'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(x1)) -> b(b(x1))
, c(c(b(x1))) -> d(c(a(x1)))
, a(x1) -> d(c(c(x1)))
, c(d(x1)) -> b(c(x1))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(a(x1)) -> c_0()
, c^#(c(b(x1))) -> c_1(c^#(a(x1)))
, a^#(x1) -> c_2(c^#(c(x1)))
, c^#(d(x1)) -> c_3(c^#(x1))}
The usable rules are:
{ a(a(x1)) -> b(b(x1))
, c(c(b(x1))) -> d(c(a(x1)))
, a(x1) -> d(c(c(x1)))
, c(d(x1)) -> b(c(x1))}
The estimated dependency graph contains the following edges:
{c^#(c(b(x1))) -> c_1(c^#(a(x1)))}
==> {c^#(d(x1)) -> c_3(c^#(x1))}
{a^#(x1) -> c_2(c^#(c(x1)))}
==> {c^#(d(x1)) -> c_3(c^#(x1))}
{a^#(x1) -> c_2(c^#(c(x1)))}
==> {c^#(c(b(x1))) -> c_1(c^#(a(x1)))}
{c^#(d(x1)) -> c_3(c^#(x1))}
==> {c^#(d(x1)) -> c_3(c^#(x1))}
{c^#(d(x1)) -> c_3(c^#(x1))}
==> {c^#(c(b(x1))) -> c_1(c^#(a(x1)))}
We consider the following path(s):
1) { a^#(x1) -> c_2(c^#(c(x1)))
, c^#(c(b(x1))) -> c_1(c^#(a(x1)))
, c^#(d(x1)) -> c_3(c^#(x1))}
The usable rules for this path are the following:
{ a(a(x1)) -> b(b(x1))
, c(c(b(x1))) -> d(c(a(x1)))
, a(x1) -> d(c(c(x1)))
, c(d(x1)) -> b(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:
{ a(a(x1)) -> b(b(x1))
, c(c(b(x1))) -> d(c(a(x1)))
, a(x1) -> d(c(c(x1)))
, c(d(x1)) -> b(c(x1))
, a^#(x1) -> c_2(c^#(c(x1)))
, c^#(c(b(x1))) -> c_1(c^#(a(x1)))
, c^#(d(x1)) -> c_3(c^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c(d(x1)) -> b(c(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(d(x1)) -> b(c(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [3]
c_3(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a^#(x1) -> c_2(c^#(c(x1)))
, c^#(d(x1)) -> c_3(c^#(x1))}
and weakly orienting the rules
{c(d(x1)) -> b(c(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a^#(x1) -> c_2(c^#(c(x1)))
, c^#(d(x1)) -> c_3(c^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [8]
c_0() = [0]
c^#(x1) = [1] x1 + [0]
c_1(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(c(b(x1))) -> d(c(a(x1)))
, c^#(c(b(x1))) -> c_1(c^#(a(x1)))}
and weakly orienting the rules
{ a^#(x1) -> c_2(c^#(c(x1)))
, c^#(d(x1)) -> c_3(c^#(x1))
, c(d(x1)) -> b(c(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(c(b(x1))) -> d(c(a(x1)))
, c^#(c(b(x1))) -> c_1(c^#(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [12]
c_0() = [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(x1)) -> b(b(x1))}
and weakly orienting the rules
{ c(c(b(x1))) -> d(c(a(x1)))
, c^#(c(b(x1))) -> c_1(c^#(a(x1)))
, a^#(x1) -> c_2(c^#(c(x1)))
, c^#(d(x1)) -> c_3(c^#(x1))
, c(d(x1)) -> b(c(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(x1)) -> b(b(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0() = [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {a(x1) -> d(c(c(x1)))}
Weak Rules:
{ a(a(x1)) -> b(b(x1))
, c(c(b(x1))) -> d(c(a(x1)))
, c^#(c(b(x1))) -> c_1(c^#(a(x1)))
, a^#(x1) -> c_2(c^#(c(x1)))
, c^#(d(x1)) -> c_3(c^#(x1))
, c(d(x1)) -> b(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: {a(x1) -> d(c(c(x1)))}
Weak Rules:
{ a(a(x1)) -> b(b(x1))
, c(c(b(x1))) -> d(c(a(x1)))
, c^#(c(b(x1))) -> c_1(c^#(a(x1)))
, a^#(x1) -> c_2(c^#(c(x1)))
, c^#(d(x1)) -> c_3(c^#(x1))
, c(d(x1)) -> b(c(x1))}
Details:
The problem is Match-bounded by 3.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 6
, a_1(2) -> 10
, a_1(8) -> 17
, a_2(8) -> 24
, b_0(2) -> 2
, b_0(4) -> 4
, b_1(8) -> 8
, b_1(8) -> 14
, b_2(20) -> 9
, b_2(28) -> 20
