'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, b(x1) -> c(c(x1))
, c(d(x1)) -> a(b(c(a(x1))))
, a(a(x1)) -> a(c(b(a(x1))))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))
, b^#(x1) -> c_1(c^#(c(x1)))
, c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))
, a^#(a(x1)) -> c_3(a^#(c(b(a(x1)))))}
The usable rules are:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, b(x1) -> c(c(x1))
, c(d(x1)) -> a(b(c(a(x1))))
, a(a(x1)) -> a(c(b(a(x1))))}
The estimated dependency graph contains the following edges:
{b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))}
==> {a^#(a(x1)) -> c_3(a^#(c(b(a(x1)))))}
{c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))}
==> {a^#(a(x1)) -> c_3(a^#(c(b(a(x1)))))}
{a^#(a(x1)) -> c_3(a^#(c(b(a(x1)))))}
==> {a^#(a(x1)) -> c_3(a^#(c(b(a(x1)))))}
We consider the following path(s):
1) { b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))
, a^#(a(x1)) -> c_3(a^#(c(b(a(x1)))))}
The usable rules for this path are the following:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, b(x1) -> c(c(x1))
, c(d(x1)) -> a(b(c(a(x1))))
, a(a(x1)) -> a(c(b(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:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, b(x1) -> c(c(x1))
, c(d(x1)) -> a(b(c(a(x1))))
, a(a(x1)) -> a(c(b(a(x1))))
, b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))
, a^#(a(x1)) -> c_3(a^#(c(b(a(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b(x1) -> c(c(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(x1) -> c(c(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [2]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [2]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))}
and weakly orienting the rules
{b(x1) -> c(c(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [10]
c_0(x1) = [1] x1 + [7]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [15]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(d(x1)) -> a(b(c(a(x1))))}
and weakly orienting the rules
{ b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))
, b(x1) -> c(c(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(d(x1)) -> a(b(c(a(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [12]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(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:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, a(a(x1)) -> a(c(b(a(x1))))
, a^#(a(x1)) -> c_3(a^#(c(b(a(x1)))))}
Weak Rules:
{ c(d(x1)) -> a(b(c(a(x1))))
, b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))
, b(x1) -> c(c(x1))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, a(a(x1)) -> a(c(b(a(x1))))
, a^#(a(x1)) -> c_3(a^#(c(b(a(x1)))))}
Weak Rules:
{ c(d(x1)) -> a(b(c(a(x1))))
, b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))
, b(x1) -> c(c(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) -> 5
, a^#_0(4) -> 7}
2) { c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))
, a^#(a(x1)) -> c_3(a^#(c(b(a(x1)))))}
The usable rules for this path are the following:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, b(x1) -> c(c(x1))
, c(d(x1)) -> a(b(c(a(x1))))
, a(a(x1)) -> a(c(b(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:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, b(x1) -> c(c(x1))
, c(d(x1)) -> a(b(c(a(x1))))
, a(a(x1)) -> a(c(b(a(x1))))
, c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))
, a^#(a(x1)) -> c_3(a^#(c(b(a(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b(x1) -> c(c(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(x1) -> c(c(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))}
and weakly orienting the rules
{b(x1) -> c(c(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(d(x1)) -> a(b(c(a(x1))))}
and weakly orienting the rules
{ c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))
