'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(b(x1))) -> a(b(c(a(x1))))
, a(c(x1)) -> b(a(a(x1)))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(a(b(x1))) -> c_0(a^#(b(c(a(x1)))))
, a^#(c(x1)) -> c_1(a^#(a(x1)))}
The usable rules are:
{ a(a(b(x1))) -> a(b(c(a(x1))))
, a(c(x1)) -> b(a(a(x1)))}
The estimated dependency graph contains the following edges:
{a^#(c(x1)) -> c_1(a^#(a(x1)))}
==> {a^#(a(b(x1))) -> c_0(a^#(b(c(a(x1)))))}
We consider the following path(s):
1) { a^#(c(x1)) -> c_1(a^#(a(x1)))
, a^#(a(b(x1))) -> c_0(a^#(b(c(a(x1)))))}
The usable rules for this path are the following:
{ a(a(b(x1))) -> a(b(c(a(x1))))
, a(c(x1)) -> b(a(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:
{ a(a(b(x1))) -> a(b(c(a(x1))))
, a(c(x1)) -> b(a(a(x1)))
, a^#(c(x1)) -> c_1(a^#(a(x1)))
, a^#(a(b(x1))) -> c_0(a^#(b(c(a(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(c(x1)) -> b(a(a(x1)))
, a^#(c(x1)) -> c_1(a^#(a(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(c(x1)) -> b(a(a(x1)))
, a^#(c(x1)) -> c_1(a^#(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [8]
c_1(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(a(b(x1))) -> a(b(c(a(x1))))
, a^#(a(b(x1))) -> c_0(a^#(b(c(a(x1)))))}
Weak Rules:
{ a(c(x1)) -> b(a(a(x1)))
, a^#(c(x1)) -> c_1(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(x1))) -> a(b(c(a(x1))))
, a^#(a(b(x1))) -> c_0(a^#(b(c(a(x1)))))}
Weak Rules:
{ a(c(x1)) -> b(a(a(x1)))
, a^#(c(x1)) -> c_1(a^#(a(x1)))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 4
, a_1(2) -> 8
, a_1(6) -> 9
, a_1(8) -> 9
, b_0(2) -> 2
, b_1(7) -> 6
, b_1(9) -> 4
, b_1(9) -> 8
, c_0(2) -> 2
, c_1(8) -> 7
, a^#_0(2) -> 1
, a^#_0(4) -> 3
, a^#_1(6) -> 5
, a^#_1(8) -> 10
, c_0_1(5) -> 3
, c_0_1(5) -> 10
, c_1_0(3) -> 1
, c_1_1(10) -> 1}
2) {a^#(c(x1)) -> c_1(a^#(a(x1)))}
The usable rules for this path are the following:
{ a(a(b(x1))) -> a(b(c(a(x1))))
, a(c(x1)) -> b(a(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:
{ a(a(b(x1))) -> a(b(c(a(x1))))
, a(c(x1)) -> b(a(a(x1)))
, a^#(c(x1)) -> c_1(a^#(a(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a(c(x1)) -> b(a(a(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(c(x1)) -> b(a(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [8]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(c(x1)) -> c_1(a^#(a(x1)))}
and weakly orienting the rules
{a(c(x1)) -> b(a(a(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(c(x1)) -> c_1(a^#(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) = [0] x1 + [0]
c_1(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(a(b(x1))) -> a(b(c(a(x1))))}
Weak Rules:
{ a^#(c(x1)) -> c_1(a^#(a(x1)))
, a(c(x1)) -> b(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(x1))) -> a(b(c(a(x1))))}
Weak Rules:
{ a^#(c(x1)) -> c_1(a^#(a(x1)))
, a(c(x1)) -> b(a(a(x1)))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 4
, a_0(4) -> 5
, a_1(2) -> 8
, a_1(6) -> 5
, a_1(6) -> 10
, a_1(8) -> 10
, b_0(2) -> 2
, b_0(5) -> 4
, b_1(7) -> 6
, b_1(10) -> 4
, b_1(10) -> 8
, c_0(2) -> 2
, c_1(8) -> 7
, a^#_0(2) -> 1
, a^#_0(4) -> 3
, a^#_1(8) -> 9
, c_1_0(3) -> 1
, c_1_1(9) -> 1}