Tool CaT
stdout:
MAYBE
Problem:
ack_in(0(),n) -> ack_out(s(n))
ack_in(s(m),0()) -> u11(ack_in(m,s(0())))
u11(ack_out(n)) -> ack_out(n)
ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
u21(ack_out(n),m) -> u22(ack_in(m,n))
u22(ack_out(n)) -> ack_out(n)
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_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, u11(ack_out(n)) -> ack_out(n)
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(n)}
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_in^#(0(), n) -> c_0()
, 2: ack_in^#(s(m), 0()) -> c_1(u11^#(ack_in(m, s(0()))))
, 3: u11^#(ack_out(n)) -> c_2()
, 4: ack_in^#(s(m), s(n)) -> c_3(u21^#(ack_in(s(m), n), m))
, 5: u21^#(ack_out(n), m) -> c_4(u22^#(ack_in(m, n)))
, 6: u22^#(ack_out(n)) -> c_5()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
`->{5} [ inherited ]
|
`->{6} [ NA ]
->{2} [ inherited ]
|
`->{3} [ NA ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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_in) = {}, Uargs(ack_out) = {}, Uargs(s) = {},
Uargs(u11) = {}, Uargs(u21) = {}, Uargs(u22) = {},
Uargs(ack_in^#) = {}, Uargs(c_1) = {}, Uargs(u11^#) = {},
Uargs(c_3) = {}, Uargs(u21^#) = {}, Uargs(c_4) = {},
Uargs(u22^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack_in(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]
ack_out(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]
u11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u21(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]
u22(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ack_in^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u11^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u21^#(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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u22^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [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: {ack_in^#(0(), n) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(ack_in^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
ack_in^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
[2 2 0] [0 0 0] [3]
[2 2 2] [0 0 0] [3]
c_0() = [0]
[1]
[1]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}.
* Path {2}->{3}: NA
-----------------
The usable rules for this path are:
{ ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(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 {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{5}->{6}.
* Path {4}->{5}: inherited
------------------------
This path is subsumed by the proof of path {4}->{5}->{6}.
* Path {4}->{5}->{6}: NA
----------------------
The usable rules for this path are:
{ ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: ack_in^#(0(), n) -> c_0()
, 2: ack_in^#(s(m), 0()) -> c_1(u11^#(ack_in(m, s(0()))))
, 3: u11^#(ack_out(n)) -> c_2()
, 4: ack_in^#(s(m), s(n)) -> c_3(u21^#(ack_in(s(m), n), m))
, 5: u21^#(ack_out(n), m) -> c_4(u22^#(ack_in(m, n)))
, 6: u22^#(ack_out(n)) -> c_5()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
`->{5} [ inherited ]
|
`->{6} [ NA ]
->{2} [ inherited ]
|
`->{3} [ MAYBE ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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_in) = {}, Uargs(ack_out) = {}, Uargs(s) = {},
Uargs(u11) = {}, Uargs(u21) = {}, Uargs(u22) = {},
Uargs(ack_in^#) = {}, Uargs(c_1) = {}, Uargs(u11^#) = {},
Uargs(c_3) = {}, Uargs(u21^#) = {}, Uargs(c_4) = {},
Uargs(u22^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack_in(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
ack_out(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
u11(x1) = [0 0] x1 + [0]
[0 0] [0]
u21(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
u22(x1) = [0 0] x1 + [0]
[0 0] [0]
ack_in^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
u11^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
u21^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
u22^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [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: {ack_in^#(0(), n) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(ack_in^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
ack_in^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_0() = [0]
[1]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}.
* Path {2}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(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_in^#(s(m), 0()) -> c_1(u11^#(ack_in(m, s(0()))))
, u11^#(ack_out(n)) -> c_2()
, ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(n)}
Proof Output:
The input cannot be shown compatible
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{5}->{6}.
* Path {4}->{5}: inherited
------------------------
This path is subsumed by the proof of path {4}->{5}->{6}.
