'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, c^#(c(x1)) -> c_3()
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, b^#(x1) -> c_5()
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c^#(d(d(x1))) -> c_7(a^#(x1))}
The usable rules are:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
The estimated dependency graph contains the following edges:
{a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
==> {b^#(b(x1)) -> c_2(c^#(c(c(x1))))}
{a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
==> {b^#(x1) -> c_5()}
{a^#(x1) -> c_1(c^#(d(x1)))}
==> {c^#(d(d(x1))) -> c_7(a^#(x1))}
{b^#(b(x1)) -> c_2(c^#(c(c(x1))))}
==> {c^#(d(d(x1))) -> c_7(a^#(x1))}
{b^#(b(x1)) -> c_2(c^#(c(c(x1))))}
==> {c^#(c(x1)) -> c_3()}
{e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
==> {a^#(x1) -> c_1(c^#(d(x1)))}
{e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
==> {a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
{e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))}
==> {b^#(x1) -> c_5()}
{e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))}
==> {b^#(b(x1)) -> c_2(c^#(c(c(x1))))}
{c^#(d(d(x1))) -> c_7(a^#(x1))}
==> {a^#(x1) -> c_1(c^#(d(x1)))}
{c^#(d(d(x1))) -> c_7(a^#(x1))}
==> {a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
We consider the following path(s):
1) { e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))}
The usable rules for this path are the following:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
e^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_1(c^#(d(x1)))}
and weakly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_1(c^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
e^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [3]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
and weakly orienting the rules
{ a^#(x1) -> c_1(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
e^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))}
and weakly orienting the rules
{ a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [12]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [5]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
e^#(x1) = [1] x1 + [3]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(b(x1)) -> c_2(c^#(c(c(x1))))}
and weakly orienting the rules
{ e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(x1)) -> c_2(c^#(c(c(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [7]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [15]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(x1) -> d(d(x1))}
and weakly orienting the rules
{ b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(x1) -> d(d(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [5]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [14]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [6]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(x1)) -> b(b(b(x1)))}
and weakly orienting the rules
{ b(x1) -> d(d(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(x1)) -> b(b(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [2]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [7]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [15]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [15]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [2]
c_3() = [0]
e^#(x1) = [1] x1 + [12]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(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:
{ a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, c^#(d(d(x1))) -> c_7(a^#(x1))}
Weak Rules:
{ a(a(x1)) -> b(b(b(x1)))
, b(x1) -> d(d(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, c^#(d(d(x1))) -> c_7(a^#(x1))}
Weak Rules:
{ a(a(x1)) -> b(b(b(x1)))
, b(x1) -> d(d(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ d_0(2) -> 2
, d_1(2) -> 5
, a^#_0(2) -> 1
, a^#_1(2) -> 3
, b^#_0(2) -> 1
, c_1_0(1) -> 1
, c_1_1(4) -> 1
, c_1_1(4) -> 3
, c^#_0(2) -> 1
, c^#_1(5) -> 4
, e^#_0(2) -> 1
, c_7_1(3) -> 1
, c_7_1(3) -> 4}
2) { e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))}
The usable rules for this path are the following:
{ b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> 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:
{ b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [10]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(d(x1))) -> c_7(a^#(x1))}
and weakly orienting the rules
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(d(x1))) -> c_7(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))
, c(d(d(x1))) -> a(x1)}
and weakly orienting the rules
{ c^#(d(d(x1))) -> c_7(a^#(x1))
