Tool CaT
stdout:
MAYBE
Problem:
minus(x,y) -> cond(min(x,y),x,y)
cond(y,x,y) -> s(minus(x,s(y)))
min(0(),v) -> 0()
min(u,0()) -> 0()
min(s(u),s(v)) -> s(min(u,v))
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:
{ minus(x, y) -> cond(min(x, y), x, y)
, cond(y, x, y) -> s(minus(x, s(y)))
, min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))}
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: minus^#(x, y) -> c_0(cond^#(min(x, y), x, y))
, 2: cond^#(y, x, y) -> c_1(minus^#(x, s(y)))
, 3: min^#(0(), v) -> c_2()
, 4: min^#(u, 0()) -> c_3()
, 5: min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))}
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:
{ minus^#(x, y) -> c_0(cond^#(min(x, y), x, y))
, cond^#(y, x, y) -> c_1(minus^#(x, s(y)))
, min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))}
Proof Output:
The input cannot be shown compatible
* 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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
cond(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]
min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
minus^#(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]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
min^#(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_2() = [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^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 3] x1 + [0]
[0 0 1] [4]
[0 0 0] [2]
min^#(x1, x2) = [1 1 0] x1 + [1 0 2] x2 + [0]
[0 0 2] [0 0 0] [2]
[1 1 0] [4 1 0] [0]
c_4(x1) = [1 2 0] x1 + [3]
[0 0 0] [2]
[0 2 0] [3]
* Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
cond(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]
min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
minus^#(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]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [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^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {min^#(0(), v) -> c_2()}
Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
0() = [0]
[2]
[0]
min^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[2 0 0] [0 4 0] [0]
c_2() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* 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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
cond(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]
min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
minus^#(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]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [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: {min^#(u, 0()) -> c_3()}
Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [0]
[0 0 0] [3]
0() = [2]
[2]
[2]
min^#(x1, x2) = [4 0 0] x1 + [3 3 0] x2 + [0]
[2 0 2] [2 0 0] [0]
[0 4 0] [2 2 3] [0]
c_3() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [7]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, y) -> c_0(cond^#(min(x, y), x, y))
, 2: cond^#(y, x, y) -> c_1(minus^#(x, s(y)))
, 3: min^#(0(), v) -> c_2()
, 4: min^#(u, 0()) -> c_3()
, 5: min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^2)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))}
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:
{ minus^#(x, y) -> c_0(cond^#(min(x, y), x, y))
, cond^#(y, x, y) -> c_1(minus^#(x, s(y)))
, min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))}
Proof Output:
The input cannot be shown compatible
* 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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2() = [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: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
min^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_4(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [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: {min^#(0(), v) -> c_2()}
Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 1] [2]
0() = [2]
[2]
min^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_2() = [1]
[0]
c_4(x1) = [1 0] x1 + [3]
[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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [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: {min^#(u, 0()) -> c_3()}
Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 3] x1 + [2]
[0 1] [2]
0() = [2]
[0]
min^#(x1, x2) = [2 0] x1 + [2 3] x2 + [0]
[0 2] [3 2] [0]
c_3() = [1]
[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: minus^#(x, y) -> c_0(cond^#(min(x, y), x, y))
, 2: cond^#(y, x, y) -> c_1(minus^#(x, s(y)))
, 3: min^#(0(), v) -> c_2()
, 4: min^#(u, 0()) -> c_3()
, 5: min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))}
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:
{ minus^#(x, y) -> c_0(cond^#(min(x, y), x, y))
, cond^#(y, x, y) -> c_1(minus^#(x, s(y)))
, min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))}
Proof Output:
The input cannot be shown compatible
* 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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
min(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
min^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [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: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
min^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4(x1) = [1] x1 + [7]
* Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
min(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [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: {min^#(0(), v) -> c_2()}
Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
min^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_2() = [1]
c_4(x1) = [1] x1 + [5]
* Path {5}->{4}: 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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
min(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [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: {min^#(u, 0()) -> c_3()}
Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
min^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_3() = [1]
c_4(x1) = [1] x1 + [6]
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:
{ minus(x, y) -> cond(min(x, y), x, y)
, cond(y, x, y) -> s(minus(x, s(y)))
, min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))}
