Tool CaT
stdout:
MAYBE
Problem:
ack(0(),n) -> s(n)
ack(s(m),0()) -> ack(m,s(0()))
ack(s(m),s(n)) -> ack(m,ack(s(m),n))
s(x()) -> r(x())
0() -> z()
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:
{ ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))
, s(x()) -> r(x())
, 0() -> 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: ack^#(0(), n) -> c_0(s^#(n))
, 2: ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, 3: ack^#(s(m), s(n)) -> c_2(ack^#(m, ack(s(m), n)))
, 4: s^#(x()) -> c_3()
, 5: 0^#() -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{2} [ inherited ]
|
|->{1} [ YES(?,O(n^1)) ]
| |
| `->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ inherited ]
|
`->{1} [ inherited ]
|
`->{4} [ NA ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{1}: YES(?,O(n^1))
----------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {}, Uargs(s^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(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() = [3]
[3]
[3]
s(x1) = [0 0 0] x1 + [3]
[0 0 0] [3]
[3 3 3] [3]
x() = [0]
[0]
[0]
r(x1) = [0 0 0] x1 + [1]
[0 0 0] [3]
[0 0 1] [3]
z() = [0]
[1]
[1]
ack^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 3 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s^#(x1) = [3 3 3] x1 + [0]
[1 1 1] [0]
[3 3 3] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
0^#() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ack^#(0(), n) -> c_0(s^#(n))}
Weak Rules:
{ ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s(x()) -> r(x())
, 0() -> z()}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {}, Uargs(c_0) = {},
Uargs(s^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[3]
[2]
s(x1) = [0 1 0] x1 + [4]
[4 3 0] [1]
[0 0 1] [0]
x() = [0]
[4]
[2]
r(x1) = [0 0 2] x1 + [3]
[0 0 2] [6]
[0 0 1] [0]
z() = [0]
[1]
[0]
ack^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [3]
[1 2 2] [2 0 0] [0]
[2 2 2] [2 0 0] [0]
c_0(x1) = [2 2 2] x1 + [0]
[0 2 2] [4]
[2 0 0] [3]
s^#(x1) = [0 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [2]
c_1(x1) = [4 0 0] x1 + [1]
[0 0 0] [7]
[0 1 0] [0]
* Path {2}->{1}->{4}: YES(?,O(n^1))
---------------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {1}, Uargs(s^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(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() = [3]
[3]
[3]
s(x1) = [0 0 0] x1 + [3]
[0 0 0] [3]
[3 3 3] [3]
x() = [0]
[0]
[0]
r(x1) = [0 0 0] x1 + [1]
[0 0 0] [3]
[0 0 1] [3]
z() = [0]
[1]
[1]
ack^#(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) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
s^#(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]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
0^#() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {s^#(x()) -> c_3()}
Weak Rules:
{ ack^#(0(), n) -> c_0(s^#(n))
, ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s(x()) -> r(x())
, 0() -> z()}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {}, Uargs(c_0) = {1},
Uargs(s^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 2 2] [0]
x() = [2]
[2]
[2]
r(x1) = [0 0 0] x1 + [2]
[0 0 0] [0]
[0 0 1] [6]
z() = [0]
[0]
[0]
ack^#(x1, x2) = [2 0 0] x1 + [4 0 4] x2 + [0]
[0 0 0] [0 0 4] [0]
[0 0 0] [0 2 2] [0]
c_0(x1) = [2 0 0] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
s^#(x1) = [2 0 0] x1 + [0]
[2 2 0] [0]
[0 0 2] [0]
c_1(x1) = [1 0 2] x1 + [0]
[0 0 0] [0]
[0 0 1] [0]
c_3() = [1]
[0]
[0]
* Path {2}->{3}: inherited
------------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{3}->{1}: inherited
-----------------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{3}->{1}->{4}: NA
---------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()
, ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {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(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {}, Uargs(s^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(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]
x() = [0]
[0]
[0]
r(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
z() = [0]
[0]
[0]
ack^#(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]
s^#(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]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
0^#() = [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: {0^#() -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0^#() = [7]
