'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))
, a(c(x)) -> x
, b(d(x)) -> x}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))
, a^#(c(x)) -> c_2()
, b^#(d(x)) -> c_3()}
The usable rules are:
{ a(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))
, a(c(x)) -> x
, b(d(x)) -> x}
The estimated dependency graph contains the following edges:
{a^#(d(x)) -> c_0(b^#(a(x)))}
==> {b^#(c(x)) -> c_1(a^#(b(x)))}
{a^#(d(x)) -> c_0(b^#(a(x)))}
==> {b^#(d(x)) -> c_3()}
{b^#(c(x)) -> c_1(a^#(b(x)))}
==> {a^#(c(x)) -> c_2()}
{b^#(c(x)) -> c_1(a^#(b(x)))}
==> {a^#(d(x)) -> c_0(b^#(a(x)))}
We consider the following path(s):
1) { a^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))}
The usable rules for this path are the following:
{ a(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))
, a(c(x)) -> x
, b(d(x)) -> x}
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(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))
, a(c(x)) -> x
, b(d(x)) -> x
, a^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a(c(x)) -> x}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(c(x)) -> x}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(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^#(c(x)) -> c_1(a^#(b(x)))}
and weakly orienting the rules
{a(c(x)) -> x}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(x)) -> c_1(a^#(b(x)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(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^#(d(x)) -> c_0(b^#(a(x)))}
and weakly orienting the rules
{ b^#(c(x)) -> c_1(a^#(b(x)))
, a(c(x)) -> x}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(d(x)) -> c_0(b^#(a(x)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(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(d(x)) -> x}
and weakly orienting the rules
{ a^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))
, a(c(x)) -> x}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(d(x)) -> x}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(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(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))}
Weak Rules:
{ b(d(x)) -> x
, a^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))
, a(c(x)) -> x}
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(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))}
Weak Rules:
{ b(d(x)) -> x
, a^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))
, a(c(x)) -> x}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 9
, a_0(3) -> 9
, a_1(2) -> 14
, a_1(3) -> 14
, a_1(17) -> 16
, d_0(2) -> 2
, d_0(2) -> 9
, d_0(2) -> 11
, d_0(2) -> 13
, d_0(2) -> 14
, d_0(2) -> 16
, d_0(2) -> 17
, d_0(3) -> 2
, d_0(3) -> 9
, d_0(3) -> 11
, d_0(3) -> 13
, d_0(3) -> 14
, d_0(3) -> 16
, d_0(3) -> 17
, d_1(12) -> 9
, d_1(12) -> 14
, d_1(12) -> 16
, d_1(16) -> 15
, d_1(16) -> 16
, c_0(2) -> 3
, c_0(2) -> 9
, c_0(2) -> 11
, c_0(2) -> 13
, c_0(2) -> 14
, c_0(2) -> 16
, c_0(2) -> 17
, c_0(3) -> 3
, c_0(3) -> 9
, c_0(3) -> 11
, c_0(3) -> 13
, c_0(3) -> 14
, c_0(3) -> 16
, c_0(3) -> 17
, c_1(13) -> 12
, c_1(13) -> 13
, c_1(15) -> 11
, c_1(15) -> 13
, c_1(15) -> 17
, b_0(2) -> 11
, b_0(3) -> 11
, b_1(2) -> 17
, b_1(3) -> 17
, b_1(14) -> 13
, a^#_0(2) -> 5
, a^#_0(3) -> 5
, a^#_0(11) -> 10
, a^#_1(17) -> 19
, c_0_0(8) -> 5
, c_0_1(18) -> 5
, c_0_1(18) -> 10
, c_0_1(18) -> 19
, b^#_0(2) -> 7
, b^#_0(3) -> 7
, b^#_0(9) -> 8
, b^#_1(14) -> 18
, c_1_0(10) -> 7
, c_1_1(19) -> 7
, c_1_1(19) -> 8
, c_1_1(19) -> 18}
2) { a^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))
, b^#(d(x)) -> c_3()}
The usable rules for this path are the following:
{ a(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))
, a(c(x)) -> x
, b(d(x)) -> x}
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(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))
, a(c(x)) -> x
, b(d(x)) -> x
, a^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))
, b^#(d(x)) -> c_3()}
Details:
We apply the weight gap principle, strictly orienting the rules
{a(c(x)) -> x}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(c(x)) -> x}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(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^#(c(x)) -> c_1(a^#(b(x)))
, b^#(d(x)) -> c_3()}
and weakly orienting the rules
{a(c(x)) -> x}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b^#(c(x)) -> c_1(a^#(b(x)))
, b^#(d(x)) -> c_3()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(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^#(d(x)) -> c_0(b^#(a(x)))}
and weakly orienting the rules
