'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(x1))))
, b(x1) -> c(c(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(x1)))))
, b^#(x1) -> c_1()
, a^#(a(x1)) -> c_2(a^#(c(b(a(x1)))))}
The usable rules are:
{ b(c(a(x1))) -> a(b(a(b(x1))))
, b(x1) -> c(c(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(x1)))))}
==> {a^#(a(x1)) -> c_2(a^#(c(b(a(x1)))))}
We consider the following path(s):
1) { b^#(c(a(x1))) -> c_0(a^#(b(a(b(x1)))))
, a^#(a(x1)) -> c_2(a^#(c(b(a(x1)))))}
The usable rules for this path are the following:
{ b(c(a(x1))) -> a(b(a(b(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:
{ b(c(a(x1))) -> a(b(a(b(x1))))
, b(x1) -> c(c(x1))
, a(a(x1)) -> a(c(b(a(x1))))
, b^#(c(a(x1))) -> c_0(a^#(b(a(b(x1)))))
, a^#(a(x1)) -> c_2(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]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
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(a(x1))) -> c_0(a^#(b(a(b(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(x1)))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2(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:
{ b(c(a(x1))) -> a(b(a(b(x1))))
, a(a(x1)) -> a(c(b(a(x1))))
, a^#(a(x1)) -> c_2(a^#(c(b(a(x1)))))}
Weak Rules:
{ b^#(c(a(x1))) -> c_0(a^#(b(a(b(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(x1))))
, a(a(x1)) -> a(c(b(a(x1))))
, a^#(a(x1)) -> c_2(a^#(c(b(a(x1)))))}
Weak Rules:
{ b^#(c(a(x1))) -> c_0(a^#(b(a(b(x1)))))
, b(x1) -> c(c(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 2
, b^#_0(2) -> 4
, a^#_0(2) -> 6}
2) {b^#(c(a(x1))) -> c_0(a^#(b(a(b(x1)))))}
The usable rules for this path are the following:
{ b(c(a(x1))) -> a(b(a(b(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:
{ b(c(a(x1))) -> a(b(a(b(x1))))
, b(x1) -> c(c(x1))
, a(a(x1)) -> a(c(b(a(x1))))
, b^#(c(a(x1))) -> c_0(a^#(b(a(b(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]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2(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(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(x1)))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_1() = [0]
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:
{ b(c(a(x1))) -> a(b(a(b(x1))))
, a(a(x1)) -> a(c(b(a(x1))))}
Weak Rules:
{ b^#(c(a(x1))) -> c_0(a^#(b(a(b(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(x1))))
, a(a(x1)) -> a(c(b(a(x1))))}
Weak Rules:
{ b^#(c(a(x1))) -> c_0(a^#(b(a(b(x1)))))
, b(x1) -> c(c(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 2
, b^#_0(2) -> 4
, a^#_0(2) -> 6}
3) {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:
b(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(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^#(x1) -> c_1()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{b^#(x1) -> c_1()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(x1) -> c_1()}
Details:
Interpretation Functions:
b(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [4]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
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^#(x1) -> c_1()}
Details:
The given problem does not contain any strict rules