* Path {4}->{5}->{6}: NA
----------------------
The usable rules for this path are:
{ ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(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.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: ack_in^#(0(), n) -> c_0()
, 2: ack_in^#(s(m), 0()) -> c_1(u11^#(ack_in(m, s(0()))))
, 3: u11^#(ack_out(n)) -> c_2()
, 4: ack_in^#(s(m), s(n)) -> c_3(u21^#(ack_in(s(m), n), m))
, 5: u21^#(ack_out(n), m) -> c_4(u22^#(ack_in(m, n)))
, 6: u22^#(ack_out(n)) -> c_5()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
`->{5} [ inherited ]
|
`->{6} [ NA ]
->{2} [ inherited ]
|
`->{3} [ MAYBE ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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_in) = {}, Uargs(ack_out) = {}, Uargs(s) = {},
Uargs(u11) = {}, Uargs(u21) = {}, Uargs(u22) = {},
Uargs(ack_in^#) = {}, Uargs(c_1) = {}, Uargs(u11^#) = {},
Uargs(c_3) = {}, Uargs(u21^#) = {}, Uargs(c_4) = {},
Uargs(u22^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack_in(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
ack_out(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
u11(x1) = [0] x1 + [0]
u21(x1, x2) = [0] x1 + [0] x2 + [0]
u22(x1) = [0] x1 + [0]
ack_in^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
u11^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
u21^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
u22^#(x1) = [0] x1 + [0]
c_5() = [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: {ack_in^#(0(), n) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(ack_in^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
ack_in^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_0() = [1]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}.
* Path {2}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(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_in^#(s(m), 0()) -> c_1(u11^#(ack_in(m, s(0()))))
, u11^#(ack_out(n)) -> c_2()
, ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(n)}
Proof Output:
The input cannot be shown compatible
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{5}->{6}.
* Path {4}->{5}: inherited
------------------------
This path is subsumed by the proof of path {4}->{5}->{6}.
* Path {4}->{5}->{6}: NA
----------------------
The usable rules for this path are:
{ ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(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.
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_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, u11(ack_out(n)) -> ack_out(n)
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(n)}
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_in^#(0(), n) -> c_0(n)
, 2: ack_in^#(s(m), 0()) -> c_1(u11^#(ack_in(m, s(0()))))
, 3: u11^#(ack_out(n)) -> c_2(n)
, 4: ack_in^#(s(m), s(n)) -> c_3(u21^#(ack_in(s(m), n), m))
, 5: u21^#(ack_out(n), m) -> c_4(u22^#(ack_in(m, n)))
, 6: u22^#(ack_out(n)) -> c_5(n)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
`->{5} [ inherited ]
|
`->{6} [ NA ]
->{2} [ inherited ]
|
`->{3} [ NA ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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_in) = {}, Uargs(ack_out) = {}, Uargs(s) = {},
Uargs(u11) = {}, Uargs(u21) = {}, Uargs(u22) = {},
Uargs(ack_in^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(u11^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(u21^#) = {}, Uargs(c_4) = {}, Uargs(u22^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack_in(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]
ack_out(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]
u11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u21(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]
u22(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ack_in^#(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) = [1 1 1] 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]
u11^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u21^#(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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u22^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ack_in^#(0(), n) -> c_0(n)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(ack_in^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[2]
ack_in^#(x1, x2) = [2 0 2] x1 + [7 7 7] x2 + [7]
[2 0 2] [7 7 7] [7]
[2 0 2] [7 7 7] [7]
c_0(x1) = [1 3 3] x1 + [0]
[1 1 1] [1]
[1 1 1] [1]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}.
* Path {2}->{3}: NA
-----------------
The usable rules for this path are:
{ ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(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 {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{5}->{6}.
* Path {4}->{5}: inherited
------------------------
This path is subsumed by the proof of path {4}->{5}->{6}.
