'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ c(c(c(a(x1)))) -> d(d(x1))
, d(b(x1)) -> c(c(x1))
, b(c(x1)) -> b(a(c(x1)))
, c(x1) -> a(a(x1))
, d(x1) -> b(c(x1))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, b^#(c(x1)) -> c_2(b^#(a(c(x1))))
, c^#(x1) -> c_3()
, d^#(x1) -> c_4(b^#(c(x1)))}
The usable rules are:
{ c(c(c(a(x1)))) -> d(d(x1))
, d(b(x1)) -> c(c(x1))
, c(x1) -> a(a(x1))
, d(x1) -> b(c(x1))
, b(c(x1)) -> b(a(c(x1)))}
The estimated dependency graph contains the following edges:
{c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
==> {d^#(x1) -> c_4(b^#(c(x1)))}
{c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
==> {d^#(b(x1)) -> c_1(c^#(c(x1)))}
{d^#(b(x1)) -> c_1(c^#(c(x1)))}
==> {c^#(x1) -> c_3()}
{d^#(b(x1)) -> c_1(c^#(c(x1)))}
==> {c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
{d^#(x1) -> c_4(b^#(c(x1)))}
==> {b^#(c(x1)) -> c_2(b^#(a(c(x1))))}
We consider the following path(s):
1) { c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
The usable rules for this path are the following:
{ c(c(c(a(x1)))) -> d(d(x1))
, d(b(x1)) -> c(c(x1))
, c(x1) -> a(a(x1))
, d(x1) -> b(c(x1))
, b(c(x1)) -> b(a(c(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:
{ c(c(c(a(x1)))) -> d(d(x1))
, d(b(x1)) -> c(c(x1))
, c(x1) -> a(a(x1))
, d(x1) -> b(c(x1))
, b(c(x1)) -> b(a(c(x1)))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{d^#(b(x1)) -> c_1(c^#(c(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(b(x1)) -> c_1(c^#(c(x1)))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(b(x1)) -> c(c(x1))}
and weakly orienting the rules
{d^#(b(x1)) -> c_1(c^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(b(x1)) -> c(c(x1))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [8]
c^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
and weakly orienting the rules
{ d(b(x1)) -> c(c(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [3]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(c(c(a(x1)))) -> d(d(x1))}
and weakly orienting the rules
{ c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d(b(x1)) -> c(c(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(c(c(a(x1)))) -> d(d(x1))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [6]
a(x1) = [1] x1 + [14]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [13]
c^#(x1) = [1] x1 + [12]
c_0(x1) = [1] x1 + [8]
d^#(x1) = [1] x1 + [12]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(x1) -> a(a(x1))}
and weakly orienting the rules
{ c(c(c(a(x1)))) -> d(d(x1))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d(b(x1)) -> c(c(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(x1) -> a(a(x1))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [9]
c^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_1(x1) = [1] x1 + [7]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(x1) -> b(c(x1))}
and weakly orienting the rules
{ c(x1) -> a(a(x1))
, c(c(c(a(x1)))) -> d(d(x1))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d(b(x1)) -> c(c(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(x1) -> b(c(x1))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [4]
a(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [7]
b(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [3]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] 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(c(x1)) -> b(a(c(x1)))}
Weak Rules:
{ d(x1) -> b(c(x1))
, c(x1) -> a(a(x1))
, c(c(c(a(x1)))) -> d(d(x1))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d(b(x1)) -> c(c(x1))
, d^#(b(x1)) -> c_1(c^#(c(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(c(x1)) -> b(a(c(x1)))}
Weak Rules:
{ d(x1) -> b(c(x1))
, c(x1) -> a(a(x1))
, c(c(c(a(x1)))) -> d(d(x1))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d(b(x1)) -> c(c(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 2
, c^#_0(2) -> 5
, d^#_0(2) -> 7}
2) { c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, d^#(x1) -> c_4(b^#(c(x1)))}
The usable rules for this path are the following:
{ c(c(c(a(x1)))) -> d(d(x1))
, d(b(x1)) -> c(c(x1))
, c(x1) -> a(a(x1))
, d(x1) -> b(c(x1))
, b(c(x1)) -> b(a(c(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:
{ c(c(c(a(x1)))) -> d(d(x1))
, d(b(x1)) -> c(c(x1))
, c(x1) -> a(a(x1))
, d(x1) -> b(c(x1))
, b(c(x1)) -> b(a(c(x1)))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, d^#(x1) -> c_4(b^#(c(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c^#(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 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(x1) -> c_4(b^#(c(x1)))}
and weakly orienting the rules
{c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(x1) -> c_4(b^#(c(x1)))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [7]