, b(x1) -> c(c(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(d(x1)) -> a(b(c(a(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [2]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [15]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [15]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [3]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, a(a(x1)) -> a(c(b(a(x1))))
, a^#(a(x1)) -> c_3(a^#(c(b(a(x1)))))}
Weak Rules:
{ c(d(x1)) -> a(b(c(a(x1))))
, c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))
, b(x1) -> c(c(x1))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, a(a(x1)) -> a(c(b(a(x1))))
, a^#(a(x1)) -> c_3(a^#(c(b(a(x1)))))}
Weak Rules:
{ c(d(x1)) -> a(b(c(a(x1))))
, c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))
, b(x1) -> c(c(x1))}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ b_0(12) -> 11
, b_1(11) -> 21
, b_1(16) -> 15
, b_1(18) -> 17
, b_1(23) -> 22
, b_2(22) -> 36
, b_2(32) -> 31
, b_2(34) -> 33
, c_0(12) -> 14
, c_0(13) -> 12
, c_0(14) -> 11
, c_1(4) -> 18
, c_1(11) -> 26
, c_1(12) -> 27
, c_1(16) -> 28
, c_1(18) -> 29
, c_1(21) -> 20
, c_1(23) -> 30
, c_1(24) -> 23
, c_1(26) -> 21
, c_1(27) -> 11
, c_1(28) -> 15
, c_1(29) -> 17
, c_1(30) -> 22
, c_2(4) -> 34
, c_2(11) -> 38
, c_2(16) -> 39
, c_2(18) -> 40
, c_2(22) -> 41
, c_2(23) -> 42
, c_2(32) -> 43
, c_2(34) -> 44
, c_2(36) -> 35
, c_2(38) -> 21
, c_2(39) -> 15
, c_2(40) -> 17
, c_2(41) -> 36
, c_2(42) -> 22
, c_2(43) -> 31
, c_2(44) -> 33
, a_0(4) -> 13
, a_1(4) -> 24
, a_1(15) -> 11
, a_1(17) -> 16
, a_1(22) -> 18
, a_1(22) -> 34
, a_2(31) -> 22
, a_2(33) -> 32
, a_2(35) -> 18
, a_2(35) -> 34
, d_0(4) -> 4
, a^#_0(4) -> 7
, a^#_0(11) -> 10
, a^#_1(20) -> 19
, a^#_1(22) -> 25
, a^#_2(35) -> 37
, c^#_0(4) -> 9
, c_2_0(10) -> 9
, c_2_1(25) -> 9
, c_3_1(19) -> 10
, c_3_2(37) -> 25}
3) {b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))}
The usable rules for this path are the following:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, b(x1) -> c(c(x1))
, c(d(x1)) -> a(b(c(a(x1))))
, a(a(x1)) -> a(c(b(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:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, b(x1) -> c(c(x1))
, c(d(x1)) -> a(b(c(a(x1))))
, a(a(x1)) -> a(c(b(a(x1))))
, b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b(x1) -> c(c(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(x1) -> c(c(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))}
and weakly orienting the rules
{b(x1) -> c(c(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(d(x1)) -> a(b(c(a(x1))))}
and weakly orienting the rules
{ b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))
, b(x1) -> c(c(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(d(x1)) -> a(b(c(a(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [3]
c_0(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, a(a(x1)) -> a(c(b(a(x1))))}
Weak Rules:
{ c(d(x1)) -> a(b(c(a(x1))))
, b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))
, b(x1) -> c(c(x1))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, a(a(x1)) -> a(c(b(a(x1))))}
Weak Rules:
{ c(d(x1)) -> a(b(c(a(x1))))
, b^#(c(a(x1))) -> c_0(a^#(b(a(b(c(x1))))))
, b(x1) -> c(c(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) -> 5
, a^#_0(4) -> 7}
4) {c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))}
The usable rules for this path are the following:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, b(x1) -> c(c(x1))
, c(d(x1)) -> a(b(c(a(x1))))
, a(a(x1)) -> a(c(b(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:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, b(x1) -> c(c(x1))
, c(d(x1)) -> a(b(c(a(x1))))
, a(a(x1)) -> a(c(b(a(x1))))
, c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b(x1) -> c(c(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(x1) -> c(c(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))}
and weakly orienting the rules
