'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(x1)) -> x1
, b(b(x1)) -> c(c(c(x1)))
, b(c(x1)) -> a(b(b(x1)))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(a(x1)) -> c_0()
, b^#(b(x1)) -> c_1()
, b^#(c(x1)) -> c_2(a^#(b(b(x1))))}
The usable rules are:
{ b(b(x1)) -> c(c(c(x1)))
, b(c(x1)) -> a(b(b(x1)))
, a(a(x1)) -> x1}
The estimated dependency graph contains the following edges:
{b^#(c(x1)) -> c_2(a^#(b(b(x1))))}
==> {a^#(a(x1)) -> c_0()}
We consider the following path(s):
1) { b^#(c(x1)) -> c_2(a^#(b(b(x1))))
, a^#(a(x1)) -> c_0()}
The usable rules for this path are the following:
{ b(b(x1)) -> c(c(c(x1)))
, b(c(x1)) -> a(b(b(x1)))
, a(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(b(x1)) -> c(c(c(x1)))
, b(c(x1)) -> a(b(b(x1)))
, a(a(x1)) -> x1
, b^#(c(x1)) -> c_2(a^#(b(b(x1))))
, a^#(a(x1)) -> c_0()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(a(x1)) -> x1
, 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)) -> x1
, a^#(a(x1)) -> c_0()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [1] x1 + [1]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(c(x1)) -> c_2(a^#(b(b(x1))))}
and weakly orienting the rules
{ a(a(x1)) -> x1
, a^#(a(x1)) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(x1)) -> c_2(a^#(b(b(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [1] x1 + [1]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(c(x1)) -> a(b(b(x1)))}
and weakly orienting the rules
{ b^#(c(x1)) -> c_2(a^#(b(b(x1))))
, a(a(x1)) -> x1
, a^#(a(x1)) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(c(x1)) -> a(b(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [1] x1 + [1]
c_1() = [0]
c_2(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(b(x1)) -> c(c(c(x1)))}
Weak Rules:
{ b(c(x1)) -> a(b(b(x1)))
, b^#(c(x1)) -> c_2(a^#(b(b(x1))))
, a(a(x1)) -> x1
, a^#(a(x1)) -> c_0()}
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(b(x1)) -> c(c(c(x1)))}
Weak Rules:
{ b(c(x1)) -> a(b(b(x1)))
, b^#(c(x1)) -> c_2(a^#(b(b(x1))))
, a(a(x1)) -> x1
, a^#(a(x1)) -> c_0()}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a_1(8) -> 5
, a_1(8) -> 9
, b_0(2) -> 5
, b_0(5) -> 4
, b_1(2) -> 9
, b_1(9) -> 8
, c_0(2) -> 2
, c_1(2) -> 7
, c_1(6) -> 4
, c_1(7) -> 6
, c_2(2) -> 12
, c_2(11) -> 8
, c_2(12) -> 11
, a^#_0(2) -> 1
, a^#_0(4) -> 3
, a^#_1(8) -> 10
, b^#_0(2) -> 1
, c_2_0(3) -> 1
, c_2_1(10) -> 1}
2) {b^#(c(x1)) -> c_2(a^#(b(b(x1))))}
The usable rules for this path are the following:
{ b(b(x1)) -> c(c(c(x1)))
, b(c(x1)) -> a(b(b(x1)))
, a(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(b(x1)) -> c(c(c(x1)))
, b(c(x1)) -> a(b(b(x1)))
, a(a(x1)) -> x1
, b^#(c(x1)) -> c_2(a^#(b(b(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a(a(x1)) -> x1}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(x1)) -> x1}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [1] x1 + [1]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(c(x1)) -> c_2(a^#(b(b(x1))))}
and weakly orienting the rules
{a(a(x1)) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(x1)) -> c_2(a^#(b(b(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [1] x1 + [1]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(c(x1)) -> a(b(b(x1)))}
and weakly orienting the rules
{ b^#(c(x1)) -> c_2(a^#(b(b(x1))))
, a(a(x1)) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(c(x1)) -> a(b(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [1] x1 + [1]
c_1() = [0]
c_2(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(b(x1)) -> c(c(c(x1)))}
Weak Rules:
{ b(c(x1)) -> a(b(b(x1)))
, b^#(c(x1)) -> c_2(a^#(b(b(x1))))
, a(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: {b(b(x1)) -> c(c(c(x1)))}
Weak Rules:
{ b(c(x1)) -> a(b(b(x1)))
, b^#(c(x1)) -> c_2(a^#(b(b(x1))))
, a(a(x1)) -> x1}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a_1(12) -> 9
, a_1(12) -> 13
, b_0(3) -> 9
, b_0(9) -> 8
, b_1(3) -> 13
, b_1(13) -> 12
, c_0(3) -> 3
, c_1(3) -> 11
, c_1(10) -> 8
, c_1(11) -> 10
, c_2(3) -> 16
, c_2(15) -> 12
, c_2(16) -> 15
, a^#_0(3) -> 4
, a^#_0(8) -> 7
, a^#_1(12) -> 14
, b^#_0(3) -> 6
, c_2_0(7) -> 6
, c_2_1(14) -> 6}
3) {b^#(b(x1)) -> c_1()}
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]
a^#(x1) = [0] x1 + [0]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(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: {b^#(b(x1)) -> c_1()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{b^#(b(x1)) -> c_1()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(x1)) -> c_1()}
Details:
Interpretation Functions:
a(x1) = [0] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0() = [0]
b^#(x1) = [1] x1 + [1]
c_1() = [0]
c_2(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: {b^#(b(x1)) -> c_1()}
Details:
The given problem does not contain any strict rules