d^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ d(b(x1)) -> c(c(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
and weakly orienting the rules
{ d^#(x1) -> c_4(b^#(c(x1)))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ d(b(x1)) -> c(c(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [8]
c^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(c(c(a(x1)))) -> d(d(x1))}
and weakly orienting the rules
{ d(b(x1)) -> c(c(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, d^#(x1) -> c_4(b^#(c(x1)))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(c(c(a(x1)))) -> d(d(x1))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [8]
c^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [4]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(x1) -> a(a(x1))}
and weakly orienting the rules
{ c(c(c(a(x1)))) -> d(d(x1))
, d(b(x1)) -> c(c(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, d^#(x1) -> c_4(b^#(c(x1)))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(x1) -> a(a(x1))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [5]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [11]
c^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [9]
b^#(x1) = [1] x1 + [2]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(x1) -> b(c(x1))}
and weakly orienting the rules
{ c(x1) -> a(a(x1))
, c(c(c(a(x1)))) -> d(d(x1))
, d(b(x1)) -> c(c(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, d^#(x1) -> c_4(b^#(c(x1)))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(x1) -> b(c(x1))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [4]
a(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [7]
b(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [12]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [4]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(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(c(x1)) -> b(a(c(x1)))}
Weak Rules:
{ d(x1) -> b(c(x1))
, c(x1) -> a(a(x1))
, c(c(c(a(x1)))) -> d(d(x1))
, d(b(x1)) -> c(c(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, d^#(x1) -> c_4(b^#(c(x1)))
, c^#(c(c(a(x1)))) -> c_0(d^#(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(c(x1)) -> b(a(c(x1)))}
Weak Rules:
{ d(x1) -> b(c(x1))
, c(x1) -> a(a(x1))
, c(c(c(a(x1)))) -> d(d(x1))
, d(b(x1)) -> c(c(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, d^#(x1) -> c_4(b^#(c(x1)))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 11
, a_0(2) -> 2
, a_0(2) -> 11
, c^#_0(2) -> 5
, d^#_0(2) -> 7
, b^#_0(2) -> 9
, b^#_0(11) -> 10
, c_4_0(10) -> 7}
3) { c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, d^#(x1) -> c_4(b^#(c(x1)))
, b^#(c(x1)) -> c_2(b^#(a(c(x1))))}
The usable rules for this path are the following:
{ c(c(c(a(x1)))) -> d(d(x1))
, d(b(x1)) -> c(c(x1))
, c(x1) -> a(a(x1))
, d(x1) -> b(c(x1))
, b(c(x1)) -> b(a(c(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:
{ c(c(c(a(x1)))) -> d(d(x1))
, d(b(x1)) -> c(c(x1))
, c(x1) -> a(a(x1))
, d(x1) -> b(c(x1))
, b(c(x1)) -> b(a(c(x1)))
, d^#(x1) -> c_4(b^#(c(x1)))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, b^#(c(x1)) -> c_2(b^#(a(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ d^#(x1) -> c_4(b^#(c(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ d^#(x1) -> c_4(b^#(c(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [3]
d^#(x1) = [1] x1 + [4]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [7]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(b(x1)) -> c(c(x1))}
and weakly orienting the rules
{ d^#(x1) -> c_4(b^#(c(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(b(x1)) -> c(c(x1))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [5]
b(x1) = [1] x1 + [8]
c^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [8]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
and weakly orienting the rules
{ d(b(x1)) -> c(c(x1))
, d^#(x1) -> c_4(b^#(c(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c^#(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 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ c(c(c(a(x1)))) -> d(d(x1))
, d(x1) -> b(c(x1))}
and weakly orienting the rules
{ c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d(b(x1)) -> c(c(x1))
, d^#(x1) -> c_4(b^#(c(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(c(c(a(x1)))) -> d(d(x1))
, d(x1) -> b(c(x1))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [6]
a(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [11]
b(x1) = [1] x1 + [3]
c^#(x1) = [1] x1 + [11]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [14]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [2]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(x1) -> a(a(x1))}
and weakly orienting the rules
{ c(c(c(a(x1)))) -> d(d(x1))
, d(x1) -> b(c(x1))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d(b(x1)) -> c(c(x1))
