'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ b(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, d(a(b(x1))) -> b(d(d(c(x1))))
, d(x1) -> a(x1)
, b(a(c(a(x1)))) -> x1}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, a^#(c(x1)) -> c_1(b^#(b(x1)))
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, d^#(x1) -> c_3(a^#(x1))
, b^#(a(c(a(x1)))) -> c_4()}
The usable rules are:
{ b(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, d(a(b(x1))) -> b(d(d(c(x1))))
, d(x1) -> a(x1)
, b(a(c(a(x1)))) -> x1}
The estimated dependency graph contains the following edges:
{b^#(a(x1)) -> c_0(a^#(a(d(x1))))}
==> {a^#(c(x1)) -> c_1(b^#(b(x1)))}
{a^#(c(x1)) -> c_1(b^#(b(x1)))}
==> {b^#(a(c(a(x1)))) -> c_4()}
{a^#(c(x1)) -> c_1(b^#(b(x1)))}
==> {b^#(a(x1)) -> c_0(a^#(a(d(x1))))}
{d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))}
==> {b^#(a(c(a(x1)))) -> c_4()}
{d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))}
==> {b^#(a(x1)) -> c_0(a^#(a(d(x1))))}
{d^#(x1) -> c_3(a^#(x1))}
==> {a^#(c(x1)) -> c_1(b^#(b(x1)))}
We consider the following path(s):
1) { d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, a^#(c(x1)) -> c_1(b^#(b(x1)))
, b^#(a(c(a(x1)))) -> c_4()}
The usable rules for this path are the following:
{ b(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, d(a(b(x1))) -> b(d(d(c(x1))))
, d(x1) -> a(x1)
, b(a(c(a(x1)))) -> 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(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, d(a(b(x1))) -> b(d(d(c(x1))))
, d(x1) -> a(x1)
, b(a(c(a(x1)))) -> x1
, b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, a^#(c(x1)) -> c_1(b^#(b(x1)))
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b^#(a(c(a(x1)))) -> c_4()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1
, b^#(a(c(a(x1)))) -> c_4()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1
, b^#(a(c(a(x1)))) -> c_4()}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ d(x1) -> a(x1)
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))}
and weakly orienting the rules
{ b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1
, b^#(a(c(a(x1)))) -> c_4()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ d(x1) -> a(x1)
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(c(x1)) -> c_1(b^#(b(x1)))}
and weakly orienting the rules
{ d(x1) -> a(x1)
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1
, b^#(a(c(a(x1)))) -> c_4()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(c(x1)) -> c_1(b^#(b(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [12]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(c(x1)) -> b(b(x1))}
and weakly orienting the rules
{ a^#(c(x1)) -> c_1(b^#(b(x1)))
, d(x1) -> a(x1)
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1
, b^#(a(c(a(x1)))) -> c_4()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(c(x1)) -> b(b(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [2]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [14]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [6]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(a(x1)) -> c_0(a^#(a(d(x1))))}
and weakly orienting the rules
{ a(c(x1)) -> b(b(x1))
, a^#(c(x1)) -> c_1(b^#(b(x1)))
, d(x1) -> a(x1)
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1
, b^#(a(c(a(x1)))) -> c_4()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(a(x1)) -> c_0(a^#(a(d(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [10]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [13]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [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: {d(a(b(x1))) -> b(d(d(c(x1))))}
Weak Rules:
{ b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, a(c(x1)) -> b(b(x1))
, a^#(c(x1)) -> c_1(b^#(b(x1)))
, d(x1) -> a(x1)
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1
, b^#(a(c(a(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: {d(a(b(x1))) -> b(d(d(c(x1))))}
Weak Rules:
{ b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, a(c(x1)) -> b(b(x1))
, a^#(c(x1)) -> c_1(b^#(b(x1)))
, d(x1) -> a(x1)
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1
, b^#(a(c(a(x1)))) -> c_4()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ b_0(4) -> 11
, c_0(4) -> 4
, b^#_0(4) -> 5
, b^#_0(11) -> 10
, a^#_0(4) -> 7
, c_1_0(10) -> 7
, d^#_0(4) -> 9}
2) { d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, a^#(c(x1)) -> c_1(b^#(b(x1)))}
The usable rules for this path are the following:
{ b(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, d(a(b(x1))) -> b(d(d(c(x1))))
, d(x1) -> a(x1)
, b(a(c(a(x1)))) -> 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(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, d(a(b(x1))) -> b(d(d(c(x1))))
, d(x1) -> a(x1)
, b(a(c(a(x1)))) -> x1
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, a^#(c(x1)) -> c_1(b^#(b(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(x1) -> a(x1)}
and weakly orienting the rules
{ b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(x1) -> a(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [6]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(a(x1)) -> c_0(a^#(a(d(x1))))}
and weakly orienting the rules
{ d(x1) -> a(x1)
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(a(x1)) -> c_0(a^#(a(d(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [6]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))}
and weakly orienting the rules
{ b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, d(x1) -> a(x1)