[7]
[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: ack^#(0(), n) -> c_0(s^#(n))
, 2: ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, 3: ack^#(s(m), s(n)) -> c_2(ack^#(m, ack(s(m), n)))
, 4: s^#(x()) -> c_3()
, 5: 0^#() -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{2} [ inherited ]
|
|->{1} [ YES(?,O(n^1)) ]
| |
| `->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ inherited ]
|
`->{1} [ inherited ]
|
`->{4} [ MAYBE ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{1}: YES(?,O(n^1))
----------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {}, Uargs(s^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [3]
[3]
s(x1) = [0 0] x1 + [3]
[3 3] [3]
x() = [0]
[0]
r(x1) = [0 3] x1 + [1]
[0 1] [3]
z() = [0]
[1]
ack^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 3] x1 + [0]
[0 0] [0]
s^#(x1) = [3 3] x1 + [0]
[1 1] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
0^#() = [0]
[0]
c_4() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ack^#(0(), n) -> c_0(s^#(n))}
Weak Rules:
{ ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s(x()) -> r(x())
, 0() -> z()}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {}, Uargs(c_0) = {},
Uargs(s^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [2 0] x1 + [4]
[0 3] [0]
x() = [0]
[0]
r(x1) = [0 0] x1 + [4]
[0 1] [0]
z() = [0]
[0]
ack^#(x1, x2) = [2 3] x1 + [0 2] x2 + [0]
[2 2] [0 0] [0]
c_0(x1) = [2 2] x1 + [1]
[0 0] [6]
s^#(x1) = [0 0] x1 + [2]
[0 0] [2]
c_1(x1) = [1 0] x1 + [0]
[0 0] [6]
* Path {2}->{1}->{4}: YES(?,O(n^1))
---------------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {1}, Uargs(s^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [3]
[3]
s(x1) = [0 0] x1 + [3]
[3 3] [3]
x() = [0]
[0]
r(x1) = [0 3] x1 + [1]
[0 1] [3]
z() = [1]
[1]
ack^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
s^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
0^#() = [0]
[0]
c_4() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {s^#(x()) -> c_3()}
Weak Rules:
{ ack^#(0(), n) -> c_0(s^#(n))
, ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s(x()) -> r(x())
, 0() -> z()}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {}, Uargs(c_0) = {1},
Uargs(s^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [0 2] x1 + [0]
[1 0] [0]
x() = [2]
[2]
r(x1) = [0 0] x1 + [0]
[0 1] [0]
z() = [0]
[0]
ack^#(x1, x2) = [2 4] x1 + [4 4] x2 + [0]
[0 0] [2 0] [0]
c_0(x1) = [2 0] x1 + [3]
[0 0] [0]
s^#(x1) = [2 2] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 3] [3]
c_3() = [1]
[0]
* Path {2}->{3}: inherited
------------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{3}->{1}: inherited
-----------------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{3}->{1}->{4}: MAYBE
------------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()
, ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))}
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:
{ ack^#(0(), n) -> c_0(s^#(n))
, ack^#(s(m), s(n)) -> c_2(ack^#(m, ack(s(m), n)))
, ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s^#(x()) -> c_3()
, s(x()) -> r(x())
, 0() -> z()
, ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))}
Proof Output:
The input cannot be shown compatible
* 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(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {}, Uargs(s^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(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]
x() = [0]
[0]
r(x1) = [0 0] x1 + [0]
[0 0] [0]
z() = [0]
[0]
ack^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
s^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
0^#() = [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: {0^#() -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0^#() = [7]
[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: ack^#(0(), n) -> c_0(s^#(n))
, 2: ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, 3: ack^#(s(m), s(n)) -> c_2(ack^#(m, ack(s(m), n)))
, 4: s^#(x()) -> c_3()
, 5: 0^#() -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{2} [ inherited ]
|
|->{1} [ YES(?,O(n^1)) ]
| |
| `->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ inherited ]
|
`->{1} [ inherited ]
|
`->{4} [ MAYBE ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{1}: YES(?,O(n^1))
----------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {}, Uargs(s^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [3]
