'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, d(x1) -> x1
, a(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(c(x1)) -> b(b(c(d(x1))))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(d(x1)) -> c_0(d^#(b(x1)))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, d^#(x1) -> c_2()
, a^#(x1) -> c_3()
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, b^#(c(x1)) -> c_5(d^#(d(x1)))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
The usable rules are:
{ d(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
The estimated dependency graph contains the following edges:
{a^#(d(x1)) -> c_0(d^#(b(x1)))}
==> {d^#(x1) -> c_2()}
{a^#(x1) -> c_1(b^#(b(b(x1))))}
==> {b^#(c(x1)) -> c_5(d^#(d(x1)))}
{a^#(x1) -> c_1(b^#(b(b(x1))))}
==> {b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
{b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
==> {a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
{b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
==> {a^#(x1) -> c_3()}
{b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
==> {a^#(x1) -> c_1(b^#(b(b(x1))))}
{b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
==> {a^#(d(x1)) -> c_0(d^#(b(x1)))}
{b^#(c(x1)) -> c_5(d^#(d(x1)))}
==> {d^#(x1) -> c_2()}
{a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
==> {b^#(c(x1)) -> c_5(d^#(d(x1)))}
{a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
==> {b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
We consider the following path(s):
1) { a^#(x1) -> c_1(b^#(b(b(x1))))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a^#(d(x1)) -> c_0(d^#(b(x1)))
, d^#(x1) -> c_2()}
The usable rules for this path are the following:
{ d(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(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:
{ d(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))
, a^#(d(x1)) -> c_0(d^#(b(x1)))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, d^#(x1) -> c_2()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
and weakly orienting the rules
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_1(b^#(b(b(x1))))}
and weakly orienting the rules
{ a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_1(b^#(b(b(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(x1) -> c_2()}
and weakly orienting the rules
{ a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(x1) -> c_2()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [3]
d^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [7]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(d(x1)) -> c_0(d^#(b(x1)))}
and weakly orienting the rules
{ d^#(x1) -> c_2()
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(d(x1)) -> c_0(d^#(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(x1) -> x1}
and weakly orienting the rules
{ a^#(d(x1)) -> c_0(d^#(b(x1)))
, d^#(x1) -> c_2()
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(x1) -> x1}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [7]
d(x1) = [1] x1 + [2]
b(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [4]
a^#(x1) = [1] x1 + [15]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [4]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(c(x1)) -> c(d(d(x1)))}
and weakly orienting the rules
{ d(x1) -> x1
, a^#(d(x1)) -> c_0(d^#(b(x1)))
, d^#(x1) -> c_2()
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(c(x1)) -> c(d(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [6]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [7]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [4]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [2]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [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:
{ b(d(b(x1))) -> a(d(x1))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
Weak Rules:
{ b(c(x1)) -> c(d(d(x1)))
, d(x1) -> x1
, a^#(d(x1)) -> c_0(d^#(b(x1)))
, d^#(x1) -> c_2()
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(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:
{ b(d(b(x1))) -> a(d(x1))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
Weak Rules:
{ b(c(x1)) -> c(d(d(x1)))
, d(x1) -> x1
, a^#(d(x1)) -> c_0(d^#(b(x1)))
, d^#(x1) -> c_2()
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ d_0(4) -> 16
, d_0(4) -> 17
, d_0(4) -> 18
, d_0(4) -> 19
, d_0(16) -> 17
, d_0(16) -> 18
, d_0(16) -> 19
, d_0(17) -> 18
, d_0(17) -> 19
, d_0(18) -> 19
, b_0(4) -> 12
, b_0(12) -> 11