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))
, c(d(d(x1))) -> a(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [14]
b(x1) = [1] x1 + [6]
c(x1) = [1] x1 + [13]
d(x1) = [1] x1 + [1]
e(x1) = [1] x1 + [15]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [9]
c^#(x1) = [1] x1 + [7]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [4]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
and weakly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))
, c(d(d(x1))) -> a(x1)
, c^#(d(d(x1))) -> c_7(a^#(x1))
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
e^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_1(c^#(d(x1)))}
and weakly orienting the rules
{ e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))
, c(d(d(x1))) -> a(x1)
, c^#(d(d(x1))) -> c_7(a^#(x1))
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_1(c^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [4]
b(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [3]
d(x1) = [1] x1 + [1]
e(x1) = [1] x1 + [4]
a^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
e^#(x1) = [1] x1 + [13]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(b(x1)) -> c_2(c^#(c(c(x1))))}
and weakly orienting the rules
{ a^#(x1) -> c_1(c^#(d(x1)))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))
, c(d(d(x1))) -> a(x1)
, c^#(d(d(x1))) -> c_7(a^#(x1))
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(x1)) -> c_2(c^#(c(c(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [7]
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [3]
d(x1) = [1] x1 + [2]
e(x1) = [1] x1 + [2]
a^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [3]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [15]
c_4(x1) = [1] x1 + [2]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(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(b(x1)) -> c(c(c(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, e(c(x1)) -> b(a(a(e(x1))))}
Weak Rules:
{ b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))
, c(d(d(x1))) -> a(x1)
, c^#(d(d(x1))) -> c_7(a^#(x1))
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(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(b(x1)) -> c(c(c(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, e(c(x1)) -> b(a(a(e(x1))))}
Weak Rules:
{ b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))
, c(d(d(x1))) -> a(x1)
, c^#(d(d(x1))) -> c_7(a^#(x1))
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a_1(20) -> 18
, a_1(20) -> 23
, a_1(21) -> 18
, a_1(21) -> 23
, a_1(30) -> 18
, a_1(30) -> 23
, b_0(16) -> 15
, b_1(21) -> 20
, c_0(17) -> 16
, c_1(20) -> 18
, c_1(20) -> 23
, c_1(22) -> 21
, c_1(30) -> 18
, c_1(30) -> 23
, c_1(33) -> 18
, c_1(33) -> 23
, d_0(4) -> 4
, d_0(16) -> 19
, d_0(18) -> 17
, d_0(19) -> 15
, d_1(4) -> 25
, d_1(15) -> 27
, d_1(16) -> 29
, d_1(19) -> 35
, d_1(20) -> 33
, d_1(21) -> 30
, d_1(23) -> 22
, d_1(29) -> 15
, d_1(30) -> 20
, e_0(4) -> 18
, e_1(4) -> 23
, a^#_0(4) -> 6
, a^#_0(15) -> 14
, a^#_1(4) -> 31
, a^#_1(16) -> 37
, a^#_1(19) -> 32
, a^#_1(20) -> 28
, a^#_1(21) -> 41
, a^#_1(29) -> 32
, a^#_1(30) -> 38
, b^#_0(4) -> 8
, c_1_0(10) -> 6
, c_1_1(24) -> 6
, c_1_1(24) -> 31
, c_1_1(26) -> 14
, c_1_1(34) -> 32
, c_1_1(36) -> 28
, c_1_1(39) -> 37
, c_1_1(40) -> 38
, c_1_1(42) -> 41
, c^#_0(4) -> 10
, c^#_1(15) -> 34
, c^#_1(20) -> 40
, c^#_1(25) -> 24
, c^#_1(27) -> 26
, c^#_1(29) -> 39
, c^#_1(30) -> 42
, c^#_1(33) -> 36
, c^#_1(35) -> 34
, e^#_0(4) -> 13
, c_4_0(14) -> 13
, c_4_1(28) -> 13
, c_7_0(6) -> 10
, c_7_1(31) -> 24
, c_7_1(32) -> 26
, c_7_1(37) -> 34
, c_7_1(38) -> 36
, c_7_1(41) -> 40}
3) { e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, b^#(x1) -> c_5()}
The usable rules for this path are the following:
{ b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> 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:
{ b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, b^#(x1) -> c_5()}
Details:
We apply the weight gap principle, strictly orienting the rules
{e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [5]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
and weakly orienting the rules
{e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [3]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [3]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a^#(x1) -> c_1(c^#(d(x1)))
, b^#(x1) -> c_5()}
and weakly orienting the rules
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a^#(x1) -> c_1(c^#(d(x1)))