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [6]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [6]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [8]
d^#(x1) = [1] x1 + [12]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(c(x1)) -> b(b(x1))
, a^#(c(x1)) -> c_1(b^#(b(x1)))}
and weakly orienting the rules
{ d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, d(x1) -> a(x1)
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(c(x1)) -> b(b(x1))
, a^#(c(x1)) -> c_1(b^#(b(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [15]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [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: {d(a(b(x1))) -> b(d(d(c(x1))))}
Weak Rules:
{ a(c(x1)) -> b(b(x1))
, a^#(c(x1)) -> c_1(b^#(b(x1)))
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, d(x1) -> a(x1)
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> 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: {d(a(b(x1))) -> b(d(d(c(x1))))}
Weak Rules:
{ a(c(x1)) -> b(b(x1))
, a^#(c(x1)) -> c_1(b^#(b(x1)))
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, d(x1) -> a(x1)
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ b_0(2) -> 4
, c_0(2) -> 2
, b^#_0(2) -> 1
, b^#_0(4) -> 3
, a^#_0(2) -> 1
, c_1_0(3) -> 1
, d^#_0(2) -> 1}
3) { d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b^#(a(c(a(x1)))) -> c_4()}
The usable rules for this path are the following:
{ d(a(b(x1))) -> b(d(d(c(x1))))
, d(x1) -> a(x1)
, b(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, b(a(c(a(x1)))) -> x1}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ d(a(b(x1))) -> b(d(d(c(x1))))
, d(x1) -> a(x1)
, b(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, b(a(c(a(x1)))) -> x1
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b^#(a(c(a(x1)))) -> c_4()}
Details:
We apply the weight gap principle, strictly orienting the rules
{d(x1) -> a(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(x1) -> a(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(a(c(a(x1)))) -> c_4()}
and weakly orienting the rules
{d(x1) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(a(c(a(x1)))) -> c_4()}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [2]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))}
and weakly orienting the rules
{ b^#(a(c(a(x1)))) -> c_4()
, d(x1) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(c(x1)) -> b(b(x1))
, b(a(c(a(x1)))) -> x1}
and weakly orienting the rules
{ d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b^#(a(c(a(x1)))) -> c_4()
, d(x1) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(c(x1)) -> b(b(x1))
, b(a(c(a(x1)))) -> x1}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [4]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(a(x1)) -> a(a(d(x1)))}
and weakly orienting the rules
{ a(c(x1)) -> b(b(x1))
, b(a(c(a(x1)))) -> x1
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b^#(a(c(a(x1)))) -> c_4()
, d(x1) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(a(x1)) -> a(a(d(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [4]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [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: {d(a(b(x1))) -> b(d(d(c(x1))))}
Weak Rules:
{ b(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, b(a(c(a(x1)))) -> x1
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b^#(a(c(a(x1)))) -> c_4()
, 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: {d(a(b(x1))) -> b(d(d(c(x1))))}
Weak Rules:
{ b(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, b(a(c(a(x1)))) -> x1
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, b^#(a(c(a(x1)))) -> c_4()
, d(x1) -> a(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(4) -> 4
, b^#_0(4) -> 5
, d^#_0(4) -> 9}
4) { d^#(x1) -> c_3(a^#(x1))
, b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, a^#(c(x1)) -> c_1(b^#(b(x1)))}
The usable rules for this path are the following:
{ b(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, d(a(b(x1))) -> b(d(d(c(x1))))
, d(x1) -> a(x1)
, b(a(c(a(x1)))) -> 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(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, d(a(b(x1))) -> b(d(d(c(x1))))
, d(x1) -> a(x1)
, b(a(c(a(x1)))) -> x1
, d^#(x1) -> c_3(a^#(x1))
, b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, a^#(c(x1)) -> c_1(b^#(b(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(a(b(x1))) -> b(d(d(c(x1))))}
and weakly orienting the rules
{ b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(a(b(x1))) -> b(d(d(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(a(x1)) -> c_0(a^#(a(d(x1))))}
and weakly orienting the rules
{ d(a(b(x1))) -> b(d(d(c(x1))))
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(a(x1)) -> c_0(a^#(a(d(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [15]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(x1) -> c_3(a^#(x1))}
and weakly orienting the rules
{ b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, d(a(b(x1))) -> b(d(d(c(x1))))
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(x1) -> c_3(a^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4() = [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(c(x1)) -> b(b(x1))
, d(x1) -> a(x1)
, a^#(c(x1)) -> c_1(b^#(b(x1)))}
Weak Rules:
{ d^#(x1) -> c_3(a^#(x1))
, b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, d(a(b(x1))) -> b(d(d(c(x1))))
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> 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(c(x1)) -> b(b(x1))