, b_0(15) -> 14
, c_0(4) -> 4
, c_0(4) -> 16
, c_0(4) -> 17
, c_0(4) -> 18
, c_0(4) -> 19
, c_0(16) -> 15
, c_0(17) -> 12
, c_0(18) -> 14
, c_0(19) -> 11
, a^#_0(4) -> 5
, d^#_0(4) -> 7
, c_1_0(10) -> 5
, b^#_0(4) -> 9
, b^#_0(11) -> 10
, b^#_0(14) -> 13
, c_2_0() -> 7
, c_6_0(13) -> 5}
2) { a^#(x1) -> c_1(b^#(b(b(x1))))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, b^#(c(x1)) -> c_5(d^#(d(x1)))
, d^#(x1) -> c_2()}
The usable rules for this path are the following:
{ d(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(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:
{ d(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))
, b^#(c(x1)) -> c_5(d^#(d(x1)))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, d^#(x1) -> c_2()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
and weakly orienting the rules
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(c(x1)) -> c_5(d^#(d(x1)))}
and weakly orienting the rules
{ b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(x1)) -> c_5(d^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(x1) -> c_2()}
and weakly orienting the rules
{ b^#(c(x1)) -> c_5(d^#(d(x1)))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(x1) -> c_2()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [8]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(x1) -> x1}
and weakly orienting the rules
{ d^#(x1) -> c_2()
, b^#(c(x1)) -> c_5(d^#(d(x1)))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(x1) -> x1}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [12]
d(x1) = [1] x1 + [12]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [11]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [9]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(c(x1)) -> c(d(d(x1)))}
and weakly orienting the rules
{ d(x1) -> x1
, d^#(x1) -> c_2()
, b^#(c(x1)) -> c_5(d^#(d(x1)))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(c(x1)) -> c(d(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [3]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [3]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [8]
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:
{ b(d(b(x1))) -> a(d(x1))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
Weak Rules:
{ b(c(x1)) -> c(d(d(x1)))
, d(x1) -> x1
, d^#(x1) -> c_2()
, b^#(c(x1)) -> c_5(d^#(d(x1)))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(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:
{ b(d(b(x1))) -> a(d(x1))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
Weak Rules:
{ b(c(x1)) -> c(d(d(x1)))
, d(x1) -> x1
, d^#(x1) -> c_2()
, b^#(c(x1)) -> c_5(d^#(d(x1)))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ d_0(4) -> 11
, d_1(4) -> 18
, d_1(4) -> 19
, d_1(4) -> 20
, d_1(4) -> 23
, d_1(4) -> 25
, d_1(18) -> 19
, d_1(18) -> 20
, d_1(18) -> 23
, d_1(18) -> 25
, d_1(19) -> 20
, d_1(19) -> 23
, d_1(19) -> 25
, d_1(20) -> 23
, d_1(20) -> 25
, d_1(23) -> 25
, b_1(4) -> 14
, b_1(14) -> 13
, b_1(17) -> 16
, c_0(4) -> 4
, c_0(4) -> 11
, c_0(4) -> 18
, c_0(4) -> 19
, c_0(4) -> 20
, c_0(4) -> 23
, c_0(4) -> 25
, c_1(18) -> 17
, c_1(19) -> 14
, c_1(20) -> 16
, c_1(23) -> 13
, a^#_0(4) -> 5
, d^#_0(4) -> 7
, d^#_0(11) -> 10
, d^#_1(20) -> 21
, d^#_1(23) -> 22
, d^#_1(25) -> 24
, c_1_1(12) -> 5
, b^#_0(4) -> 9
, b^#_1(13) -> 12
, b^#_1(16) -> 15
, c_2_0() -> 7
, c_2_0() -> 10
, c_2_1() -> 21
, c_2_1() -> 22
, c_2_1() -> 24
, c_5_0(10) -> 9
, c_5_1(21) -> 9
, c_5_1(22) -> 15
, c_5_1(24) -> 12
, c_6_1(15) -> 5}
3) { a^#(x1) -> c_1(b^#(b(b(x1))))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, b^#(c(x1)) -> c_5(d^#(d(x1)))}
The usable rules for this path are the following:
{ d(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(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:
{ d(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, b^#(c(x1)) -> c_5(d^#(d(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
and weakly orienting the rules
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [4]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [7]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(c(x1)) -> c_5(d^#(d(x1)))}
and weakly orienting the rules
{ a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(x1)) -> c_5(d^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [7]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(x1) -> x1}
and weakly orienting the rules