s(x1) = [3] x1 + [2]
x() = [3]
r(x1) = [1] x1 + [3]
z() = [1]
ack^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_0(x1) = [3] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
0^#() = [0]
c_4() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ack^#(0(), n) -> c_0(s^#(n))}
Weak Rules:
{ ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s(x()) -> r(x())
, 0() -> z()}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {}, Uargs(c_0) = {},
Uargs(s^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [2] x1 + [2]
x() = [0]
r(x1) = [1] x1 + [2]
z() = [0]
ack^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_0(x1) = [2] x1 + [3]
s^#(x1) = [0] x1 + [2]
c_1(x1) = [1] x1 + [3]
* Path {2}->{1}->{4}: YES(?,O(n^1))
---------------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {1}, Uargs(s^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [3]
s(x1) = [3] x1 + [2]
x() = [3]
r(x1) = [1] x1 + [3]
z() = [0]
ack^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
0^#() = [0]
c_4() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {s^#(x()) -> c_3()}
Weak Rules:
{ ack^#(0(), n) -> c_0(s^#(n))
, ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s(x()) -> r(x())
, 0() -> z()}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {}, Uargs(c_0) = {1},
Uargs(s^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
x() = [3]
r(x1) = [1] x1 + [0]
z() = [0]
ack^#(x1, x2) = [2] x1 + [4] x2 + [4]
c_0(x1) = [2] x1 + [3]
s^#(x1) = [2] x1 + [2]
c_1(x1) = [1] x1 + [0]
c_3() = [1]
* Path {2}->{3}: inherited
------------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{3}->{1}: inherited
-----------------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{3}->{1}->{4}: MAYBE
------------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()
, ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))}
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:
{ ack^#(0(), n) -> c_0(s^#(n))
, ack^#(s(m), s(n)) -> c_2(ack^#(m, ack(s(m), n)))
, ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s^#(x()) -> c_3()
, s(x()) -> r(x())
, 0() -> z()
, ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))}
Proof Output:
The input cannot be shown compatible
* 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(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {}, Uargs(s^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
x() = [0]
r(x1) = [0] x1 + [0]
z() = [0]
ack^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
0^#() = [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: {0^#() -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0^#() = [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:
{ ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))
, s(x()) -> r(x())
, 0() -> 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: ack^#(0(), n) -> c_0(s^#(n))
, 2: ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, 3: ack^#(s(m), s(n)) -> c_2(ack^#(m, ack(s(m), n)))
, 4: s^#(x()) -> c_3()
, 5: 0^#() -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{2} [ inherited ]
|
|->{1} [ YES(?,O(n^1)) ]
| |
| `->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ inherited ]
|
`->{1} [ inherited ]
|
`->{4} [ NA ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{1}: YES(?,O(n^1))
----------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {}, Uargs(s^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(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() = [3]
[3]
[3]
s(x1) = [0 0 0] x1 + [3]
[0 0 0] [3]
[3 3 3] [3]
x() = [0]
[0]
[0]
r(x1) = [0 0 0] x1 + [1]
[0 0 0] [3]
[0 0 1] [3]
z() = [0]
[1]
[1]
ack^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 3 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s^#(x1) = [3 3 3] x1 + [0]
[1 1 1] [0]
[3 3 3] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
0^#() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ack^#(0(), n) -> c_0(s^#(n))}
Weak Rules:
{ ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s(x()) -> r(x())
, 0() -> z()}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {}, Uargs(c_0) = {},
Uargs(s^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[3]
[2]
s(x1) = [0 1 0] x1 + [4]
[4 3 0] [1]
[0 0 1] [0]
x() = [0]
[4]
[2]
r(x1) = [0 0 2] x1 + [3]
[0 0 2] [6]
[0 0 1] [0]
z() = [0]
[1]
[0]
ack^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [3]