* Path {4}->{5}->{6}: NA
----------------------
The usable rules for this path are:
{ ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: ack_in^#(0(), n) -> c_0(n)
, 2: ack_in^#(s(m), 0()) -> c_1(u11^#(ack_in(m, s(0()))))
, 3: u11^#(ack_out(n)) -> c_2(n)
, 4: ack_in^#(s(m), s(n)) -> c_3(u21^#(ack_in(s(m), n), m))
, 5: u21^#(ack_out(n), m) -> c_4(u22^#(ack_in(m, n)))
, 6: u22^#(ack_out(n)) -> c_5(n)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
`->{5} [ inherited ]
|
`->{6} [ NA ]
->{2} [ inherited ]
|
`->{3} [ MAYBE ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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_in) = {}, Uargs(ack_out) = {}, Uargs(s) = {},
Uargs(u11) = {}, Uargs(u21) = {}, Uargs(u22) = {},
Uargs(ack_in^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(u11^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(u21^#) = {}, Uargs(c_4) = {}, Uargs(u22^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack_in(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
ack_out(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
u11(x1) = [0 0] x1 + [0]
[0 0] [0]
u21(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
u22(x1) = [0 0] x1 + [0]
[0 0] [0]
ack_in^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 1] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
u11^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
u21^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
u22^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ack_in^#(0(), n) -> c_0(n)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(ack_in^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
ack_in^#(x1, x2) = [2 2] x1 + [7 7] x2 + [7]
[2 2] [7 7] [3]
c_0(x1) = [1 3] x1 + [0]
[1 1] [1]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}.
* Path {2}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(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_in^#(s(m), 0()) -> c_1(u11^#(ack_in(m, s(0()))))
, u11^#(ack_out(n)) -> c_2(n)
, ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(n)}
Proof Output:
The input cannot be shown compatible
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{5}->{6}.
* Path {4}->{5}: inherited
------------------------
This path is subsumed by the proof of path {4}->{5}->{6}.
* Path {4}->{5}->{6}: NA
----------------------
The usable rules for this path are:
{ ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(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.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: ack_in^#(0(), n) -> c_0(n)
, 2: ack_in^#(s(m), 0()) -> c_1(u11^#(ack_in(m, s(0()))))
, 3: u11^#(ack_out(n)) -> c_2(n)
, 4: ack_in^#(s(m), s(n)) -> c_3(u21^#(ack_in(s(m), n), m))
, 5: u21^#(ack_out(n), m) -> c_4(u22^#(ack_in(m, n)))
, 6: u22^#(ack_out(n)) -> c_5(n)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
`->{5} [ inherited ]
|
`->{6} [ NA ]
->{2} [ inherited ]
|
`->{3} [ MAYBE ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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_in) = {}, Uargs(ack_out) = {}, Uargs(s) = {},
Uargs(u11) = {}, Uargs(u21) = {}, Uargs(u22) = {},
Uargs(ack_in^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(u11^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(u21^#) = {}, Uargs(c_4) = {}, Uargs(u22^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
ack_in(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
ack_out(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
u11(x1) = [0] x1 + [0]
u21(x1, x2) = [0] x1 + [0] x2 + [0]
u22(x1) = [0] x1 + [0]
ack_in^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
u11^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
u21^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
u22^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ack_in^#(0(), n) -> c_0(n)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(ack_in^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [5]
ack_in^#(x1, x2) = [3] x1 + [7] x2 + [0]
c_0(x1) = [1] x1 + [0]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}.
* Path {2}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(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_in^#(s(m), 0()) -> c_1(u11^#(ack_in(m, s(0()))))
, u11^#(ack_out(n)) -> c_2(n)
, ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(n)}
Proof Output:
The input cannot be shown compatible
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{5}->{6}.
* Path {4}->{5}: inherited
------------------------
This path is subsumed by the proof of path {4}->{5}->{6}.
* Path {4}->{5}->{6}: NA
----------------------
The usable rules for this path are:
{ ack_in(0(), n) -> ack_out(s(n))
, ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
, ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
, u11(ack_out(n)) -> ack_out(n)
, u21(ack_out(n), m) -> u22(ack_in(m, n))
, u22(ack_out(n)) -> ack_out(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.
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.