'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ A(b(x1)) -> b(a(B(A(x1))))
, B(a(x1)) -> a(b(A(B(x1))))
, A(a(x1)) -> x1
, B(b(x1)) -> x1}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))
, A^#(a(x1)) -> c_2()
, B^#(b(x1)) -> c_3()}
The usable rules are:
{ A(b(x1)) -> b(a(B(A(x1))))
, B(a(x1)) -> a(b(A(B(x1))))
, A(a(x1)) -> x1
, B(b(x1)) -> x1}
The estimated dependency graph contains the following edges:
{A^#(b(x1)) -> c_0(B^#(A(x1)))}
==> {B^#(a(x1)) -> c_1(A^#(B(x1)))}
{A^#(b(x1)) -> c_0(B^#(A(x1)))}
==> {B^#(b(x1)) -> c_3()}
{B^#(a(x1)) -> c_1(A^#(B(x1)))}
==> {A^#(a(x1)) -> c_2()}
{B^#(a(x1)) -> c_1(A^#(B(x1)))}
==> {A^#(b(x1)) -> c_0(B^#(A(x1)))}
We consider the following path(s):
1) { A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))}
The usable rules for this path are the following:
{ A(b(x1)) -> b(a(B(A(x1))))
, B(a(x1)) -> a(b(A(B(x1))))
, A(a(x1)) -> x1
, B(b(x1)) -> x1}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ A(b(x1)) -> b(a(B(A(x1))))
, B(a(x1)) -> a(b(A(B(x1))))
, A(a(x1)) -> x1
, B(b(x1)) -> x1
, A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{A(a(x1)) -> x1}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{A(a(x1)) -> x1}
Details:
Interpretation Functions:
A(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
B(x1) = [1] x1 + [0]
A^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
B^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{B^#(a(x1)) -> c_1(A^#(B(x1)))}
and weakly orienting the rules
{A(a(x1)) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{B^#(a(x1)) -> c_1(A^#(B(x1)))}
Details:
Interpretation Functions:
A(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
B(x1) = [1] x1 + [0]
A^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [3]
B^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{A^#(b(x1)) -> c_0(B^#(A(x1)))}
and weakly orienting the rules
{ B^#(a(x1)) -> c_1(A^#(B(x1)))
, A(a(x1)) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{A^#(b(x1)) -> c_0(B^#(A(x1)))}
Details:
Interpretation Functions:
A(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [8]
B(x1) = [1] x1 + [0]
A^#(x1) = [1] x1 + [3]
c_0(x1) = [1] x1 + [0]
B^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{B(b(x1)) -> x1}
and weakly orienting the rules
{ A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))
, A(a(x1)) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{B(b(x1)) -> x1}
Details:
Interpretation Functions:
A(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [5]
B(x1) = [1] x1 + [1]
A^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [3]
B^#(x1) = [1] x1 + [4]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ A(b(x1)) -> b(a(B(A(x1))))
, B(a(x1)) -> a(b(A(B(x1))))}
Weak Rules:
{ B(b(x1)) -> x1
, A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))
, A(a(x1)) -> x1}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ A(b(x1)) -> b(a(B(A(x1))))
, B(a(x1)) -> a(b(A(B(x1))))}
Weak Rules:
{ B(b(x1)) -> x1
, A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))
, A(a(x1)) -> x1}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ A_0(2) -> 4
, A_1(2) -> 9
, A_1(12) -> 11
, b_0(2) -> 2
, b_0(2) -> 4
, b_0(2) -> 6
, b_0(2) -> 8
, b_0(2) -> 9
, b_0(2) -> 11
, b_0(2) -> 12
, b_1(7) -> 4
, b_1(7) -> 9
, b_1(7) -> 11
, b_1(11) -> 10
, b_1(11) -> 11
, a_0(2) -> 2
, a_0(2) -> 4
, a_0(2) -> 6
, a_0(2) -> 8
, a_0(2) -> 9
, a_0(2) -> 11
, a_0(2) -> 12
, a_1(8) -> 7
, a_1(8) -> 8
, a_1(10) -> 6
, a_1(10) -> 8
, a_1(10) -> 12
, B_0(2) -> 6