, b^#(x1) -> c_5()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [3]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [13]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [9]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(b(x1)) -> c_2(c^#(c(c(x1))))}
and weakly orienting the rules
{ a^#(x1) -> c_1(c^#(d(x1)))
, b^#(x1) -> c_5()
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(x1)) -> c_2(c^#(c(c(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [5]
a^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [2]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [13]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))}
and weakly orienting the rules
{ b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, b^#(x1) -> c_5()
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [4]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
e^#(x1) = [1] x1 + [8]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(x1)) -> c(c(c(x1)))}
and weakly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, b^#(x1) -> c_5()
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(x1)) -> c(c(c(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
e^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(d(x1))) -> c_7(a^#(x1))}
and weakly orienting the rules
{ b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, b^#(x1) -> c_5()
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(d(x1))) -> c_7(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [10]
b(x1) = [1] x1 + [5]
c(x1) = [1] x1 + [3]
d(x1) = [1] x1 + [1]
e(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [11]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(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:
{ e(d(x1)) -> a(b(c(d(e(x1)))))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
Weak Rules:
{ c^#(d(d(x1))) -> c_7(a^#(x1))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, b^#(x1) -> c_5()
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(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:
{ e(d(x1)) -> a(b(c(d(e(x1)))))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
Weak Rules:
{ c^#(d(d(x1))) -> c_7(a^#(x1))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, b^#(x1) -> c_5()
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
Details:
The problem is Match-bounded by 3.
The enriched problem is compatible with the following automaton:
{ a_1(19) -> 18
, a_1(19) -> 22
, a_2(20) -> 18
, a_2(20) -> 22
, a_2(24) -> 18
, a_2(24) -> 22
, a_2(33) -> 18
, a_3(20) -> 18
, a_3(20) -> 22
, a_3(33) -> 18
, a_3(33) -> 22
, b_0(16) -> 15
, b_1(20) -> 19
, c_0(17) -> 16
, c_1(21) -> 20
, c_1(29) -> 18
, c_2(37) -> 18
, c_2(37) -> 22
, c_3(45) -> 18
, c_3(45) -> 22
, d_0(4) -> 4
, d_0(18) -> 17
, d_1(4) -> 26
, d_1(15) -> 28
, d_1(16) -> 23
, d_1(19) -> 29
, d_1(20) -> 24
, d_1(22) -> 21
, d_1(23) -> 15
, d_1(24) -> 19
, d_2(4) -> 35
, d_2(19) -> 37
, d_2(20) -> 33
, d_2(23) -> 39
, d_2(24) -> 37
, d_2(33) -> 19
, d_3(16) -> 43
, d_3(20) -> 45
, d_3(24) -> 45
, d_3(33) -> 45
, e_0(4) -> 18
, e_1(4) -> 22
, a^#_0(4) -> 6
, a^#_0(15) -> 14
, a^#_1(4) -> 31
, a^#_1(19) -> 30
, a^#_1(23) -> 32
, a^#_2(16) -> 40
, a^#_2(20) -> 41
, a^#_2(24) -> 41
, a^#_2(33) -> 41
, a^#_3(20) -> 46
, b^#_0(4) -> 8
, c_1_0(10) -> 6
, c_1_1(25) -> 6
, c_1_1(27) -> 14
, c_1_2(34) -> 31
, c_1_2(36) -> 30
, c_1_2(38) -> 32
, c_1_3(42) -> 40
, c_1_3(44) -> 41
, c_1_3(44) -> 46
, c^#_0(4) -> 10
, c^#_1(26) -> 25
, c^#_1(28) -> 27
, c^#_2(35) -> 34
, c^#_2(37) -> 36
, c^#_2(39) -> 38
, c^#_3(43) -> 42
, c^#_3(45) -> 44
, e^#_0(4) -> 13
, c_4_0(14) -> 13
, c_4_1(30) -> 13
, c_5_0() -> 8
, c_7_0(6) -> 10
, c_7_1(31) -> 25
, c_7_1(31) -> 34
, c_7_1(32) -> 27
, c_7_2(40) -> 38
, c_7_2(41) -> 36
, c_7_2(41) -> 44
, c_7_3(46) -> 44}
4) { e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, c^#(c(x1)) -> c_3()}
The usable rules for this path are the following:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c^#(c(x1)) -> c_3()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, c^#(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(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, c^#(c(x1)) -> c_3()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [2]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [3]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))}
and weakly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, c^#(c(x1)) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [6]
c_3() = [0]
e^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(d(x1))) -> c_7(a^#(x1))}
and weakly orienting the rules