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: minus^#(x, y) -> c_0(cond^#(min(x, y), x, y))
, 2: cond^#(y, x, y) -> c_1(minus^#(x, s(y)))
, 3: min^#(0(), v) -> c_2()
, 4: min^#(u, 0()) -> c_3()
, 5: min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))}
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:
{ minus^#(x, y) -> c_0(cond^#(min(x, y), x, y))
, cond^#(y, x, y) -> c_1(minus^#(x, s(y)))
, min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))}
Proof Output:
The input cannot be shown compatible
* 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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
cond(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]
min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
minus^#(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]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
min^#(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_2() = [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^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 3] x1 + [0]
[0 0 1] [4]
[0 0 0] [2]
min^#(x1, x2) = [1 1 0] x1 + [1 0 2] x2 + [0]
[0 0 2] [0 0 0] [2]
[1 1 0] [4 1 0] [0]
c_4(x1) = [1 2 0] x1 + [3]
[0 0 0] [2]
[0 2 0] [3]
* Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
cond(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]
min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
minus^#(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]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [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^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {min^#(0(), v) -> c_2()}
Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
0() = [0]
[2]
[0]
min^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[2 0 0] [0 4 0] [0]
c_2() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* 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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
cond(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]
min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
minus^#(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]
cond^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [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: DP runtime-complexity with respect to
Strict Rules: {min^#(u, 0()) -> c_3()}
Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [0]
[0 0 0] [3]
0() = [2]
[2]
[2]
min^#(x1, x2) = [4 0 0] x1 + [3 3 0] x2 + [0]
[2 0 2] [2 0 0] [0]
[0 4 0] [2 2 3] [0]
c_3() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [7]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, y) -> c_0(cond^#(min(x, y), x, y))
, 2: cond^#(y, x, y) -> c_1(minus^#(x, s(y)))
, 3: min^#(0(), v) -> c_2()
, 4: min^#(u, 0()) -> c_3()
, 5: min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^2)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))}
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:
{ minus^#(x, y) -> c_0(cond^#(min(x, y), x, y))
, cond^#(y, x, y) -> c_1(minus^#(x, s(y)))
, min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))}
Proof Output:
The input cannot be shown compatible
* 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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2() = [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: DP runtime-complexity with respect to
Strict Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
min^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_4(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [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: DP runtime-complexity with respect to
Strict Rules: {min^#(0(), v) -> c_2()}
Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 1] [2]
0() = [2]
[2]
min^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_2() = [1]
[0]
c_4(x1) = [1 0] x1 + [3]
[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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [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: DP runtime-complexity with respect to
Strict Rules: {min^#(u, 0()) -> c_3()}
Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 3] x1 + [2]
[0 1] [2]
0() = [2]
[0]
min^#(x1, x2) = [2 0] x1 + [2 3] x2 + [0]
[0 2] [3 2] [0]
c_3() = [1]
[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: minus^#(x, y) -> c_0(cond^#(min(x, y), x, y))
, 2: cond^#(y, x, y) -> c_1(minus^#(x, s(y)))
, 3: min^#(0(), v) -> c_2()
, 4: min^#(u, 0()) -> c_3()
, 5: min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))}
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:
{ minus^#(x, y) -> c_0(cond^#(min(x, y), x, y))
, cond^#(y, x, y) -> c_1(minus^#(x, s(y)))
, min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))}
Proof Output:
The input cannot be shown compatible
* 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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
min(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
min^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [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: DP runtime-complexity with respect to
Strict Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
min^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4(x1) = [1] x1 + [7]
* Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
min(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [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: DP runtime-complexity with respect to
Strict Rules: {min^#(0(), v) -> c_2()}
Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
min^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_2() = [1]
c_4(x1) = [1] x1 + [5]
* Path {5}->{4}: 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(minus) = {}, Uargs(cond) = {}, Uargs(min) = {},
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(cond^#) = {}, Uargs(c_1) = {}, Uargs(min^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
min(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [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: DP runtime-complexity with respect to
Strict Rules: {min^#(u, 0()) -> c_3()}
Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
min^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_3() = [1]
c_4(x1) = [1] x1 + [6]
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.