, d(x1) -> a(x1)
, a^#(c(x1)) -> c_1(b^#(b(x1)))}
Weak Rules:
{ d^#(x1) -> c_3(a^#(x1))
, b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, d(a(b(x1))) -> b(d(d(c(x1))))
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ b_1(4) -> 11
, c_0(4) -> 4
, b^#_0(4) -> 5
, b^#_1(11) -> 10
, a^#_0(4) -> 7
, a^#_1(4) -> 12
, c_1_1(10) -> 7
, c_1_1(10) -> 12
, d^#_0(4) -> 9
, c_3_0(7) -> 9
, c_3_1(12) -> 9}
5) { d^#(x1) -> c_3(a^#(x1))
, b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, a^#(c(x1)) -> c_1(b^#(b(x1)))
, b^#(a(c(a(x1)))) -> c_4()}
The usable rules for this path are the following:
{ b(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, d(a(b(x1))) -> b(d(d(c(x1))))
, d(x1) -> a(x1)
, b(a(c(a(x1)))) -> 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(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, d(a(b(x1))) -> b(d(d(c(x1))))
, d(x1) -> a(x1)
, b(a(c(a(x1)))) -> x1
, b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, a^#(c(x1)) -> c_1(b^#(b(x1)))
, d^#(x1) -> c_3(a^#(x1))
, b^#(a(c(a(x1)))) -> c_4()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1
, b^#(a(c(a(x1)))) -> c_4()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1
, b^#(a(c(a(x1)))) -> c_4()}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ d(a(b(x1))) -> b(d(d(c(x1))))
, d^#(x1) -> c_3(a^#(x1))}
and weakly orienting the rules
{ b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1
, b^#(a(c(a(x1)))) -> c_4()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ d(a(b(x1))) -> b(d(d(c(x1))))
, d^#(x1) -> c_3(a^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(a(x1)) -> c_0(a^#(a(d(x1))))}
and weakly orienting the rules
{ d(a(b(x1))) -> b(d(d(c(x1))))
, d^#(x1) -> c_3(a^#(x1))
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1
, b^#(a(c(a(x1)))) -> c_4()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(a(x1)) -> c_0(a^#(a(d(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [12]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [9]
c_4() = [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(c(x1)) -> b(b(x1))
, d(x1) -> a(x1)
, a^#(c(x1)) -> c_1(b^#(b(x1)))}
Weak Rules:
{ b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, d(a(b(x1))) -> b(d(d(c(x1))))
, d^#(x1) -> c_3(a^#(x1))
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1
, b^#(a(c(a(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:
{ a(c(x1)) -> b(b(x1))
, d(x1) -> a(x1)
, a^#(c(x1)) -> c_1(b^#(b(x1)))}
Weak Rules:
{ b^#(a(x1)) -> c_0(a^#(a(d(x1))))
, d(a(b(x1))) -> b(d(d(c(x1))))
, d^#(x1) -> c_3(a^#(x1))
, b(a(x1)) -> a(a(d(x1)))
, b(a(c(a(x1)))) -> x1
, b^#(a(c(a(x1)))) -> c_4()}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ b_1(4) -> 11
, c_0(4) -> 4
, b^#_0(4) -> 5
, b^#_1(11) -> 10
, a^#_0(4) -> 7
, a^#_1(4) -> 12
, c_1_1(10) -> 7
, c_1_1(10) -> 12
, d^#_0(4) -> 9
, c_3_0(7) -> 9
, c_3_1(12) -> 9}
6) {d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))}
The usable rules for this path are the following:
{ d(a(b(x1))) -> b(d(d(c(x1))))
, d(x1) -> a(x1)
, b(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, b(a(c(a(x1)))) -> x1}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ d(a(b(x1))) -> b(d(d(c(x1))))
, d(x1) -> a(x1)
, b(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, b(a(c(a(x1)))) -> x1
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{d(x1) -> a(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(x1) -> a(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [2]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))}
and weakly orienting the rules
{d(x1) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(c(x1)) -> b(b(x1))
, b(a(c(a(x1)))) -> x1}
and weakly orienting the rules
{ d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, d(x1) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(c(x1)) -> b(b(x1))
, b(a(c(a(x1)))) -> x1}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [4]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(a(x1)) -> a(a(d(x1)))}
and weakly orienting the rules
{ a(c(x1)) -> b(b(x1))
, b(a(c(a(x1)))) -> x1
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, d(x1) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(a(x1)) -> a(a(d(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [4]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [11]
c_2(x1) = [1] x1 + [2]
c_3(x1) = [0] x1 + [0]
c_4() = [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: {d(a(b(x1))) -> b(d(d(c(x1))))}
Weak Rules:
{ b(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, b(a(c(a(x1)))) -> x1
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, 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: {d(a(b(x1))) -> b(d(d(c(x1))))}
Weak Rules:
{ b(a(x1)) -> a(a(d(x1)))
, a(c(x1)) -> b(b(x1))
, b(a(c(a(x1)))) -> x1
, d^#(a(b(x1))) -> c_2(b^#(d(d(c(x1)))))
, d(x1) -> a(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(4) -> 4
, b^#_0(4) -> 5
, d^#_0(4) -> 9}
7) {d^#(x1) -> c_3(a^#(x1))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {d^#(x1) -> c_3(a^#(x1))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{d^#(x1) -> c_3(a^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(x1) -> c_3(a^#(x1))}
Details:
Interpretation Functions:
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4() = [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {d^#(x1) -> c_3(a^#(x1))}
Details:
The given problem does not contain any strict rules