{ e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, c^#(c(x1)) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(d(x1))) -> c_7(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [7]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(b(x1)) -> c_2(c^#(c(c(x1))))}
and weakly orienting the rules
{ c^#(d(d(x1))) -> c_7(a^#(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, c^#(c(x1)) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(x1)) -> c_2(c^#(c(c(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [4]
b^#(x1) = [1] x1 + [7]
c_1(x1) = [1] x1 + [3]
c^#(x1) = [1] x1 + [2]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
e^#(x1) = [1] x1 + [15]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_1(c^#(d(x1)))}
and weakly orienting the rules
{ b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, c^#(c(x1)) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_1(c^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [7]
c(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [5]
e(x1) = [1] x1 + [3]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [9]
b^#(x1) = [1] x1 + [12]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [7]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(x1) -> d(d(x1))}
and weakly orienting the rules
{ a^#(x1) -> c_1(c^#(d(x1)))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, c^#(c(x1)) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(x1) -> d(d(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [3]
c(x1) = [1] x1 + [3]
d(x1) = [1] x1 + [1]
e(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [14]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [6]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
e^#(x1) = [1] x1 + [14]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(x1)) -> b(b(b(x1)))}
and weakly orienting the rules
{ b(x1) -> d(d(x1))
, a^#(x1) -> c_1(c^#(d(x1)))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, c^#(c(x1)) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(x1)) -> b(b(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [2]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [5]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [15]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
e^#(x1) = [1] x1 + [15]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(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:
{ a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
Weak Rules:
{ a(a(x1)) -> b(b(b(x1)))
, b(x1) -> d(d(x1))
, a^#(x1) -> c_1(c^#(d(x1)))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, c^#(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:
{ a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
Weak Rules:
{ a(a(x1)) -> b(b(b(x1)))
, b(x1) -> d(d(x1))
, a^#(x1) -> c_1(c^#(d(x1)))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, c^#(c(x1)) -> c_3()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ d_0(4) -> 4
, a^#_0(4) -> 6
, b^#_0(4) -> 8
, c_1_0(10) -> 6
, c^#_0(4) -> 10
, e^#_0(4) -> 13
, c_7_0(6) -> 10}
5) { e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, b^#(x1) -> c_5()}
The usable rules for this path are the following:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, b^#(x1) -> c_5()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))}
and weakly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(d(x1))) -> c_7(a^#(x1))}
and weakly orienting the rules
{ e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(d(x1))) -> c_7(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [15]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, b^#(x1) -> c_5()}
and weakly orienting the rules
{ c^#(d(d(x1))) -> c_7(a^#(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, b^#(x1) -> c_5()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [3]
c^#(x1) = [1] x1 + [2]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
e^#(x1) = [1] x1 + [12]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_1(c^#(d(x1)))}
and weakly orienting the rules
{ b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, b^#(x1) -> c_5()
, c^#(d(d(x1))) -> c_7(a^#(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_1(c^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [2]
e(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [3]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [12]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [11]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(x1) -> d(d(x1))}
and weakly orienting the rules
{ a^#(x1) -> c_1(c^#(d(x1)))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, b^#(x1) -> c_5()
, c^#(d(d(x1))) -> c_7(a^#(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(x1) -> d(d(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [11]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [12]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(x1)) -> b(b(b(x1)))}