{ b^#(c(x)) -> c_1(a^#(b(x)))
, b^#(d(x)) -> c_3()
, a(c(x)) -> x}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(d(x)) -> c_0(b^#(a(x)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(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(d(x)) -> x}
and weakly orienting the rules
{ a^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))
, b^#(d(x)) -> c_3()
, a(c(x)) -> x}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(d(x)) -> x}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(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(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))}
Weak Rules:
{ b(d(x)) -> x
, a^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))
, b^#(d(x)) -> c_3()
, a(c(x)) -> x}
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(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))}
Weak Rules:
{ b(d(x)) -> x
, a^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))
, b^#(d(x)) -> c_3()
, a(c(x)) -> x}
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
, d_0(2) -> 2
, d_0(2) -> 4
, d_0(2) -> 6
, d_0(2) -> 8
, d_0(2) -> 9
, d_0(2) -> 11
, d_0(2) -> 12
, d_1(7) -> 4
, d_1(7) -> 9
, d_1(7) -> 11
, d_1(11) -> 10
, d_1(11) -> 11
, c_0(2) -> 2
, c_0(2) -> 4
, c_0(2) -> 6
, c_0(2) -> 8
, c_0(2) -> 9
, c_0(2) -> 11
, c_0(2) -> 12
, c_1(8) -> 7
, c_1(8) -> 8
, c_1(10) -> 6
, c_1(10) -> 8
, c_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^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))
, a^#(c(x)) -> c_2()}
The usable rules for this path are the following:
{ a(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))
, a(c(x)) -> x
, b(d(x)) -> x}
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(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))
, a(c(x)) -> x
, b(d(x)) -> x
, a^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))
, a^#(c(x)) -> c_2()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(c(x)) -> x
, a^#(c(x)) -> c_2()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(c(x)) -> x
, a^#(c(x)) -> c_2()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(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^#(c(x)) -> c_1(a^#(b(x)))}
and weakly orienting the rules
{ a(c(x)) -> x
, a^#(c(x)) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(x)) -> c_1(a^#(b(x)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(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^#(d(x)) -> c_0(b^#(a(x)))}
and weakly orienting the rules
{ b^#(c(x)) -> c_1(a^#(b(x)))
, a(c(x)) -> x
, a^#(c(x)) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(d(x)) -> c_0(b^#(a(x)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(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(d(x)) -> x}
and weakly orienting the rules
{ a^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))
, a(c(x)) -> x
, a^#(c(x)) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(d(x)) -> x}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(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(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))}
Weak Rules:
{ b(d(x)) -> x
, a^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))
, a(c(x)) -> x
, a^#(c(x)) -> 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(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))}
Weak Rules:
{ b(d(x)) -> x
, a^#(d(x)) -> c_0(b^#(a(x)))
, b^#(c(x)) -> c_1(a^#(b(x)))
, a(c(x)) -> x
, a^#(c(x)) -> c_2()}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 9
, a_0(3) -> 9
, a_1(2) -> 14
, a_1(3) -> 14
, a_1(17) -> 16
, d_0(2) -> 2
, d_0(2) -> 9
, d_0(2) -> 11
, d_0(2) -> 13
, d_0(2) -> 14
, d_0(2) -> 16
, d_0(2) -> 17
, d_0(3) -> 2
, d_0(3) -> 9
, d_0(3) -> 11
, d_0(3) -> 13
, d_0(3) -> 14
, d_0(3) -> 16
, d_0(3) -> 17
, d_1(12) -> 9
, d_1(12) -> 14
, d_1(12) -> 16
, d_1(16) -> 15
, d_1(16) -> 16
, c_0(2) -> 3
, c_0(2) -> 9
, c_0(2) -> 11
, c_0(2) -> 13
, c_0(2) -> 14
, c_0(2) -> 16
, c_0(2) -> 17
, c_0(3) -> 3
, c_0(3) -> 9
, c_0(3) -> 11
, c_0(3) -> 13
, c_0(3) -> 14
, c_0(3) -> 16
, c_0(3) -> 17
, c_1(13) -> 12
, c_1(13) -> 13
, c_1(15) -> 11
, c_1(15) -> 13
, c_1(15) -> 17
, b_0(2) -> 11
, b_0(3) -> 11
, b_1(2) -> 17
, b_1(3) -> 17
, b_1(14) -> 13
, a^#_0(2) -> 5
, a^#_0(3) -> 5
, a^#_0(11) -> 10
, a^#_1(17) -> 19
, c_0_0(8) -> 5
, c_0_1(18) -> 5
, c_0_1(18) -> 10
, c_0_1(18) -> 19
, b^#_0(2) -> 7
, b^#_0(3) -> 7
, b^#_0(9) -> 8
, b^#_1(14) -> 18
, c_1_0(10) -> 7
, c_1_1(19) -> 7
, c_1_1(19) -> 8
, c_1_1(19) -> 18
, c_2_0() -> 5
, c_2_0() -> 10
, c_2_1() -> 10
, c_2_1() -> 19}