'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))
, b(d(b(x1))) -> a(d(x1))
, d(f(x1)) -> b(d(x1))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(d(x1)) -> c_0(d^#(b(c(b(d(x1))))))
, a^#(x1) -> c_1(b^#(b(f(b(b(x1))))))
, b^#(d(b(x1))) -> c_2(a^#(d(x1)))
, d^#(f(x1)) -> c_3(b^#(d(x1)))}
The usable rules are:
{ b(d(b(x1))) -> a(d(x1))
, d(f(x1)) -> b(d(x1))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))}
The estimated dependency graph contains the following edges:
{b^#(d(b(x1))) -> c_2(a^#(d(x1)))}
==> {a^#(x1) -> c_1(b^#(b(f(b(b(x1))))))}
{b^#(d(b(x1))) -> c_2(a^#(d(x1)))}
==> {a^#(d(x1)) -> c_0(d^#(b(c(b(d(x1))))))}
{d^#(f(x1)) -> c_3(b^#(d(x1)))}
==> {b^#(d(b(x1))) -> c_2(a^#(d(x1)))}
We consider the following path(s):
1) { d^#(f(x1)) -> c_3(b^#(d(x1)))
, b^#(d(b(x1))) -> c_2(a^#(d(x1)))
, a^#(d(x1)) -> c_0(d^#(b(c(b(d(x1))))))}
The usable rules for this path are the following:
{ b(d(b(x1))) -> a(d(x1))
, d(f(x1)) -> b(d(x1))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(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:
{ b(d(b(x1))) -> a(d(x1))
, d(f(x1)) -> b(d(x1))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))
, b^#(d(b(x1))) -> c_2(a^#(d(x1)))
, d^#(f(x1)) -> c_3(b^#(d(x1)))
, a^#(d(x1)) -> c_0(d^#(b(c(b(d(x1))))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(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(c(b(d(x1)))))
, a(x1) -> b(b(f(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]
f(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(f(x1)) -> c_3(b^#(d(x1)))}
and weakly orienting the rules
{ a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(f(x1)) -> c_3(b^#(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]
f(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [8]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(d(b(x1))) -> c_2(a^#(d(x1)))}
and weakly orienting the rules
{ d^#(f(x1)) -> c_3(b^#(d(x1)))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(d(b(x1))) -> c_2(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]
f(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [3]
d^#(x1) = [1] x1 + [8]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [7]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(f(x1)) -> b(d(x1))}
and weakly orienting the rules
{ b^#(d(b(x1))) -> c_2(a^#(d(x1)))
, d^#(f(x1)) -> c_3(b^#(d(x1)))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(f(x1)) -> b(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]
f(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [4]
d^#(x1) = [1] x1 + [13]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [2]
c_3(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(c(b(d(x1))))))}
and weakly orienting the rules
{ d(f(x1)) -> b(d(x1))
, b^#(d(b(x1))) -> c_2(a^#(d(x1)))
, d^#(f(x1)) -> c_3(b^#(d(x1)))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(d(x1)) -> c_0(d^#(b(c(b(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]
f(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(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))}
Weak Rules:
{ a^#(d(x1)) -> c_0(d^#(b(c(b(d(x1))))))
, d(f(x1)) -> b(d(x1))
, b^#(d(b(x1))) -> c_2(a^#(d(x1)))
, d^#(f(x1)) -> c_3(b^#(d(x1)))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(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))}
Weak Rules:
{ a^#(d(x1)) -> c_0(d^#(b(c(b(d(x1))))))
, d(f(x1)) -> b(d(x1))
, b^#(d(b(x1))) -> c_2(a^#(d(x1)))
, d^#(f(x1)) -> c_3(b^#(d(x1)))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ d_0(2) -> 4
, b_0(4) -> 4
, c_0(2) -> 2
, f_0(2) -> 2
, a^#_0(2) -> 1
, d^#_0(2) -> 1
, b^#_0(2) -> 1
, b^#_0(4) -> 3
, c_3_0(3) -> 1}
2) { d^#(f(x1)) -> c_3(b^#(d(x1)))
, b^#(d(b(x1))) -> c_2(a^#(d(x1)))
, a^#(x1) -> c_1(b^#(b(f(b(b(x1))))))}
The usable rules for this path are the following:
{ b(d(b(x1))) -> a(d(x1))
, d(f(x1)) -> b(d(x1))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(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:
{ b(d(b(x1))) -> a(d(x1))
, d(f(x1)) -> b(d(x1))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))
, b^#(d(b(x1))) -> c_2(a^#(d(x1)))
, d^#(f(x1)) -> c_3(b^#(d(x1)))
, a^#(x1) -> c_1(b^#(b(f(b(b(x1))))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(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(c(b(d(x1)))))
, a(x1) -> b(b(f(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]
f(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 + [0]
b^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(f(x1)) -> c_3(b^#(d(x1)))}
and weakly orienting the rules
{ a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(f(x1)) -> c_3(b^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(d(b(x1))) -> c_2(a^#(d(x1)))}
and weakly orienting the rules
{ d^#(f(x1)) -> c_3(b^#(d(x1)))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(d(b(x1))) -> c_2(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]
f(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 + [0]
b^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(f(x1)) -> b(d(x1))}
and weakly orienting the rules
{ b^#(d(b(x1))) -> c_2(a^#(d(x1)))