[1 2 2] [2 0 0] [0]
[2 2 2] [2 0 0] [0]
c_0(x1) = [2 2 2] x1 + [0]
[0 2 2] [4]
[2 0 0] [3]
s^#(x1) = [0 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [2]
c_1(x1) = [4 0 0] x1 + [1]
[0 0 0] [7]
[0 1 0] [0]
* Path {2}->{1}->{4}: YES(?,O(n^1))
---------------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {1}, Uargs(s^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(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() = [3]
[3]
[3]
s(x1) = [0 0 0] x1 + [3]
[0 0 0] [3]
[3 3 3] [3]
x() = [0]
[0]
[0]
r(x1) = [0 0 0] x1 + [1]
[0 0 0] [3]
[0 0 1] [3]
z() = [0]
[1]
[1]
ack^#(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) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
s^#(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]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
0^#() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {s^#(x()) -> c_3()}
Weak Rules:
{ ack^#(0(), n) -> c_0(s^#(n))
, ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s(x()) -> r(x())
, 0() -> z()}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {}, Uargs(c_0) = {1},
Uargs(s^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 2 2] [0]
x() = [2]
[2]
[2]
r(x1) = [0 0 0] x1 + [2]
[0 0 0] [0]
[0 0 1] [6]
z() = [0]
[0]
[0]
ack^#(x1, x2) = [2 0 0] x1 + [4 0 4] x2 + [0]
[0 0 0] [0 0 4] [0]
[0 0 0] [0 2 2] [0]
c_0(x1) = [2 0 0] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
s^#(x1) = [2 0 0] x1 + [0]
[2 2 0] [0]
[0 0 2] [0]
c_1(x1) = [1 0 2] x1 + [0]
[0 0 0] [0]
[0 0 1] [0]
c_3() = [1]
[0]
[0]
* Path {2}->{3}: inherited
------------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{3}->{1}: inherited
-----------------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{3}->{1}->{4}: NA
---------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()
, ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {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(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {}, Uargs(s^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(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]
x() = [0]
[0]
[0]
r(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
z() = [0]
[0]
[0]
ack^#(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]
s^#(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]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
0^#() = [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: DP runtime-complexity with respect to
Strict Rules: {0^#() -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0^#() = [7]
[7]
[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: ack^#(0(), n) -> c_0(s^#(n))
, 2: ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, 3: ack^#(s(m), s(n)) -> c_2(ack^#(m, ack(s(m), n)))
, 4: s^#(x()) -> c_3()
, 5: 0^#() -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{2} [ inherited ]
|
|->{1} [ YES(?,O(n^1)) ]
| |
| `->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ inherited ]
|
`->{1} [ inherited ]
|
`->{4} [ MAYBE ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{1}: YES(?,O(n^1))
----------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {}, Uargs(s^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [3]
[3]
s(x1) = [0 0] x1 + [3]
[3 3] [3]
x() = [0]
[0]
r(x1) = [0 3] x1 + [1]
[0 1] [3]
z() = [0]
[1]
ack^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 3] x1 + [0]
[0 0] [0]
s^#(x1) = [3 3] x1 + [0]
[1 1] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
0^#() = [0]
[0]
c_4() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ack^#(0(), n) -> c_0(s^#(n))}
Weak Rules:
{ ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s(x()) -> r(x())
, 0() -> z()}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {}, Uargs(c_0) = {},
Uargs(s^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [2 0] x1 + [4]
[0 3] [0]
x() = [0]
[0]
r(x1) = [0 0] x1 + [4]
[0 1] [0]
z() = [0]
[0]
ack^#(x1, x2) = [2 3] x1 + [0 2] x2 + [0]
[2 2] [0 0] [0]
c_0(x1) = [2 2] x1 + [1]
[0 0] [6]
s^#(x1) = [0 0] x1 + [2]
[0 0] [2]
c_1(x1) = [1 0] x1 + [0]
[0 0] [6]
* Path {2}->{1}->{4}: YES(?,O(n^1))
---------------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {1}, Uargs(s^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [3]
[3]
s(x1) = [0 0] x1 + [3]
[3 3] [3]
x() = [0]
[0]
r(x1) = [0 3] x1 + [1]
[0 1] [3]
z() = [1]
[1]
ack^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