and weakly orienting the rules
{ b(x1) -> d(d(x1))
, a^#(x1) -> c_1(c^#(d(x1)))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, b^#(x1) -> c_5()
, c^#(d(d(x1))) -> c_7(a^#(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(x1)) -> b(b(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [5]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
e^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(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:
{ a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
Weak Rules:
{ a(a(x1)) -> b(b(b(x1)))
, b(x1) -> d(d(x1))
, a^#(x1) -> c_1(c^#(d(x1)))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, b^#(x1) -> c_5()
, c^#(d(d(x1))) -> c_7(a^#(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))}
Weak Rules:
{ a(a(x1)) -> b(b(b(x1)))
, b(x1) -> d(d(x1))
, a^#(x1) -> c_1(c^#(d(x1)))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, b^#(x1) -> c_5()
, c^#(d(d(x1))) -> c_7(a^#(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, c(c(x1)) -> d(d(d(x1)))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ d_0(4) -> 4
, a^#_0(4) -> 6
, b^#_0(4) -> 8
, c_1_0(10) -> 6
, c^#_0(4) -> 10
, e^#_0(4) -> 13
, c_5_0() -> 8
, c_7_0(6) -> 10}
6) { e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, c^#(c(x1)) -> c_3()}
The usable rules for this path are the following:
{ b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> 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:
{ b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, a^#(x1) -> c_1(c^#(d(x1)))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, c^#(c(x1)) -> c_3()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(c(x1)) -> c_3()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(c(x1)) -> c_3()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [15]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(d(x1))) -> c_7(a^#(x1))}
and weakly orienting the rules
{ a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(c(x1)) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(d(x1))) -> c_7(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))}
and weakly orienting the rules
{ c^#(d(d(x1))) -> c_7(a^#(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(c(x1)) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [11]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [3]
a^#(x1) = [1] x1 + [5]
c_0(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [6]
c_2(x1) = [1] x1 + [11]
c_3() = [0]
e^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
and weakly orienting the rules
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(c(x1)) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
e^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(b(x1)) -> c(c(c(x1)))
, c(d(d(x1))) -> a(x1)
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))}
and weakly orienting the rules
{ e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(c(x1)) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(x1)) -> c(c(c(x1)))
, c(d(d(x1))) -> a(x1)
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [15]
b(x1) = [1] x1 + [3]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [15]
e(x1) = [1] x1 + [4]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
e^#(x1) = [1] x1 + [8]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_1(c^#(d(x1)))}
and weakly orienting the rules
{ b(b(x1)) -> c(c(c(x1)))
, c(d(d(x1))) -> a(x1)
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(c(x1)) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_1(c^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [12]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
e(x1) = [1] x1 + [4]
a^#(x1) = [1] x1 + [11]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [5]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [2]
c_3() = [0]
e^#(x1) = [1] x1 + [15]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(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:
{ c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))}
Weak Rules:
{ a^#(x1) -> c_1(c^#(d(x1)))
, b(b(x1)) -> c(c(c(x1)))
, c(d(d(x1))) -> a(x1)
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(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:
{ c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))}
Weak Rules:
{ a^#(x1) -> c_1(c^#(d(x1)))
, b(b(x1)) -> c(c(c(x1)))
, c(d(d(x1))) -> a(x1)
, b^#(b(x1)) -> c_2(c^#(c(c(x1))))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, c^#(d(d(x1))) -> c_7(a^#(x1))