, d^#(f(x1)) -> c_3(b^#(d(x1)))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(f(x1)) -> b(d(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [14]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [15]
c_2(x1) = [1] x1 + [1]
c_3(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))
, a^#(x1) -> c_1(b^#(b(f(b(b(x1))))))}
Weak Rules:
{ d(f(x1)) -> b(d(x1))
, b^#(d(b(x1))) -> c_2(a^#(d(x1)))
, d^#(f(x1)) -> c_3(b^#(d(x1)))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(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(f(b(b(x1))))))}
Weak Rules:
{ d(f(x1)) -> b(d(x1))
, b^#(d(b(x1))) -> c_2(a^#(d(x1)))
, d^#(f(x1)) -> c_3(b^#(d(x1)))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ d_0(4) -> 12
, d_0(5) -> 12
, d_1(4) -> 19
, d_1(5) -> 19
, b_0(12) -> 12
, b_1(4) -> 17
, b_1(5) -> 17
, b_1(15) -> 14
, b_1(17) -> 16
, b_1(19) -> 19
, c_0(4) -> 4
, c_0(5) -> 4
, f_0(4) -> 5
, f_0(5) -> 5
, f_1(16) -> 15
, a^#_0(4) -> 6
, a^#_0(5) -> 6
, d^#_0(4) -> 8
, d^#_0(5) -> 8
, c_1_1(13) -> 6
, b^#_0(4) -> 10
, b^#_0(5) -> 10
, b^#_0(12) -> 11
, b^#_1(14) -> 13
, b^#_1(19) -> 18
, c_3_0(11) -> 8
, c_3_1(18) -> 8}
3) { d^#(f(x1)) -> c_3(b^#(d(x1)))
, b^#(d(b(x1))) -> c_2(a^#(d(x1)))}
The usable rules for this path are the following:
{ d(f(x1)) -> b(d(x1))
, b(d(b(x1))) -> a(d(x1))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(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(f(x1)) -> b(d(x1))
, b(d(b(x1))) -> a(d(x1))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))
, d^#(f(x1)) -> c_3(b^#(d(x1)))
, b^#(d(b(x1))) -> c_2(a^#(d(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b^#(d(b(x1))) -> c_2(a^#(d(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(d(b(x1))) -> c_2(a^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [7]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(f(x1)) -> c_3(b^#(d(x1)))}
and weakly orienting the rules
{b^#(d(b(x1))) -> c_2(a^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(f(x1)) -> c_3(b^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))}
and weakly orienting the rules
{ d^#(f(x1)) -> c_3(b^#(d(x1)))
, b^#(d(b(x1))) -> c_2(a^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [3]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(f(x1)) -> b(d(x1))}
and weakly orienting the rules
{ a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))
, d^#(f(x1)) -> c_3(b^#(d(x1)))
, b^#(d(b(x1))) -> c_2(a^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(f(x1)) -> b(d(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [2]
a^#(x1) = [1] x1 + [2]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [3]
c_2(x1) = [1] x1 + [1]
c_3(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))}
Weak Rules:
{ d(f(x1)) -> b(d(x1))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))
, d^#(f(x1)) -> c_3(b^#(d(x1)))
, b^#(d(b(x1))) -> c_2(a^#(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))}
Weak Rules:
{ d(f(x1)) -> b(d(x1))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))
, d^#(f(x1)) -> c_3(b^#(d(x1)))
, b^#(d(b(x1))) -> c_2(a^#(d(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ d_0(4) -> 12
, d_0(5) -> 12
, b_0(12) -> 12
, c_0(4) -> 4
, c_0(5) -> 4
, f_0(4) -> 5
, f_0(5) -> 5
, a^#_0(4) -> 6
, a^#_0(5) -> 6
, d^#_0(4) -> 8
, d^#_0(5) -> 8
, b^#_0(4) -> 10
, b^#_0(5) -> 10
, b^#_0(12) -> 11
, c_3_0(11) -> 8}
4) {d^#(f(x1)) -> c_3(b^#(d(x1)))}
The usable rules for this path are the following:
{ d(f(x1)) -> b(d(x1))
, b(d(b(x1))) -> a(d(x1))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(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(f(x1)) -> b(d(x1))
, b(d(b(x1))) -> a(d(x1))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))
, d^#(f(x1)) -> c_3(b^#(d(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{d^#(f(x1)) -> c_3(b^#(d(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(f(x1)) -> c_3(b^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))}
and weakly orienting the rules
{d^#(f(x1)) -> c_3(b^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [2]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [6]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(f(x1)) -> b(d(x1))}
and weakly orienting the rules
{ a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))
, d^#(f(x1)) -> c_3(b^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(f(x1)) -> b(d(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [2]
f(x1) = [1] x1 + [3]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(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))}
Weak Rules:
{ d(f(x1)) -> b(d(x1))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))
, d^#(f(x1)) -> c_3(b^#(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))}
Weak Rules:
{ d(f(x1)) -> b(d(x1))
, a(d(x1)) -> d(b(c(b(d(x1)))))
, a(x1) -> b(b(f(b(b(x1)))))
, d^#(f(x1)) -> c_3(b^#(d(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ d_0(4) -> 12
, d_0(5) -> 12
, b_0(12) -> 12
, c_0(4) -> 4
, c_0(5) -> 4
, f_0(4) -> 5
, f_0(5) -> 5
, d^#_0(4) -> 8
, d^#_0(5) -> 8
, b^#_0(4) -> 10
, b^#_0(5) -> 10
, b^#_0(12) -> 11
, c_3_0(11) -> 8}