s^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
0^#() = [0]
[0]
c_4() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {s^#(x()) -> c_3()}
Weak Rules:
{ ack^#(0(), n) -> c_0(s^#(n))
, ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s(x()) -> r(x())
, 0() -> z()}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {}, Uargs(c_0) = {1},
Uargs(s^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [0 2] x1 + [0]
[1 0] [0]
x() = [2]
[2]
r(x1) = [0 0] x1 + [0]
[0 1] [0]
z() = [0]
[0]
ack^#(x1, x2) = [2 4] x1 + [4 4] x2 + [0]
[0 0] [2 0] [0]
c_0(x1) = [2 0] x1 + [3]
[0 0] [0]
s^#(x1) = [2 2] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 3] [3]
c_3() = [1]
[0]
* Path {2}->{3}: inherited
------------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{3}->{1}: inherited
-----------------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{3}->{1}->{4}: MAYBE
------------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()
, ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))}
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:
{ ack^#(0(), n) -> c_0(s^#(n))
, ack^#(s(m), s(n)) -> c_2(ack^#(m, ack(s(m), n)))
, ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s^#(x()) -> c_3()
, s(x()) -> r(x())
, 0() -> z()
, ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))}
Proof Output:
The input cannot be shown compatible
* 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(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {}, Uargs(s^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(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]
x() = [0]
[0]
r(x1) = [0 0] x1 + [0]
[0 0] [0]
z() = [0]
[0]
ack^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
s^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
0^#() = [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: DP runtime-complexity with respect to
Strict Rules: {0^#() -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0^#() = [7]
[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: ack^#(0(), n) -> c_0(s^#(n))
, 2: ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, 3: ack^#(s(m), s(n)) -> c_2(ack^#(m, ack(s(m), n)))
, 4: s^#(x()) -> c_3()
, 5: 0^#() -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{2} [ inherited ]
|
|->{1} [ YES(?,O(n^1)) ]
| |
| `->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ inherited ]
|
`->{1} [ inherited ]
|
`->{4} [ MAYBE ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{1}: YES(?,O(n^1))
----------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {}, Uargs(s^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [3]
s(x1) = [3] x1 + [2]
x() = [3]
r(x1) = [1] x1 + [3]
z() = [1]
ack^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_0(x1) = [3] x1 + [0]
s^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
0^#() = [0]
c_4() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ack^#(0(), n) -> c_0(s^#(n))}
Weak Rules:
{ ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s(x()) -> r(x())
, 0() -> z()}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {}, Uargs(c_0) = {},
Uargs(s^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [2] x1 + [2]
x() = [0]
r(x1) = [1] x1 + [2]
z() = [0]
ack^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_0(x1) = [2] x1 + [3]
s^#(x1) = [0] x1 + [2]
c_1(x1) = [1] x1 + [3]
* Path {2}->{1}->{4}: YES(?,O(n^1))
---------------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {1}, Uargs(s^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [3]
s(x1) = [3] x1 + [2]
x() = [3]
r(x1) = [1] x1 + [3]
z() = [0]
ack^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
0^#() = [0]
c_4() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {s^#(x()) -> c_3()}
Weak Rules:
{ ack^#(0(), n) -> c_0(s^#(n))
, ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s(x()) -> r(x())
, 0() -> z()}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {}, Uargs(c_0) = {1},
Uargs(s^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
x() = [3]
r(x1) = [1] x1 + [0]
z() = [0]
ack^#(x1, x2) = [2] x1 + [4] x2 + [4]
c_0(x1) = [2] x1 + [3]
s^#(x1) = [2] x1 + [2]
c_1(x1) = [1] x1 + [0]
c_3() = [1]
* Path {2}->{3}: inherited
------------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{3}->{1}: inherited
-----------------------------
This path is subsumed by the proof of path {2}->{3}->{1}->{4}.