, a^#(a(x1)) -> c_0(b^#(b(b(x1))))
, c^#(c(x1)) -> c_3()}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a_1(19) -> 18
, a_1(19) -> 22
, a_2(20) -> 18
, a_2(20) -> 22
, a_2(24) -> 18
, a_2(24) -> 22
, b_0(16) -> 15
, b_1(20) -> 19
, c_0(17) -> 16
, c_1(21) -> 20
, c_2(19) -> 18
, c_2(19) -> 22
, c_2(24) -> 18
, c_2(24) -> 22
, c_2(30) -> 18
, c_2(30) -> 22
, d_0(4) -> 4
, d_0(18) -> 17
, d_1(4) -> 26
, d_1(15) -> 28
, d_1(16) -> 23
, d_1(22) -> 21
, d_1(23) -> 15
, d_2(4) -> 34
, d_2(16) -> 41
, d_2(19) -> 30
, d_2(20) -> 24
, d_2(23) -> 37
, d_2(24) -> 19
, e_0(4) -> 18
, e_1(4) -> 22
, a^#_0(4) -> 6
, a^#_0(15) -> 14
, a^#_1(4) -> 31
, a^#_1(19) -> 29
, a^#_1(23) -> 32
, a^#_2(16) -> 38
, a^#_2(20) -> 43
, a^#_2(24) -> 39
, b^#_0(4) -> 8
, c_1_0(10) -> 6
, c_1_1(25) -> 6
, c_1_1(27) -> 14
, c_1_2(33) -> 31
, c_1_2(35) -> 29
, c_1_2(36) -> 32
, c_1_2(40) -> 38
, c_1_2(42) -> 39
, c_1_2(44) -> 43
, c^#_0(4) -> 10
, c^#_1(26) -> 25
, c^#_1(28) -> 27
, c^#_2(19) -> 42
, c^#_2(24) -> 44
, c^#_2(30) -> 35
, c^#_2(34) -> 33
, c^#_2(37) -> 36
, c^#_2(41) -> 40
, e^#_0(4) -> 13
, c_4_0(14) -> 13
, c_4_1(29) -> 13
, c_7_0(6) -> 10
, c_7_1(31) -> 25
, c_7_1(31) -> 33
, c_7_1(32) -> 27
, c_7_2(38) -> 36
, c_7_2(39) -> 35
, c_7_2(43) -> 42}
7) {e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
The usable rules for this path are the following:
{ b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> 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:
{ b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)
, a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c(d(d(x1))) -> a(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(d(d(x1))) -> a(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [8]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
and weakly orienting the rules
{c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(x1)) -> b(b(b(x1)))}
and weakly orienting the rules
{ e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(x1)) -> b(b(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
e(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(x1) -> c(d(x1))}
and weakly orienting the rules
{ a(a(x1)) -> b(b(b(x1)))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(x1) -> c(d(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [2]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
e(x1) = [1] x1 + [3]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [8]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(c(x1)) -> d(d(d(x1)))}
and weakly orienting the rules
{ a(x1) -> c(d(x1))
, a(a(x1)) -> b(b(b(x1)))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(c(x1)) -> d(d(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [2]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [10]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(x1) -> d(d(x1))}
and weakly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, a(x1) -> c(d(x1))
, a(a(x1)) -> b(b(b(x1)))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(x1) -> d(d(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [10]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(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(b(x1)) -> c(c(c(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, e(c(x1)) -> b(a(a(e(x1))))}
Weak Rules:
{ b(x1) -> d(d(x1))
, c(c(x1)) -> d(d(d(x1)))
, a(x1) -> c(d(x1))
, a(a(x1)) -> b(b(b(x1)))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, c(d(d(x1))) -> a(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(b(x1)) -> c(c(c(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, e(c(x1)) -> b(a(a(e(x1))))}
Weak Rules:
{ b(x1) -> d(d(x1))
, c(c(x1)) -> d(d(d(x1)))
, a(x1) -> c(d(x1))
, a(a(x1)) -> b(b(b(x1)))
, e^#(d(x1)) -> c_4(a^#(b(c(d(e(x1))))))
, c(d(d(x1))) -> a(x1)}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a_1(19) -> 18
, a_1(19) -> 22
, a_1(20) -> 18
, a_1(20) -> 22
, a_1(24) -> 18
, a_1(24) -> 22
, b_0(16) -> 15
, b_1(20) -> 19
, c_0(17) -> 16
, c_1(19) -> 18
, c_1(19) -> 22
, c_1(21) -> 20
, c_1(24) -> 18
, c_1(24) -> 22
, c_1(25) -> 18
, c_1(25) -> 22
, d_0(4) -> 4
, d_0(18) -> 17
, d_1(16) -> 23
, d_1(19) -> 25
, d_1(20) -> 24
, d_1(22) -> 21
, d_1(23) -> 15
, d_1(24) -> 19
, e_0(4) -> 18
, e_1(4) -> 22
, a^#_0(4) -> 6
, a^#_0(15) -> 14
, a^#_1(19) -> 26
, e^#_0(4) -> 13
, c_4_0(14) -> 13
, c_4_1(26) -> 13}
8) { e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, b^#(x1) -> c_5()}
The usable rules for this path are the following:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, e(c(x1)) -> b(a(a(e(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))
