'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(x1)) -> b(c(c(c(x1))))
, b(c(x1)) -> d(d(d(d(x1))))
, a(x1) -> d(c(d(x1)))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(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^#(c(c(c(x1)))))
, b^#(c(x1)) -> c_1()
, a^#(x1) -> c_2(c^#(d(x1)))
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, c^#(c(x1)) -> c_4()
, c^#(d(d(x1))) -> c_5(a^#(x1))}
The usable rules are:
{ c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)
, a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))}
The estimated dependency graph contains the following edges:
{a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))}
==> {b^#(b(x1)) -> c_3(c^#(c(c(x1))))}
{a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))}
==> {b^#(c(x1)) -> c_1()}
{a^#(x1) -> c_2(c^#(d(x1)))}
==> {c^#(d(d(x1))) -> c_5(a^#(x1))}
{b^#(b(x1)) -> c_3(c^#(c(c(x1))))}
==> {c^#(d(d(x1))) -> c_5(a^#(x1))}
{b^#(b(x1)) -> c_3(c^#(c(c(x1))))}
==> {c^#(c(x1)) -> c_4()}
{c^#(d(d(x1))) -> c_5(a^#(x1))}
==> {a^#(x1) -> c_2(c^#(d(x1)))}
{c^#(d(d(x1))) -> c_5(a^#(x1))}
==> {a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))}
We consider the following path(s):
1) { a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(x1) -> c_2(c^#(d(x1)))}
The usable rules for this path are the following:
{ c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)
, a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(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(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)
, a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(x1) -> c_2(c^#(d(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(d(d(x1))) -> c_5(a^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(d(d(x1))) -> c_5(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [4]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [5]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(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^#(c(c(c(x1)))))}
and weakly orienting the rules
{ b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(d(d(x1))) -> c_5(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [7]
c^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(b(x1)) -> c_3(c^#(c(c(x1))))}
and weakly orienting the rules
{ a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(d(d(x1))) -> c_5(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(x1)) -> c_3(c^#(c(c(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(d(d(x1))) -> a(x1)}
and weakly orienting the rules
{ b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(d(d(x1))) -> c_5(a^#(x1))}
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 + [1]
b(x1) = [1] x1 + [14]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [2]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [14]
c_3(x1) = [1] x1 + [1]
c_4() = [0]
c_5(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_2(c^#(d(x1)))}
and weakly orienting the rules
{ c(d(d(x1))) -> a(x1)
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(d(d(x1))) -> c_5(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_2(c^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [4]
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [2]
a^#(x1) = [1] x1 + [12]
c_0(x1) = [1] x1 + [3]
b^#(x1) = [1] x1 + [8]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [1] 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_2(c^#(d(x1)))
, c(d(d(x1))) -> a(x1)
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(d(d(x1))) -> c_5(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 + [15]
c(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [2]
a^#(x1) = [1] x1 + [11]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(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(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))}
Weak Rules:
{ c(c(x1)) -> d(d(d(x1)))
, a^#(x1) -> c_2(c^#(d(x1)))
, c(d(d(x1))) -> a(x1)
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(d(d(x1))) -> c_5(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(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))}
Weak Rules:
{ c(c(x1)) -> d(d(d(x1)))
, a^#(x1) -> c_2(c^#(d(x1)))
, c(d(d(x1))) -> a(x1)
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(d(d(x1))) -> c_5(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) -> 5
, b^#_0(4) -> 7
, c_2_0(10) -> 5
, c^#_0(4) -> 10
, c_5_0(5) -> 10}
2) { a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(x1) -> c_2(c^#(d(x1)))
, c^#(c(x1)) -> c_4()}
The usable rules for this path are the following:
{ c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)
, a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(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(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)
, a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(x1) -> c_2(c^#(d(x1)))
, c^#(c(x1)) -> c_4()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(c(x1)) -> c_4()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(c(x1)) -> c_4()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(d(x1))) -> c_5(a^#(x1))}
and weakly orienting the rules
{ b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(c(x1)) -> c_4()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(d(x1))) -> c_5(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [11]
c_4() = [0]
c_5(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^#(c(c(c(x1)))))}
and weakly orienting the rules
{ c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(c(x1)) -> c_4()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [4]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(b(x1)) -> c_3(c^#(c(c(x1))))}
and weakly orienting the rules