* Path {2}->{3}->{1}->{4}: MAYBE
------------------------------
The usable rules for this path are:
{ s(x()) -> r(x())
, 0() -> z()
, ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))}
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:
{ ack^#(0(), n) -> c_0(s^#(n))
, ack^#(s(m), s(n)) -> c_2(ack^#(m, ack(s(m), n)))
, ack^#(s(m), 0()) -> c_1(ack^#(m, s(0())))
, s^#(x()) -> c_3()
, s(x()) -> r(x())
, 0() -> z()
, ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))}
Proof Output:
The input cannot be shown compatible
* 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(ack) = {}, Uargs(s) = {}, Uargs(r) = {}, Uargs(ack^#) = {},
Uargs(c_0) = {}, Uargs(s^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
x() = [0]
r(x1) = [0] x1 + [0]
z() = [0]
ack^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
0^#() = [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: DP runtime-complexity with respect to
Strict Rules: {0^#() -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0^#() = [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 pair1rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))
, s(x()) -> r(x())
, 0() -> z()}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair1 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair2rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))
, s(x()) -> r(x())
, 0() -> z()}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3irc
stdout:
YES(?,O(n^1))
We consider the following Problem:
Strict Trs:
{ ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))
, s(x()) -> r(x())
, 0() -> z()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Following rules were removed:
The rule 0() -> z()
makes following rules inapplicable:
ack(0(), n) -> s(n)ack(s(m), 0()) -> ack(m, s(0()))
We consider the following Problem:
Strict Trs:
{ ack(s(m), s(n)) -> ack(m, ack(s(m), n))
, s(x()) -> r(x())
, 0() -> z()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Application of 'Fastest':
-------------------------
'Fastest' proved the goal fastest:
'Sequentially' proved the goal fastest:
'Fastest' succeeded:
'matrix-interpretation of dimension 2 (timeout of 100.0 seconds)' proved the goal fastest:
The following argument positions are usable:
Uargs(ack) = {2}, Uargs(s) = {}, Uargs(r) = {}
We have the following constructor-restricted (at most 1 in the main diagonals) matrix interpretation:
Interpretation Functions:
ack(x1, x2) = [0 0] x1 + [2 0] x2 + [0]
[0 0] [0 0] [0]
0() = [2]
[2]
s(x1) = [2 0] x1 + [2]
[0 0] [0]
x() = [1]
[0]
r(x1) = [0 2] x1 + [1]
[0 1] [0]
z() = [1]
[0]
Hurray, we answered YES(?,O(n^1))Tool pair3rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))
, s(x()) -> r(x())
, 0() -> z()}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))
, s(x()) -> r(x())
, 0() -> z()}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool tup3irc
stdout:
YES(?,O(n^1))
We consider the following Problem:
Strict Trs:
{ ack(0(), n) -> s(n)
, ack(s(m), 0()) -> ack(m, s(0()))
, ack(s(m), s(n)) -> ack(m, ack(s(m), n))
, s(x()) -> r(x())
, 0() -> z()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Application of 'tup3 (timeout of 60.0 seconds)':
------------------------------------------------
Following rules were removed:
The rule 0() -> z()
makes following rules inapplicable:
ack(0(), n) -> s(n)ack(s(m), 0()) -> ack(m, s(0()))
We consider the following Problem:
Strict Trs:
{ ack(s(m), s(n)) -> ack(m, ack(s(m), n))
, s(x()) -> r(x())
, 0() -> z()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Application of 'Fastest':
-------------------------
'Fastest' proved the goal fastest:
'Sequentially' proved the goal fastest:
'Fastest' succeeded:
'matrix-interpretation of dimension 2 (timeout of 100.0 seconds)' proved the goal fastest:
The following argument positions are usable:
Uargs(ack) = {2}, Uargs(s) = {}, Uargs(r) = {}
We have the following constructor-restricted (at most 1 in the main diagonals) matrix interpretation:
Interpretation Functions:
ack(x1, x2) = [0 0] x1 + [2 0] x2 + [0]
[0 0] [0 0] [0]
0() = [2]
[2]
s(x1) = [2 0] x1 + [2]
[0 0] [0]
x() = [1]
[0]
r(x1) = [0 2] x1 + [1]
[0 1] [0]
z() = [1]
[0]
Hurray, we answered YES(?,O(n^1))