, B_1(2) -> 12
, B_1(9) -> 8
, A^#_0(2) -> 1
, A^#_0(6) -> 5
, A^#_1(12) -> 14
, c_0_0(3) -> 1
, c_0_1(13) -> 1
, c_0_1(13) -> 5
, c_0_1(13) -> 14
, B^#_0(2) -> 1
, B^#_0(4) -> 3
, B^#_1(9) -> 13
, c_1_0(5) -> 1
, c_1_1(14) -> 1
, c_1_1(14) -> 3
, c_1_1(14) -> 13}
2) { A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))
, B^#(b(x1)) -> c_3()}
The usable rules for this path are the following:
{ A(b(x1)) -> b(a(B(A(x1))))
, B(a(x1)) -> a(b(A(B(x1))))
, A(a(x1)) -> x1
, B(b(x1)) -> x1}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ A(b(x1)) -> b(a(B(A(x1))))
, B(a(x1)) -> a(b(A(B(x1))))
, A(a(x1)) -> x1
, B(b(x1)) -> x1
, A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))
, B^#(b(x1)) -> c_3()}
Details:
We apply the weight gap principle, strictly orienting the rules
{A(a(x1)) -> x1}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{A(a(x1)) -> x1}
Details:
Interpretation Functions:
A(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
B(x1) = [1] x1 + [0]
A^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
B^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ B^#(a(x1)) -> c_1(A^#(B(x1)))
, B^#(b(x1)) -> c_3()}
and weakly orienting the rules
{A(a(x1)) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ B^#(a(x1)) -> c_1(A^#(B(x1)))
, B^#(b(x1)) -> c_3()}
Details:
Interpretation Functions:
A(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
B(x1) = [1] x1 + [0]
A^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [3]
B^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{A^#(b(x1)) -> c_0(B^#(A(x1)))}
and weakly orienting the rules
{ B^#(a(x1)) -> c_1(A^#(B(x1)))
, B^#(b(x1)) -> c_3()
, A(a(x1)) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{A^#(b(x1)) -> c_0(B^#(A(x1)))}
Details:
Interpretation Functions:
A(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [8]
B(x1) = [1] x1 + [0]
A^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [0]
B^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{B(b(x1)) -> x1}
and weakly orienting the rules
{ A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))
, B^#(b(x1)) -> c_3()
, A(a(x1)) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{B(b(x1)) -> x1}
Details:
Interpretation Functions:
A(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [9]
B(x1) = [1] x1 + [8]
A^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
B^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ A(b(x1)) -> b(a(B(A(x1))))
, B(a(x1)) -> a(b(A(B(x1))))}
Weak Rules:
{ B(b(x1)) -> x1
, A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))
, B^#(b(x1)) -> c_3()
, A(a(x1)) -> x1}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ A(b(x1)) -> b(a(B(A(x1))))
, B(a(x1)) -> a(b(A(B(x1))))}
Weak Rules:
{ B(b(x1)) -> x1
, A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))
, B^#(b(x1)) -> c_3()
, A(a(x1)) -> x1}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ A_0(2) -> 4
, A_1(2) -> 9
, A_1(12) -> 11
, b_0(2) -> 2
, b_0(2) -> 4
, b_0(2) -> 6
, b_0(2) -> 8
, b_0(2) -> 9
, b_0(2) -> 11
, b_0(2) -> 12
, b_1(7) -> 4
, b_1(7) -> 9
, b_1(7) -> 11
, b_1(11) -> 10
, b_1(11) -> 11
, a_0(2) -> 2
, a_0(2) -> 4
, a_0(2) -> 6
, a_0(2) -> 8
, a_0(2) -> 9
, a_0(2) -> 11
, a_0(2) -> 12
, a_1(8) -> 7
, a_1(8) -> 8
, a_1(10) -> 6
, a_1(10) -> 8
, a_1(10) -> 12
, B_0(2) -> 6
, B_1(2) -> 12
, B_1(9) -> 8
, A^#_0(2) -> 1
, A^#_0(6) -> 5
, A^#_1(12) -> 14
, c_0_0(3) -> 1
, c_0_1(13) -> 1
, c_0_1(13) -> 5
, c_0_1(13) -> 14
, B^#_0(2) -> 1
, B^#_0(4) -> 3
, B^#_1(9) -> 13
