'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(c(x1)) -> a(x1)
, d(a(x1)) -> a(c(b(c(d(x1)))))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(a(x1))))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(c(x1)) -> c_0(a^#(x1))
, d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))
, a^#(c(b(c(x1)))) -> c_2(c^#(b(c(c(x1)))))
, c^#(x1) -> c_3(a^#(a(x1)))
, d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))}
The usable rules are:
{ a(c(x1)) -> a(x1)
, d(a(x1)) -> a(c(b(c(d(x1)))))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(a(x1))))}
The estimated dependency graph contains the following edges:
{a^#(c(x1)) -> c_0(a^#(x1))}
==> {a^#(c(b(c(x1)))) -> c_2(c^#(b(c(c(x1)))))}
{a^#(c(x1)) -> c_0(a^#(x1))}
==> {a^#(c(x1)) -> c_0(a^#(x1))}
{d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))}
==> {a^#(c(b(c(x1)))) -> c_2(c^#(b(c(c(x1)))))}
{d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))}
==> {a^#(c(x1)) -> c_0(a^#(x1))}
{a^#(c(b(c(x1)))) -> c_2(c^#(b(c(c(x1)))))}
==> {c^#(x1) -> c_3(a^#(a(x1)))}
{c^#(x1) -> c_3(a^#(a(x1)))}
==> {a^#(c(b(c(x1)))) -> c_2(c^#(b(c(c(x1)))))}
{c^#(x1) -> c_3(a^#(a(x1)))}
==> {a^#(c(x1)) -> c_0(a^#(x1))}
{d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))}
==> {a^#(c(b(c(x1)))) -> c_2(c^#(b(c(c(x1)))))}
{d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))}
==> {a^#(c(x1)) -> c_0(a^#(x1))}
We consider the following path(s):
1) { d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))
, a^#(c(x1)) -> c_0(a^#(x1))
, c^#(x1) -> c_3(a^#(a(x1)))
, a^#(c(b(c(x1)))) -> c_2(c^#(b(c(c(x1)))))}
The usable rules for this path are the following:
{ a(c(x1)) -> a(x1)
, d(a(x1)) -> a(c(b(c(d(x1)))))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(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(c(x1)) -> a(x1)
, d(a(x1)) -> a(c(b(c(d(x1)))))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(a(x1))))
, d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))
, a^#(c(x1)) -> c_0(a^#(x1))
, c^#(x1) -> c_3(a^#(a(x1)))
, a^#(c(b(c(x1)))) -> c_2(c^#(b(c(c(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a(c(b(c(x1)))) -> c(b(c(c(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(c(b(c(x1)))) -> c(b(c(c(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [15]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(x1) -> c_3(a^#(a(x1)))}
and weakly orienting the rules
{a(c(b(c(x1)))) -> c(b(c(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(x1) -> c_3(a^#(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [4]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))}
and weakly orienting the rules
{ c^#(x1) -> c_3(a^#(a(x1)))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [2]
c^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [7]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(c(x1)) -> a(x1)}
and weakly orienting the rules
{ d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))
, c^#(x1) -> c_3(a^#(a(x1)))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(c(x1)) -> a(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [2]
d^#(x1) = [1] x1 + [15]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(c(x1)) -> c_0(a^#(x1))}
and weakly orienting the rules
{ a(c(x1)) -> a(x1)
, d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))
, c^#(x1) -> c_3(a^#(a(x1)))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(c(x1)) -> c_0(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [6]
b(x1) = [1] x1 + [2]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_1(x1) = [1] x1 + [4]
c_2(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [4]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ d(a(x1)) -> a(c(b(c(d(x1)))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(a(x1))))
, a^#(c(b(c(x1)))) -> c_2(c^#(b(c(c(x1)))))}
Weak Rules:
{ a^#(c(x1)) -> c_0(a^#(x1))
, a(c(x1)) -> a(x1)
, d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))
, c^#(x1) -> c_3(a^#(a(x1)))
, a(c(b(c(x1)))) -> c(b(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:
{ d(a(x1)) -> a(c(b(c(d(x1)))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(a(x1))))
, a^#(c(b(c(x1)))) -> c_2(c^#(b(c(c(x1)))))}
Weak Rules:
{ a^#(c(x1)) -> c_0(a^#(x1))
, a(c(x1)) -> a(x1)
, d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))
, c^#(x1) -> c_3(a^#(a(x1)))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(4) -> 12
, b_0(4) -> 4
, a^#_0(4) -> 5
, a^#_0(12) -> 11
, d^#_0(4) -> 7
, c^#_0(4) -> 10
, c_3_0(11) -> 10}
2) { d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))
, a^#(c(x1)) -> c_0(a^#(x1))
, c^#(x1) -> c_3(a^#(a(x1)))
, a^#(c(b(c(x1)))) -> c_2(c^#(b(c(c(x1)))))}
The usable rules for this path are the following:
{ a(c(x1)) -> a(x1)
, d(a(x1)) -> a(c(b(c(d(x1)))))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(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(c(x1)) -> a(x1)
, d(a(x1)) -> a(c(b(c(d(x1)))))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(a(x1))))