{b(x1) -> c(c(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(d(x1)) -> a(b(c(a(x1))))}
and weakly orienting the rules
{ c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))
, b(x1) -> c(c(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(d(x1)) -> a(b(c(a(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, a(a(x1)) -> a(c(b(a(x1))))}
Weak Rules:
{ c(d(x1)) -> a(b(c(a(x1))))
, c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))
, b(x1) -> c(c(x1))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, a(a(x1)) -> a(c(b(a(x1))))}
Weak Rules:
{ c(d(x1)) -> a(b(c(a(x1))))
, c^#(d(x1)) -> c_2(a^#(b(c(a(x1)))))
, b(x1) -> c(c(x1))}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ b_0(12) -> 11
, b_1(16) -> 15
, b_1(18) -> 17
, b_1(20) -> 19
, b_2(19) -> 32
, b_2(28) -> 27
, b_2(30) -> 29
, c_0(12) -> 14
, c_0(13) -> 12
, c_0(14) -> 11
, c_1(4) -> 18
, c_1(12) -> 23
, c_1(16) -> 24
, c_1(18) -> 25
, c_1(20) -> 26
, c_1(21) -> 20
, c_1(23) -> 11
, c_1(24) -> 15
, c_1(25) -> 17
, c_1(26) -> 19
, c_2(4) -> 30
, c_2(16) -> 33
, c_2(18) -> 34
, c_2(19) -> 35
, c_2(20) -> 36
, c_2(28) -> 37
, c_2(30) -> 38
, c_2(32) -> 31
, c_2(33) -> 15
, c_2(34) -> 17
, c_2(35) -> 32
, c_2(36) -> 19
, c_2(37) -> 27
, c_2(38) -> 29
, a_0(4) -> 13
, a_1(4) -> 21
, a_1(15) -> 11
, a_1(17) -> 16
, a_1(19) -> 18
, a_1(19) -> 30
, a_2(27) -> 19
, a_2(29) -> 28
, a_2(31) -> 18
, a_2(31) -> 30
, d_0(4) -> 4
, a^#_0(4) -> 7
, a^#_0(11) -> 10
, a^#_1(19) -> 22
, c^#_0(4) -> 9
, c_2_0(10) -> 9
, c_2_1(22) -> 9}
5) {b^#(x1) -> c_1(c^#(c(x1)))}
The usable rules for this path are the following:
{ c(d(x1)) -> a(b(c(a(x1))))
, b(c(a(x1))) -> a(b(a(b(c(x1)))))
, b(x1) -> c(c(x1))
, a(a(x1)) -> a(c(b(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:
{ c(d(x1)) -> a(b(c(a(x1))))
, b(c(a(x1))) -> a(b(a(b(c(x1)))))
, b(x1) -> c(c(x1))
, a(a(x1)) -> a(c(b(a(x1))))
, b^#(x1) -> c_1(c^#(c(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b^#(x1) -> c_1(c^#(c(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(x1) -> c_1(c^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(x1) -> c(c(x1))}
and weakly orienting the rules
{b^#(x1) -> c_1(c^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(x1) -> c(c(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(d(x1)) -> a(b(c(a(x1))))}
and weakly orienting the rules
{ b(x1) -> c(c(x1))
, b^#(x1) -> c_1(c^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(d(x1)) -> a(b(c(a(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, a(a(x1)) -> a(c(b(a(x1))))}
Weak Rules:
{ c(d(x1)) -> a(b(c(a(x1))))
, b(x1) -> c(c(x1))
, b^#(x1) -> c_1(c^#(c(x1)))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ b(c(a(x1))) -> a(b(a(b(c(x1)))))
, a(a(x1)) -> a(c(b(a(x1))))}
Weak Rules:
{ c(d(x1)) -> a(b(c(a(x1))))
, b(x1) -> c(c(x1))
, b^#(x1) -> c_1(c^#(c(x1)))}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ b_0(13) -> 12
, b_1(12) -> 21
, b_1(17) -> 16
, b_1(19) -> 18
, b_1(22) -> 20
, b_2(20) -> 34
, b_2(30) -> 29
, b_2(32) -> 31
, c_0(4) -> 11
, c_0(13) -> 15
, c_0(14) -> 13
, c_0(15) -> 12
, c_1(4) -> 19
, c_1(12) -> 24
, c_1(13) -> 25
, c_1(17) -> 26
, c_1(19) -> 27
, c_1(21) -> 20
, c_1(22) -> 21
, c_1(23) -> 22
, c_1(24) -> 21
, c_1(25) -> 12
, c_1(26) -> 16
, c_1(27) -> 18
, c_2(4) -> 32
, c_2(12) -> 35
, c_2(17) -> 36
, c_2(19) -> 37
, c_2(20) -> 38
, c_2(22) -> 39
, c_2(30) -> 40
, c_2(32) -> 41
, c_2(34) -> 33
, c_2(35) -> 21
, c_2(36) -> 16
, c_2(37) -> 18
, c_2(38) -> 34
, c_2(39) -> 20
, c_2(40) -> 29
, c_2(41) -> 31
, a_0(4) -> 14
, a_0(12) -> 11
, a_1(4) -> 23
, a_1(16) -> 12
, a_1(18) -> 17
, a_1(20) -> 11
, a_1(20) -> 19
, a_1(20) -> 32
, a_2(29) -> 20
, a_2(31) -> 30
, a_2(33) -> 11
, a_2(33) -> 19
, a_2(33) -> 32
, d_0(4) -> 4
, b^#_0(4) -> 5
, c_1_0(10) -> 5
, c_1_1(28) -> 5
, c^#_0(4) -> 9
, c^#_0(11) -> 10
, c^#_1(19) -> 28}