'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(x1) -> b(x1)
, b(a(c(x1))) -> c(b(a(a(x1))))
, b(b(x1)) -> c(x1)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(x1) -> c_0(b^#(x1))
, b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))
, b^#(b(x1)) -> c_2()}
The usable rules are:
{ a(x1) -> b(x1)
, b(a(c(x1))) -> c(b(a(a(x1))))
, b(b(x1)) -> c(x1)}
The estimated dependency graph contains the following edges:
{a^#(x1) -> c_0(b^#(x1))}
==> {b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))}
{a^#(x1) -> c_0(b^#(x1))}
==> {b^#(b(x1)) -> c_2()}
{b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))}
==> {b^#(b(x1)) -> c_2()}
{b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))}
==> {b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))}
We consider the following path(s):
1) { a^#(x1) -> c_0(b^#(x1))
, b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))
, b^#(b(x1)) -> c_2()}
The usable rules for this path are the following:
{ a(x1) -> b(x1)
, b(a(c(x1))) -> c(b(a(a(x1))))
, b(b(x1)) -> 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(x1) -> b(x1)
, b(a(c(x1))) -> c(b(a(a(x1))))
, b(b(x1)) -> c(x1)
, b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))
, a^#(x1) -> c_0(b^#(x1))
, b^#(b(x1)) -> c_2()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(b(x1)) -> c(x1)
, b^#(b(x1)) -> c_2()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(x1)) -> c(x1)
, b^#(b(x1)) -> c_2()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [2]
c_2() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_0(b^#(x1))}
and weakly orienting the rules
{ b(b(x1)) -> c(x1)
, b^#(b(x1)) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_0(b^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(x1) -> b(x1)}
and weakly orienting the rules
{ a^#(x1) -> c_0(b^#(x1))
, b(b(x1)) -> c(x1)
, b^#(b(x1)) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(x1) -> b(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [2]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [2]
a^#(x1) = [1] x1 + [12]
c_0(x1) = [1] x1 + [9]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))}
and weakly orienting the rules
{ a(x1) -> b(x1)
, a^#(x1) -> c_0(b^#(x1))
, b(b(x1)) -> c(x1)
, b^#(b(x1)) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [4]
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [4]
c_1(x1) = [1] x1 + [0]
c_2() = [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(a(c(x1))) -> c(b(a(a(x1))))}
Weak Rules:
{ b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))
, a(x1) -> b(x1)
, a^#(x1) -> c_0(b^#(x1))
, b(b(x1)) -> c(x1)
, b^#(b(x1)) -> c_2()}
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(a(c(x1))) -> c(b(a(a(x1))))}
Weak Rules:
{ b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))
, a(x1) -> b(x1)
, a^#(x1) -> c_0(b^#(x1))
, b(b(x1)) -> c(x1)
, b^#(b(x1)) -> c_2()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 2
, a^#_0(2) -> 1
, c_0_0(1) -> 1
, b^#_0(2) -> 1}
2) { a^#(x1) -> c_0(b^#(x1))
, b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))}
The usable rules for this path are the following:
{ a(x1) -> b(x1)
, b(a(c(x1))) -> c(b(a(a(x1))))
, b(b(x1)) -> 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(x1) -> b(x1)
, b(a(c(x1))) -> c(b(a(a(x1))))
, b(b(x1)) -> c(x1)
, a^#(x1) -> c_0(b^#(x1))
, b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b(b(x1)) -> c(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(x1)) -> c(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_0(b^#(x1))}
and weakly orienting the rules
{b(b(x1)) -> c(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_0(b^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(x1) -> b(x1)}
and weakly orienting the rules
{ a^#(x1) -> c_0(b^#(x1))
, b(b(x1)) -> c(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(x1) -> b(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [4]
b(x1) = [1] x1 + [3]
c(x1) = [1] x1 + [4]
a^#(x1) = [1] x1 + [12]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [3]
c_2() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))}
and weakly orienting the rules
{ a(x1) -> b(x1)
, a^#(x1) -> c_0(b^#(x1))
, b(b(x1)) -> c(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [4]
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [2]
b^#(x1) = [1] x1 + [4]
c_1(x1) = [1] x1 + [0]
c_2() = [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(a(c(x1))) -> c(b(a(a(x1))))}
Weak Rules:
{ b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))
, a(x1) -> b(x1)
, a^#(x1) -> c_0(b^#(x1))
, b(b(x1)) -> 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(a(c(x1))) -> c(b(a(a(x1))))}
Weak Rules:
{ b^#(a(c(x1))) -> c_1(b^#(a(a(x1))))
, a(x1) -> b(x1)
, a^#(x1) -> c_0(b^#(x1))
, b(b(x1)) -> c(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(3) -> 3
, a^#_0(3) -> 4
, c_0_0(6) -> 4
, b^#_0(3) -> 6}
3) { a^#(x1) -> c_0(b^#(x1))
, b^#(b(x1)) -> c_2()}
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(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [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_2()}
Weak Rules: {a^#(x1) -> c_0(b^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b^#(b(x1)) -> c_2()}
and weakly orienting the rules
{a^#(x1) -> c_0(b^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(x1)) -> c_2()}
Details:
Interpretation Functions:
a(x1) = [0] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2() = [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_2()
, a^#(x1) -> c_0(b^#(x1))}
Details:
The given problem does not contain any strict rules
4) {a^#(x1) -> c_0(b^#(x1))}
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(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [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^#(x1) -> c_0(b^#(x1))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_0(b^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_0(b^#(x1))}
Details:
Interpretation Functions:
a(x1) = [0] x1 + [0]
b(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [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^#(x1) -> c_0(b^#(x1))}
Details:
The given problem does not contain any strict rules