, c(d(d(x1))) -> a(x1)}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, e(c(x1)) -> b(a(a(e(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))
, c(d(d(x1))) -> a(x1)
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, b^#(x1) -> c_5()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ e(c(x1)) -> b(a(a(e(x1))))
, c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ e(c(x1)) -> b(a(a(e(x1))))
, c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(x1) -> c_5()}
and weakly orienting the rules
{ e(c(x1)) -> b(a(a(e(x1))))
, c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(x1) -> c_5()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [3]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))}
and weakly orienting the rules
{ b^#(x1) -> c_5()
, e(c(x1)) -> b(a(a(e(x1))))
, c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(x1) -> d(d(x1))}
and weakly orienting the rules
{ e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, b^#(x1) -> c_5()
, e(c(x1)) -> b(a(a(e(x1))))
, c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(x1) -> d(d(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [5]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(x1)) -> b(b(b(x1)))}
and weakly orienting the rules
{ b(x1) -> d(d(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, b^#(x1) -> c_5()
, e(c(x1)) -> b(a(a(e(x1))))
, c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(x1)) -> b(b(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [2]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [6]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(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:
{ a(x1) -> c(d(x1))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(b(x1)) -> c(c(c(x1)))}
Weak Rules:
{ a(a(x1)) -> b(b(b(x1)))
, b(x1) -> d(d(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, b^#(x1) -> c_5()
, e(c(x1)) -> b(a(a(e(x1))))
, c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ a(x1) -> c(d(x1))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(b(x1)) -> c(c(c(x1)))}
Weak Rules:
{ a(a(x1)) -> b(b(b(x1)))
, b(x1) -> d(d(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, b^#(x1) -> c_5()
, e(c(x1)) -> b(a(a(e(x1))))
, c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ d_0(4) -> 4
, b^#_0(4) -> 8
, e^#_0(4) -> 13
, c_5_0() -> 8}
9) {e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))}
The usable rules for this path are the following:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, e(c(x1)) -> b(a(a(e(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))
, c(d(d(x1))) -> a(x1)}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, e(c(x1)) -> b(a(a(e(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, b(x1) -> d(d(x1))
, c(d(d(x1))) -> a(x1)
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ e(c(x1)) -> b(a(a(e(x1))))
, c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ e(c(x1)) -> b(a(a(e(x1))))
, c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))}
and weakly orienting the rules
{ e(c(x1)) -> b(a(a(e(x1))))
, c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(x1) -> d(d(x1))}
and weakly orienting the rules
{ e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, e(c(x1)) -> b(a(a(e(x1))))
, c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(x1) -> d(d(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(x1)) -> b(b(b(x1)))}
and weakly orienting the rules
{ b(x1) -> d(d(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, e(c(x1)) -> b(a(a(e(x1))))
, c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(x1)) -> b(b(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [2]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [6]
d(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [6]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
e^#(x1) = [1] x1 + [13]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(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:
{ a(x1) -> c(d(x1))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(b(x1)) -> c(c(c(x1)))}
Weak Rules:
{ a(a(x1)) -> b(b(b(x1)))
, b(x1) -> d(d(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, e(c(x1)) -> b(a(a(e(x1))))
, c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ a(x1) -> c(d(x1))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(b(x1)) -> c(c(c(x1)))}
Weak Rules:
{ a(a(x1)) -> b(b(b(x1)))
, b(x1) -> d(d(x1))
, e^#(c(x1)) -> c_6(b^#(a(a(e(x1)))))
, e(c(x1)) -> b(a(a(e(x1))))
, c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ d_0(4) -> 4
, b^#_0(4) -> 8
, e^#_0(4) -> 13}