, d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))
, a^#(c(x1)) -> c_0(a^#(x1))
, c^#(x1) -> c_3(a^#(a(x1)))
, a^#(c(b(c(x1)))) -> c_2(c^#(b(c(c(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(c(b(c(x1)))) -> c(b(c(c(x1))))
, c^#(x1) -> c_3(a^#(a(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(c(b(c(x1)))) -> c(b(c(c(x1))))
, c^#(x1) -> c_3(a^#(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))}
and weakly orienting the rules
{ a(c(b(c(x1)))) -> c(b(c(c(x1))))
, c^#(x1) -> c_3(a^#(a(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [3]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(c(x1)) -> a(x1)
, a^#(c(x1)) -> c_0(a^#(x1))}
and weakly orienting the rules
{ d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))
, c^#(x1) -> c_3(a^#(a(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(c(x1)) -> a(x1)
, a^#(c(x1)) -> c_0(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [4]
b(x1) = [1] x1 + [9]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [15]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ d(a(x1)) -> a(c(b(c(d(x1)))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(a(x1))))
, a^#(c(b(c(x1)))) -> c_2(c^#(b(c(c(x1)))))}
Weak Rules:
{ a(c(x1)) -> a(x1)
, a^#(c(x1)) -> c_0(a^#(x1))
, d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))
, c^#(x1) -> c_3(a^#(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(x1)) -> a(c(b(c(d(x1)))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(a(x1))))
, a^#(c(b(c(x1)))) -> c_2(c^#(b(c(c(x1)))))}
Weak Rules:
{ a(c(x1)) -> a(x1)
, a^#(c(x1)) -> c_0(a^#(x1))
, d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))
, c^#(x1) -> c_3(a^#(a(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(4) -> 12
, b_0(4) -> 4
, a^#_0(4) -> 5
, a^#_0(12) -> 11
, d^#_0(4) -> 7
, c^#_0(4) -> 10
, c_3_0(11) -> 10}
3) {d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))}
The usable rules for this path are the following:
{ d(a(x1)) -> a(c(b(c(d(x1)))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(a(x1))))
, a(c(x1)) -> a(x1)
, a(c(b(c(x1)))) -> c(b(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:
{ d(a(x1)) -> a(c(b(c(d(x1)))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(a(x1))))
, a(c(x1)) -> a(x1)
, a(c(b(c(x1)))) -> c(b(c(c(x1))))
, d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(c(x1)) -> a(x1)}
and weakly orienting the rules
{d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(c(x1)) -> a(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [2]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [7]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(x1) -> b(a(a(x1)))}
and weakly orienting the rules
{ a(c(x1)) -> a(x1)
, d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(x1) -> b(a(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [13]
c_1(x1) = [1] x1 + [4]
c_2(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ d(a(x1)) -> a(c(b(c(d(x1)))))
, d(c(x1)) -> a(c(d(a(x1))))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))}
Weak Rules:
{ c(x1) -> b(a(a(x1)))
, a(c(x1)) -> a(x1)
, d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ d(a(x1)) -> a(c(b(c(d(x1)))))
, d(c(x1)) -> a(c(d(a(x1))))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))}
Weak Rules:
{ c(x1) -> b(a(a(x1)))
, a(c(x1)) -> a(x1)
, d^#(a(x1)) -> c_1(a^#(c(b(c(d(x1))))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ b_0(4) -> 4
, a^#_0(4) -> 5
, d^#_0(4) -> 7}
4) {d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))}
The usable rules for this path are the following:
{ a(c(x1)) -> a(x1)
, d(a(x1)) -> a(c(b(c(d(x1)))))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(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(c(x1)) -> a(x1)
, d(a(x1)) -> a(c(b(c(d(x1)))))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(a(x1))))
, d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a(c(b(c(x1)))) -> c(b(c(c(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(c(b(c(x1)))) -> c(b(c(c(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))}
and weakly orienting the rules
{a(c(b(c(x1)))) -> c(b(c(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(c(x1)) -> a(x1)}
and weakly orienting the rules
{ d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(c(x1)) -> a(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [7]
b(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [7]
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(x1)) -> a(c(b(c(d(x1)))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(a(x1))))}
Weak Rules:
{ a(c(x1)) -> a(x1)
, d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))
, a(c(b(c(x1)))) -> c(b(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:
{ d(a(x1)) -> a(c(b(c(d(x1)))))
, c(x1) -> b(a(a(x1)))
, d(c(x1)) -> a(c(d(a(x1))))}
Weak Rules:
{ a(c(x1)) -> a(x1)
, d^#(c(x1)) -> c_4(a^#(c(d(a(x1)))))
, a(c(b(c(x1)))) -> c(b(c(c(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ b_0(4) -> 4
, a^#_0(4) -> 5
, d^#_0(4) -> 7}