#(g(X)))
, 2: g^#(0()) -> c_1()
, 3: g^#(s(X)) -> c_2(g^#(X))
, 4: sel^#(0(), cons(X, Y)) -> c_3(X)
, 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^1)) ]
|
`->{4} [ NA ]
->{3} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ g(0()) -> s(0())
, g(s(X)) -> s(s(g(X)))}
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^#(X) -> c_0(X, f^#(g(X)))
, g(0()) -> s(0())
, g(s(X)) -> s(s(g(X)))}
Proof Output:
The input cannot be shown compatible
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
g^#(x1) = [3] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
sel^#(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: {g^#(s(X)) -> c_2(g^#(X))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
g^#(x1) = [2] x1 + [0]
c_2(x1) = [1] x1 + [7]
* Path {3}->{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(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
g^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
sel^#(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: {g^#(0()) -> c_1()}
Weak Rules: {g^#(s(X)) -> c_2(g^#(X))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
g^#(x1) = [2] x1 + [0]
c_1() = [1]
c_2(x1) = [1] x1 + [0]
* Path {5}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
g^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
sel^#(x1, x2) = [3] x1 + [3] 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(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [2]
sel^#(x1, x2) = [3] x1 + [0] x2 + [2]
c_4(x1) = [1] x1 + [5]
* Path {5}->{4}: NA
-----------------
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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
g^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
sel^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
We have not generated a proof for the resulting sub-problem.
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.