, d^#(x1) -> c_4(b^#(c(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(x1) -> a(a(x1))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [10]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [15]
b(x1) = [1] x1 + [5]
c^#(x1) = [1] x1 + [10]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [2]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(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(c(x1)) -> b(a(c(x1)))
, b^#(c(x1)) -> c_2(b^#(a(c(x1))))}
Weak Rules:
{ c(x1) -> a(a(x1))
, c(c(c(a(x1)))) -> d(d(x1))
, d(x1) -> b(c(x1))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d(b(x1)) -> c(c(x1))
, d^#(x1) -> c_4(b^#(c(x1)))
, d^#(b(x1)) -> c_1(c^#(c(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(c(x1)) -> b(a(c(x1)))
, b^#(c(x1)) -> c_2(b^#(a(c(x1))))}
Weak Rules:
{ c(x1) -> a(a(x1))
, c(c(c(a(x1)))) -> d(d(x1))
, d(x1) -> b(c(x1))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d(b(x1)) -> c(c(x1))
, d^#(x1) -> c_4(b^#(c(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 11
, c_1(2) -> 14
, c_2(2) -> 19
, a_0(2) -> 2
, a_1(2) -> 15
, a_1(14) -> 13
, a_1(15) -> 11
, a_1(15) -> 14
, a_2(2) -> 20
, a_2(19) -> 18
, a_2(20) -> 14
, a_2(20) -> 19
, c^#_0(2) -> 5
, d^#_0(2) -> 7
, b^#_0(2) -> 9
, b^#_0(11) -> 10
, b^#_1(13) -> 12
, b^#_1(14) -> 16
, b^#_2(18) -> 17
, c_2_1(12) -> 10
, c_2_2(17) -> 16
, c_4_0(10) -> 7
, c_4_1(16) -> 7}
4) { c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, c^#(x1) -> c_3()}
The usable rules for this path are the following:
{ c(c(c(a(x1)))) -> d(d(x1))
, d(b(x1)) -> c(c(x1))
, c(x1) -> a(a(x1))
, d(x1) -> b(c(x1))
, b(c(x1)) -> b(a(c(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:
{ c(c(c(a(x1)))) -> d(d(x1))
, d(b(x1)) -> c(c(x1))
, c(x1) -> a(a(x1))
, d(x1) -> b(c(x1))
, b(c(x1)) -> b(a(c(x1)))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, c^#(x1) -> c_3()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, c^#(x1) -> c_3()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, c^#(x1) -> c_3()}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [4]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(b(x1)) -> c(c(x1))}
and weakly orienting the rules
{ c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, c^#(x1) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(b(x1)) -> c(c(x1))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [8]
c^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [8]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(b(x1)) -> c_1(c^#(c(x1)))}
and weakly orienting the rules
{ d(b(x1)) -> c(c(x1))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, c^#(x1) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(b(x1)) -> c_1(c^#(c(x1)))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(c(c(a(x1)))) -> d(d(x1))}
and weakly orienting the rules
{ d^#(b(x1)) -> c_1(c^#(c(x1)))
, d(b(x1)) -> c(c(x1))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, c^#(x1) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(c(c(a(x1)))) -> d(d(x1))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [2]
a(x1) = [1] x1 + [12]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [7]
c^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(x1) -> a(a(x1))}
and weakly orienting the rules
{ c(c(c(a(x1)))) -> d(d(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, d(b(x1)) -> c(c(x1))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, c^#(x1) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(x1) -> a(a(x1))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [9]
c^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(x1) -> b(c(x1))}
and weakly orienting the rules
{ c(x1) -> a(a(x1))
, c(c(c(a(x1)))) -> d(d(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, d(b(x1)) -> c(c(x1))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, c^#(x1) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(x1) -> b(c(x1))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [3]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [5]
b(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [3]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] 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(c(x1)) -> b(a(c(x1)))}
Weak Rules:
{ d(x1) -> b(c(x1))
, c(x1) -> a(a(x1))
, c(c(c(a(x1)))) -> d(d(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, d(b(x1)) -> c(c(x1))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, c^#(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(c(x1)) -> b(a(c(x1)))}
Weak Rules:
{ d(x1) -> b(c(x1))
, c(x1) -> a(a(x1))
, c(c(c(a(x1)))) -> d(d(x1))
, d^#(b(x1)) -> c_1(c^#(c(x1)))
, d(b(x1)) -> c(c(x1))
, c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
, c^#(x1) -> c_3()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 2
, c^#_0(2) -> 5
, d^#_0(2) -> 7
, c_3_0() -> 5}