{ a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(c(x1)) -> c_4()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(x1)) -> c_3(c^#(c(c(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(x1) -> d(c(d(x1)))}
and weakly orienting the rules
{ b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(c(x1)) -> c_4()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(x1) -> d(c(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1() = [0]
c_2(x1) = [1] x1 + [8]
c^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
c_4() = [0]
c_5(x1) = [1] x1 + [4]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(x1)) -> b(c(c(c(x1))))}
and weakly orienting the rules
{ a(x1) -> d(c(d(x1)))
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(c(x1)) -> c_4()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(x1)) -> b(c(c(c(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [15]
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
c_4() = [0]
c_5(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_2(c^#(d(x1)))}
and weakly orienting the rules
{ a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(c(x1)) -> c_4()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_2(c^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [9]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [2]
a^#(x1) = [1] x1 + [15]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [15]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [12]
c_3(x1) = [1] x1 + [1]
c_4() = [0]
c_5(x1) = [1] 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_2(c^#(d(x1)))
, a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(c(x1)) -> c_4()}
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 + [15]
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [5]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [5]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(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(d(d(x1))) -> a(x1)}
Weak Rules:
{ c(c(x1)) -> d(d(d(x1)))
, a^#(x1) -> c_2(c^#(d(x1)))
, a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(c(x1)) -> c_4()}
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(d(d(x1))) -> a(x1)}
Weak Rules:
{ c(c(x1)) -> d(d(d(x1)))
, a^#(x1) -> c_2(c^#(d(x1)))
, a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, c^#(c(x1)) -> c_4()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ d_0(4) -> 4
, a^#_0(4) -> 5
, b^#_0(4) -> 7
, c_2_0(10) -> 5
, c^#_0(4) -> 10
, c_5_0(5) -> 10}
3) { a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(x1) -> c_2(c^#(d(x1)))
, b^#(c(x1)) -> c_1()}
The usable rules for this path are the following:
{ c(c(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)
, a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(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(x1)) -> d(d(d(x1)))
, c(d(d(x1))) -> a(x1)
, a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(x1) -> c_2(c^#(d(x1)))
, b^#(c(x1)) -> c_1()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(d(x1))) -> c_5(a^#(x1))}
and weakly orienting the rules
{ b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(d(x1))) -> c_5(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [8]
c^#(x1) = [1] x1 + [5]
c_3(x1) = [1] x1 + [3]
c_4() = [0]
c_5(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^#(c(c(c(x1)))))}
and weakly orienting the rules
{ c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [2]
c^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(b(x1)) -> c_3(c^#(c(c(x1))))}
and weakly orienting the rules
{ a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(x1)) -> c_3(c^#(c(c(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(c(x1)) -> c_1()}
and weakly orienting the rules
{ b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(x1)) -> c_1()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [7]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))}
and weakly orienting the rules
{ b^#(c(x1)) -> c_1()
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [7]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
c_4() = [0]
c_5(x1) = [1] 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(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))
, b^#(c(x1)) -> c_1()
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(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 + [15]
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [5]
d(x1) = [1] x1 + [3]
a^#(x1) = [1] x1 + [15]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [15]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [13]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_2(c^#(d(x1)))}
and weakly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))
, b^#(c(x1)) -> c_1()
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_2(c^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [14]
b(x1) = [1] x1 + [10]
c(x1) = [1] x1 + [6]
d(x1) = [1] x1 + [4]
a^#(x1) = [1] x1 + [15]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [10]
c_1() = [0]
c_2(x1) = [1] x1 + [2]
c^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(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(d(d(x1))) -> a(x1)}
Weak Rules:
{ a^#(x1) -> c_2(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))
, b^#(c(x1)) -> c_1()
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(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: {c(d(d(x1))) -> a(x1)}
Weak Rules:
{ a^#(x1) -> c_2(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, a(a(x1)) -> b(c(c(c(x1))))
, a(x1) -> d(c(d(x1)))
, b^#(c(x1)) -> c_1()
, b^#(b(x1)) -> c_3(c^#(c(c(x1))))
, a^#(a(x1)) -> c_0(b^#(c(c(c(x1)))))
, c^#(d(d(x1))) -> c_5(a^#(x1))
, b(c(x1)) -> d(d(d(d(x1))))
, b(b(x1)) -> c(c(c(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) -> 5
, b^#_0(4) -> 7
, c_2_0(10) -> 5
, c^#_0(4) -> 10
, c_5_0(5) -> 10}