{ b^#(c(x1)) -> c_5(d^#(d(x1)))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(x1) -> x1}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [4]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(c(x1)) -> c(d(d(x1)))}
and weakly orienting the rules
{ d(x1) -> x1
, b^#(c(x1)) -> c_5(d^#(d(x1)))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(c(x1)) -> c(d(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [10]
d(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [3]
c(x1) = [1] x1 + [13]
a^#(x1) = [1] x1 + [12]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [3]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [4]
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:
{ b(d(b(x1))) -> a(d(x1))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
Weak Rules:
{ b(c(x1)) -> c(d(d(x1)))
, d(x1) -> x1
, b^#(c(x1)) -> c_5(d^#(d(x1)))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(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:
{ b(d(b(x1))) -> a(d(x1))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
Weak Rules:
{ b(c(x1)) -> c(d(d(x1)))
, d(x1) -> x1
, b^#(c(x1)) -> c_5(d^#(d(x1)))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ d_0(4) -> 11
, d_0(4) -> 18
, d_0(4) -> 19
, d_0(4) -> 21
, d_0(4) -> 23
, d_0(11) -> 18
, d_0(11) -> 19
, d_0(11) -> 21
, d_0(11) -> 23
, d_0(18) -> 19
, d_0(18) -> 21
, d_0(18) -> 23
, d_0(19) -> 21
, d_0(19) -> 23
, d_0(21) -> 23
, b_0(4) -> 14
, b_0(14) -> 13
, b_0(17) -> 16
, c_0(4) -> 4
, c_0(4) -> 11
, c_0(4) -> 18
, c_0(4) -> 19
, c_0(4) -> 21
, c_0(4) -> 23
, c_0(11) -> 17
, c_0(18) -> 14
, c_0(19) -> 16
, c_0(21) -> 13
, a^#_0(4) -> 5
, d^#_0(4) -> 7
, d^#_0(11) -> 10
, d^#_0(21) -> 20
, d^#_0(23) -> 22
, c_1_0(12) -> 5
, b^#_0(4) -> 9
, b^#_0(13) -> 12
, b^#_0(16) -> 15
, c_5_0(10) -> 9
, c_5_0(20) -> 15
, c_5_0(22) -> 12
, c_6_0(15) -> 5}
4) { a^#(x1) -> c_1(b^#(b(b(x1))))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a^#(d(x1)) -> c_0(d^#(b(x1)))}
The usable rules for this path are the following:
{ d(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(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:
{ d(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a^#(d(x1)) -> c_0(d^#(b(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
and weakly orienting the rules
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [7]
d^#(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(d(x1)) -> c_0(d^#(b(x1)))}
and weakly orienting the rules
{ a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(d(x1)) -> c_0(d^#(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [7]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(c(x1)) -> c(d(d(x1)))}
and weakly orienting the rules
{ a^#(d(x1)) -> c_0(d^#(b(x1)))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(c(x1)) -> c(d(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [13]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [2]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [4]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(x1) -> x1}
and weakly orienting the rules
{ b(c(x1)) -> c(d(d(x1)))
, a^#(d(x1)) -> c_0(d^#(b(x1)))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(x1) -> x1}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [13]
d(x1) = [1] x1 + [2]
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [14]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [2]
b^#(x1) = [1] x1 + [2]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [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:
{ b(d(b(x1))) -> a(d(x1))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
Weak Rules:
{ d(x1) -> x1
, b(c(x1)) -> c(d(d(x1)))
, a^#(d(x1)) -> c_0(d^#(b(x1)))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(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:
{ b(d(b(x1))) -> a(d(x1))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
Weak Rules:
{ d(x1) -> x1
, b(c(x1)) -> c(d(d(x1)))
, a^#(d(x1)) -> c_0(d^#(b(x1)))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ d_0(4) -> 16
, d_0(4) -> 17
, d_0(4) -> 18
, d_0(4) -> 19
, d_0(16) -> 17
, d_0(16) -> 18
, d_0(16) -> 19
, d_0(17) -> 18
, d_0(17) -> 19
, d_0(18) -> 19
, b_0(4) -> 12
, b_0(12) -> 11
, b_0(15) -> 14
, c_0(4) -> 4
, c_0(4) -> 16
, c_0(4) -> 17
, c_0(4) -> 18
, c_0(4) -> 19
, c_0(16) -> 15
, c_0(17) -> 12
, c_0(18) -> 14
, c_0(19) -> 11
, a^#_0(4) -> 5
, d^#_0(4) -> 7
, c_1_0(10) -> 5
, b^#_0(4) -> 9
, b^#_0(11) -> 10