, c_1_0(5) -> 1
, c_1_1(14) -> 1
, c_1_1(14) -> 3
, c_1_1(14) -> 13
, c_3_0() -> 1
, c_3_0() -> 3
, c_3_1() -> 3
, c_3_1() -> 13}
3) { A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))
, A^#(a(x1)) -> c_2()}
The usable rules for this path are the following:
{ A(b(x1)) -> b(a(B(A(x1))))
, B(a(x1)) -> a(b(A(B(x1))))
, A(a(x1)) -> x1
, B(b(x1)) -> x1}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ A(b(x1)) -> b(a(B(A(x1))))
, B(a(x1)) -> a(b(A(B(x1))))
, A(a(x1)) -> x1
, B(b(x1)) -> x1
, A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))
, A^#(a(x1)) -> c_2()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ A(a(x1)) -> x1
, A^#(a(x1)) -> c_2()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ A(a(x1)) -> x1
, A^#(a(x1)) -> c_2()}
Details:
Interpretation Functions:
A(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
B(x1) = [1] x1 + [0]
A^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
B^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{B^#(a(x1)) -> c_1(A^#(B(x1)))}
and weakly orienting the rules
{ A(a(x1)) -> x1
, A^#(a(x1)) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{B^#(a(x1)) -> c_1(A^#(B(x1)))}
Details:
Interpretation Functions:
A(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
B(x1) = [1] x1 + [0]
A^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [10]
B^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{A^#(b(x1)) -> c_0(B^#(A(x1)))}
and weakly orienting the rules
{ B^#(a(x1)) -> c_1(A^#(B(x1)))
, A(a(x1)) -> x1
, A^#(a(x1)) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{A^#(b(x1)) -> c_0(B^#(A(x1)))}
Details:
Interpretation Functions:
A(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [8]
B(x1) = [1] x1 + [0]
A^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [0]
B^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{B(b(x1)) -> x1}
and weakly orienting the rules
{ A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))
, A(a(x1)) -> x1
, A^#(a(x1)) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{B(b(x1)) -> x1}
Details:
Interpretation Functions:
A(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [10]
B(x1) = [1] x1 + [8]
A^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
B^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ A(b(x1)) -> b(a(B(A(x1))))
, B(a(x1)) -> a(b(A(B(x1))))}
Weak Rules:
{ B(b(x1)) -> x1
, A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))
, A(a(x1)) -> x1
, A^#(a(x1)) -> c_2()}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ A(b(x1)) -> b(a(B(A(x1))))
, B(a(x1)) -> a(b(A(B(x1))))}
Weak Rules:
{ B(b(x1)) -> x1
, A^#(b(x1)) -> c_0(B^#(A(x1)))
, B^#(a(x1)) -> c_1(A^#(B(x1)))
, A(a(x1)) -> x1
, A^#(a(x1)) -> c_2()}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ A_0(2) -> 4
, A_1(2) -> 9
, A_1(12) -> 11
, b_0(2) -> 2
, b_0(2) -> 4
, b_0(2) -> 6
, b_0(2) -> 8
, b_0(2) -> 9
, b_0(2) -> 11
, b_0(2) -> 12
, b_1(7) -> 4
, b_1(7) -> 9
, b_1(7) -> 11
, b_1(11) -> 10
, b_1(11) -> 11
, a_0(2) -> 2
, a_0(2) -> 4
, a_0(2) -> 6
, a_0(2) -> 8
, a_0(2) -> 9
, a_0(2) -> 11
, a_0(2) -> 12
, a_1(8) -> 7
, a_1(8) -> 8
, a_1(10) -> 6
, a_1(10) -> 8
, a_1(10) -> 12
, B_0(2) -> 6
, B_1(2) -> 12
, B_1(9) -> 8
, A^#_0(2) -> 1
, A^#_0(6) -> 5
, A^#_1(12) -> 14
, c_0_0(3) -> 1
, c_0_1(13) -> 1
, c_0_1(13) -> 5
, c_0_1(13) -> 14
, B^#_0(2) -> 1
, B^#_0(4) -> 3
, B^#_1(9) -> 13
, c_1_0(5) -> 1
, c_1_1(14) -> 1
, c_1_1(14) -> 3
, c_1_1(14) -> 13
, c_2_0() -> 1
, c_2_0() -> 5
, c_2_1() -> 5
, c_2_1() -> 14}