, b_2(28) -> 21
, b_2(28) -> 29
, b_3(33) -> 23
, b_3(35) -> 33
, c_0(2) -> 4
, c_1(2) -> 8
, c_1(8) -> 7
, c_1(10) -> 9
, c_2(2) -> 14
, c_2(8) -> 22
, c_2(9) -> 28
, c_2(13) -> 20
, c_2(14) -> 13
, c_2(22) -> 21
, c_2(24) -> 23
, c_3(8) -> 30
, c_3(23) -> 35
, c_3(29) -> 33
, c_3(30) -> 29
, d_0(2) -> 2
, d_1(7) -> 6
, d_1(9) -> 7
, d_1(9) -> 13
, d_1(9) -> 22
, d_1(9) -> 30
, d_2(13) -> 10
, d_2(21) -> 17
, d_2(23) -> 20
, d_2(23) -> 21
, d_2(23) -> 29
, d_3(29) -> 24
, a^#_0(2) -> 1
, c^#_0(2) -> 1
, c^#_0(4) -> 3
, c^#_0(6) -> 5
, c^#_1(7) -> 12
, c^#_1(8) -> 11
, c^#_1(9) -> 18
, c^#_1(10) -> 15
, c^#_1(17) -> 16
, c^#_2(9) -> 26
, c^#_2(13) -> 19
, c^#_2(21) -> 27
, c^#_2(23) -> 31
, c^#_2(24) -> 25
, c^#_3(23) -> 34
, c^#_3(29) -> 32
, c_1_0(5) -> 3
, c_1_1(15) -> 11
, c_1_1(16) -> 12
, c_1_2(25) -> 19
, c_2_0(3) -> 1
, c_2_1(11) -> 1
, c_3_0(1) -> 1
, c_3_1(12) -> 5
, c_3_1(18) -> 12
, c_3_2(19) -> 15
, c_3_2(26) -> 19
, c_3_2(27) -> 16
, c_3_2(31) -> 27
, c_3_3(32) -> 25
, c_3_3(34) -> 32}
2) {a^#(x1) -> c_2(c^#(c(x1)))}
The usable rules for this path are the following:
{ c(c(b(x1))) -> d(c(a(x1)))
, c(d(x1)) -> b(c(x1))
, a(a(x1)) -> b(b(x1))
, a(x1) -> d(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(b(x1))) -> d(c(a(x1)))
, c(d(x1)) -> b(c(x1))
, a(a(x1)) -> b(b(x1))
, a(x1) -> d(c(c(x1)))
, a^#(x1) -> c_2(c^#(c(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c(d(x1)) -> b(c(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(d(x1)) -> b(c(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_2(c^#(c(x1)))}
and weakly orienting the rules
{c(d(x1)) -> b(c(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_2(c^#(c(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [8]
c_0() = [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(a(x1)) -> b(b(x1))
, a(x1) -> d(c(c(x1)))}
and weakly orienting the rules
{ a^#(x1) -> c_2(c^#(c(x1)))
, c(d(x1)) -> b(c(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(a(x1)) -> b(b(x1))
, a(x1) -> d(c(c(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [8]
c_0() = [0]
c^#(x1) = [1] x1 + [2]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [3]
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: {c(c(b(x1))) -> d(c(a(x1)))}
Weak Rules:
{ a(a(x1)) -> b(b(x1))
, a(x1) -> d(c(c(x1)))
, a^#(x1) -> c_2(c^#(c(x1)))
, c(d(x1)) -> b(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(c(b(x1))) -> d(c(a(x1)))}
Weak Rules:
{ a(a(x1)) -> b(b(x1))
, a(x1) -> d(c(c(x1)))
, a^#(x1) -> c_2(c^#(c(x1)))
, c(d(x1)) -> b(c(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ b_0(2) -> 2
, b_0(4) -> 2
, b_0(9) -> 9
, c_0(2) -> 9
, c_0(4) -> 9
, d_0(2) -> 4
, d_0(4) -> 4
, a^#_0(2) -> 5
, a^#_0(4) -> 5
, c^#_0(2) -> 7
, c^#_0(4) -> 7
, c^#_0(9) -> 8
, c_2_0(8) -> 5}
3) {a^#(a(x1)) -> c_0()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a(x1) = [0] x1 + [0]
b(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0() = [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [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: {a^#(a(x1)) -> c_0()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#(a(x1)) -> c_0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(x1)) -> c_0()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [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: {a^#(a(x1)) -> c_0()}
Details:
The given problem does not contain any strict rules