, b^#_0(14) -> 13
, c_6_0(13) -> 5}
5) { a^#(x1) -> c_1(b^#(b(b(x1))))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
The usable rules for this path are the following:
{ d(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(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:
{ d(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
and weakly orienting the rules
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(x1) -> x1}
and weakly orienting the rules
{ a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(x1) -> x1}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [11]
d(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(c(x1)) -> c(d(d(x1)))}
and weakly orienting the rules
{ d(x1) -> x1
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(c(x1)) -> c(d(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [12]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [4]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
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:
{ b(d(b(x1))) -> a(d(x1))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
Weak Rules:
{ b(c(x1)) -> c(d(d(x1)))
, d(x1) -> x1
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(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:
{ b(d(b(x1))) -> a(d(x1))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
Weak Rules:
{ b(c(x1)) -> c(d(d(x1)))
, d(x1) -> x1
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ d_0(4) -> 16
, d_0(4) -> 17
, d_0(4) -> 18
, d_0(4) -> 19
, d_0(16) -> 17
, d_0(16) -> 18
, d_0(16) -> 19
, d_0(17) -> 18
, d_0(17) -> 19
, d_0(18) -> 19
, b_0(4) -> 12
, b_0(12) -> 11
, b_0(15) -> 14
, c_0(4) -> 4
, c_0(4) -> 16
, c_0(4) -> 17
, c_0(4) -> 18
, c_0(4) -> 19
, c_0(16) -> 15
, c_0(17) -> 12
, c_0(18) -> 14
, c_0(19) -> 11
, a^#_0(4) -> 5
, c_1_0(10) -> 5
, b^#_0(4) -> 9
, b^#_0(11) -> 10
, b^#_0(14) -> 13
, c_6_0(13) -> 5}
6) { a^#(x1) -> c_1(b^#(b(b(x1))))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a^#(x1) -> c_3()}
The usable rules for this path are the following:
{ d(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(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:
{ d(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a^#(x1) -> c_3()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a^#(x1) -> c_3()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a^#(x1) -> c_3()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [9]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(x1) -> x1}
and weakly orienting the rules
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a^#(x1) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(x1) -> x1}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [15]
d(x1) = [1] x1 + [3]
b(x1) = [1] x1 + [5]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [12]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(c(x1)) -> c(d(d(x1)))}
and weakly orienting the rules
{ d(x1) -> x1
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a^#(x1) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(c(x1)) -> c(d(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [6]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [7]
a^#(x1) = [1] x1 + [5]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [12]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
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:
{ b(d(b(x1))) -> a(d(x1))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
Weak Rules:
{ b(c(x1)) -> c(d(d(x1)))
, d(x1) -> x1
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a^#(x1) -> c_3()}
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:
{ b(d(b(x1))) -> a(d(x1))
, b^#(d(b(x1))) -> c_4(a^#(d(x1)))}
Weak Rules:
{ b(c(x1)) -> c(d(d(x1)))
, d(x1) -> x1
, a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, a(x1) -> x1
, a(c(x1)) -> b(b(c(d(x1))))
, a^#(x1) -> c_1(b^#(b(b(x1))))
, a^#(c(x1)) -> c_6(b^#(b(c(d(x1)))))
, a^#(x1) -> c_3()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ d_0(4) -> 16
, d_0(4) -> 17
, d_0(4) -> 18
, d_0(4) -> 19
, d_0(16) -> 17
, d_0(16) -> 18
, d_0(16) -> 19
, d_0(17) -> 18
, d_0(17) -> 19
, d_0(18) -> 19
, b_0(4) -> 12
, b_0(12) -> 11
, b_0(15) -> 14
, c_0(4) -> 4
, c_0(4) -> 16
, c_0(4) -> 17
, c_0(4) -> 18
, c_0(4) -> 19
, c_0(16) -> 15
, c_0(17) -> 12
, c_0(18) -> 14
, c_0(19) -> 11
, a^#_0(4) -> 5
, c_1_0(10) -> 5
, b^#_0(4) -> 9
, b^#_0(11) -> 10
, b^#_0(14) -> 13
, c_3_0() -> 5
, c_6_0(13) -> 5}