Tool CaT
stdout:
MAYBE
Problem:
*(X,+(Y,1())) -> +(*(X,+(Y,*(1(),0()))),X)
*(X,1()) -> X
*(X,0()) -> X
*(X,0()) -> 0()
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
, *(X, 1()) -> X
, *(X, 0()) -> X
, *(X, 0()) -> 0()}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))))
, 2: *^#(X, 1()) -> c_1()
, 3: *^#(X, 0()) -> c_2()
, 4: *^#(X, 0()) -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
, *(X, 1()) -> X
, *(X, 0()) -> X
, *(X, 0()) -> 0()}
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 {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
+(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]
1() = [0]
[0]
[0]
0() = [0]
[0]
[0]
*^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3() = [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: {*^#(X, 1()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1() = [2]
[2]
[2]
*^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
[0 0 0] [2 2 0] [3]
[0 0 0] [2 2 2] [3]
c_1() = [0]
[1]
[1]
* Path {3}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
+(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]
1() = [0]
[0]
[0]
0() = [0]
[0]
[0]
*^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3() = [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: {*^#(X, 0()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
*^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
[0 0 0] [2 2 0] [3]
[0 0 0] [2 2 2] [3]
c_2() = [0]
[1]
[1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
+(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]
1() = [0]
[0]
[0]
0() = [0]
[0]
[0]
*^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3() = [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: {*^#(X, 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
*^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
[0 0 0] [2 2 0] [3]
[0 0 0] [2 2 2] [3]
c_3() = [0]
[1]
[1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))))
, 2: *^#(X, 1()) -> c_1()
, 3: *^#(X, 0()) -> c_2()
, 4: *^#(X, 0()) -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
, *(X, 1()) -> X
, *(X, 0()) -> X
, *(X, 0()) -> 0()}
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:
{ *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))))
, *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
, *(X, 1()) -> X
, *(X, 0()) -> X
, *(X, 0()) -> 0()}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
1() = [0]
[0]
0() = [0]
[0]
*^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3() = [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: {*^#(X, 1()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1() = [2]
[2]
*^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_1() = [0]
[1]
* Path {3}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
1() = [0]
[0]
0() = [0]
[0]
*^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3() = [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: {*^#(X, 0()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
*^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_2() = [0]
[1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
1() = [0]
[0]
0() = [0]
[0]
*^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3() = [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: {*^#(X, 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
*^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_3() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))))
, 2: *^#(X, 1()) -> c_1()
, 3: *^#(X, 0()) -> c_2()
, 4: *^#(X, 0()) -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
, *(X, 1()) -> X
, *(X, 0()) -> X
, *(X, 0()) -> 0()}
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:
{ *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))))
, *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
, *(X, 1()) -> X
, *(X, 0()) -> X
, *(X, 0()) -> 0()}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
1() = [0]
0() = [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [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: {*^#(X, 1()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1() = [7]
*^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_1() = [1]
* Path {3}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
1() = [0]
0() = [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [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: {*^#(X, 0()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
*^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_2() = [1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
1() = [0]
0() = [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [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: {*^#(X, 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
*^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_3() = [1]
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
, *(X, 1()) -> X
, *(X, 0()) -> X
, *(X, 0()) -> 0()}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))), X)
, 2: *^#(X, 1()) -> c_1(X)
, 3: *^#(X, 0()) -> c_2(X)
, 4: *^#(X, 0()) -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
, *(X, 1()) -> X
, *(X, 0()) -> X
, *(X, 0()) -> 0()}
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 {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
+(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]
1() = [0]
[0]
[0]
0() = [0]
[0]
[0]
*^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1, 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_1(x1) = [1 1 1] 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]
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: {*^#(X, 1()) -> c_1(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1() = [2]
[0]
[2]
*^#(x1, x2) = [7 7 7] x1 + [2 0 2] x2 + [7]
[7 7 7] [2 0 2] [7]
[7 7 7] [2 0 2] [7]
c_1(x1) = [1 3 3] x1 + [0]
[1 1 1] [1]
[1 1 1] [1]
* Path {3}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
+(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]
1() = [0]
[0]
[0]
0() = [0]
[0]
[0]
*^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1, 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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [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: {*^#(X, 0()) -> c_2(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[2]
*^#(x1, x2) = [7 7 7] x1 + [2 0 2] x2 + [7]
[7 7 7] [2 0 2] [7]
[7 7 7] [2 0 2] [7]
c_2(x1) = [1 3 3] x1 + [0]
[1 1 1] [1]
[1 1 1] [1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
+(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]
1() = [0]
[0]
[0]
0() = [0]
[0]
[0]
*^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1, 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_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]
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: {*^#(X, 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
*^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
[0 0 0] [2 2 0] [3]
[0 0 0] [2 2 2] [3]
c_3() = [0]
[1]
[1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))), X)
, 2: *^#(X, 1()) -> c_1(X)
, 3: *^#(X, 0()) -> c_2(X)
, 4: *^#(X, 0()) -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
, *(X, 1()) -> X
, *(X, 0()) -> X
, *(X, 0()) -> 0()}
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:
{ *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))), X)
, *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
, *(X, 1()) -> X
, *(X, 0()) -> X
, *(X, 0()) -> 0()}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
1() = [0]
[0]
0() = [0]
[0]
*^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [1 1] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [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: {*^#(X, 1()) -> c_1(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1() = [2]
[2]
*^#(x1, x2) = [7 7] x1 + [2 2] x2 + [7]
[7 7] [2 2] [3]
c_1(x1) = [1 3] x1 + [0]
[1 1] [1]
* Path {3}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
1() = [0]
[0]
0() = [0]
[0]
*^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [1 1] x1 + [0]
[0 0] [0]
c_3() = [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: {*^#(X, 0()) -> c_2(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
*^#(x1, x2) = [7 7] x1 + [2 2] x2 + [7]
[7 7] [2 2] [3]
c_2(x1) = [1 3] x1 + [0]
[1 1] [1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
1() = [0]
[0]
0() = [0]
[0]
*^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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]
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: {*^#(X, 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
*^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_3() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))), X)
, 2: *^#(X, 1()) -> c_1(X)
, 3: *^#(X, 0()) -> c_2(X)
, 4: *^#(X, 0()) -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
, *(X, 1()) -> X
, *(X, 0()) -> X
, *(X, 0()) -> 0()}
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:
{ *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))), X)
, *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
, *(X, 1()) -> X
, *(X, 0()) -> X
, *(X, 0()) -> 0()}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
1() = [0]
0() = [0]
*^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [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: {*^#(X, 1()) -> c_1(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1() = [5]
*^#(x1, x2) = [7] x1 + [3] x2 + [0]
c_1(x1) = [1] x1 + [0]
* Path {3}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
1() = [0]
0() = [0]
*^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [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: {*^#(X, 0()) -> c_2(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [5]
*^#(x1, x2) = [7] x1 + [3] x2 + [0]
c_2(x1) = [1] x1 + [0]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
1() = [0]
0() = [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [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: {*^#(X, 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(*^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
*^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_3() = [1]
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.