'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
, a(c(x1)) -> c(a(x1))
, b(c(x1)) -> c(b(x1))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(a(a(x1)))))))
, a^#(c(x1)) -> c_1(a^#(x1))
, b^#(c(x1)) -> c_2(b^#(x1))}
The usable rules are:
{ a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
, a(c(x1)) -> c(a(x1))
, b(c(x1)) -> c(b(x1))}
The estimated dependency graph contains the following edges:
{a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(a(a(x1)))))))}
==> {b^#(c(x1)) -> c_2(b^#(x1))}
{a^#(c(x1)) -> c_1(a^#(x1))}
==> {a^#(c(x1)) -> c_1(a^#(x1))}
{a^#(c(x1)) -> c_1(a^#(x1))}
==> {a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(a(a(x1)))))))}
{b^#(c(x1)) -> c_2(b^#(x1))}
==> {b^#(c(x1)) -> c_2(b^#(x1))}
We consider the following path(s):
1) { a^#(c(x1)) -> c_1(a^#(x1))
, a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(a(a(x1)))))))
, b^#(c(x1)) -> c_2(b^#(x1))}
The usable rules for this path are the following:
{ a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
, a(c(x1)) -> c(a(x1))
, b(c(x1)) -> c(b(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(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
, a(c(x1)) -> c(a(x1))
, b(c(x1)) -> c(b(x1))
, a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(a(a(x1)))))))
, a^#(c(x1)) -> c_1(a^#(x1))
, b^#(c(x1)) -> c_2(b^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(a(a(x1)))))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(a(a(x1)))))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [7]
c_2(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a^#(c(x1)) -> c_1(a^#(x1))
, b^#(c(x1)) -> c_2(b^#(x1))}
and weakly orienting the rules
{a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(a(a(x1)))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a^#(c(x1)) -> c_1(a^#(x1))
, b^#(c(x1)) -> c_2(b^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [15]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [5]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [5]
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(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
, a(c(x1)) -> c(a(x1))
, b(c(x1)) -> c(b(x1))}
Weak Rules:
{ a^#(c(x1)) -> c_1(a^#(x1))
, b^#(c(x1)) -> c_2(b^#(x1))
, a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(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:
{ a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
, a(c(x1)) -> c(a(x1))
, b(c(x1)) -> c(b(x1))}
Weak Rules:
{ a^#(c(x1)) -> c_1(a^#(x1))
, b^#(c(x1)) -> c_2(b^#(x1))
, a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(a(a(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
, b^#_0(3) -> 6
, c_1_0(4) -> 4
, c_2_0(6) -> 6}
2) { a^#(c(x1)) -> c_1(a^#(x1))
, a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(a(a(x1)))))))}
The usable rules for this path are the following:
{ a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
, a(c(x1)) -> c(a(x1))
, b(c(x1)) -> c(b(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(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
, a(c(x1)) -> c(a(x1))
, b(c(x1)) -> c(b(x1))
, a^#(c(x1)) -> c_1(a^#(x1))
, a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(a(a(x1)))))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(a(a(x1)))))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(a(a(x1)))))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(c(x1)) -> c_1(a^#(x1))}
and weakly orienting the rules
{a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(a(a(x1)))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(c(x1)) -> c_1(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2(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:
{ a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
, a(c(x1)) -> c(a(x1))
, b(c(x1)) -> c(b(x1))}
Weak Rules:
{ a^#(c(x1)) -> c_1(a^#(x1))
, a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(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:
{ a(a(b(b(x1)))) -> b(b(b(a(a(a(x1))))))
, a(c(x1)) -> c(a(x1))
, b(c(x1)) -> c(b(x1))}
Weak Rules:
{ a^#(c(x1)) -> c_1(a^#(x1))
, a^#(a(b(b(x1)))) -> c_0(b^#(b(b(a(a(a(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
, b^#_0(3) -> 6
, c_1_0(4) -> 4}
3) {a^#(c(x1)) -> c_1(a^#(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(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^#(c(x1)) -> c_1(a^#(x1))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#(c(x1)) -> c_1(a^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(c(x1)) -> c_1(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [0] x1 + [0]
b(x1) = [0] x1 + [0]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [3]
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: {a^#(c(x1)) -> c_1(a^#(x1))}
Details:
The given problem does not contain any strict rules