'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(a(x1))) -> a(b(x1))
, b(c(x1)) -> c(c(a(a(a(x1)))))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(a(a(x1))) -> c_0(a^#(b(x1)))
, b^#(c(x1)) -> c_1(a^#(a(a(x1))))}
The usable rules are:
{ a(a(a(x1))) -> a(b(x1))
, b(c(x1)) -> c(c(a(a(a(x1)))))}
The estimated dependency graph contains the following edges:
{a^#(a(a(x1))) -> c_0(a^#(b(x1)))}
==> {a^#(a(a(x1))) -> c_0(a^#(b(x1)))}
{b^#(c(x1)) -> c_1(a^#(a(a(x1))))}
==> {a^#(a(a(x1))) -> c_0(a^#(b(x1)))}
We consider the following path(s):
1) {b^#(c(x1)) -> c_1(a^#(a(a(x1))))}
The usable rules for this path are the following:
{ a(a(a(x1))) -> a(b(x1))
, b(c(x1)) -> c(c(a(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(a(x1))) -> a(b(x1))
, b(c(x1)) -> c(c(a(a(a(x1)))))
, b^#(c(x1)) -> c_1(a^#(a(a(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b^#(c(x1)) -> c_1(a^#(a(a(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(x1)) -> c_1(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 + [0]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(c(x1)) -> c(c(a(a(a(x1)))))}
and weakly orienting the rules
{b^#(c(x1)) -> c_1(a^#(a(a(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(c(x1)) -> c(c(a(a(a(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [9]
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(a(x1))) -> a(b(x1))}
Weak Rules:
{ b(c(x1)) -> c(c(a(a(a(x1)))))
, b^#(c(x1)) -> c_1(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(a(x1))) -> a(b(x1))}
Weak Rules:
{ b(c(x1)) -> c(c(a(a(a(x1)))))
, b^#(c(x1)) -> c_1(a^#(a(a(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 5
, a_0(5) -> 4
, c_0(2) -> 2
, a^#_0(2) -> 1
, a^#_0(4) -> 3
, b^#_0(2) -> 1
, c_1_0(3) -> 1}
2) { b^#(c(x1)) -> c_1(a^#(a(a(x1))))
, a^#(a(a(x1))) -> c_0(a^#(b(x1)))}
The usable rules for this path are the following:
{ a(a(a(x1))) -> a(b(x1))
, b(c(x1)) -> c(c(a(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(a(x1))) -> a(b(x1))
, b(c(x1)) -> c(c(a(a(a(x1)))))
, b^#(c(x1)) -> c_1(a^#(a(a(x1))))
, a^#(a(a(x1))) -> c_0(a^#(b(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b^#(c(x1)) -> c_1(a^#(a(a(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(x1)) -> c_1(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 + [0]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(c(x1)) -> c(c(a(a(a(x1)))))}
and weakly orienting the rules
{b^#(c(x1)) -> c_1(a^#(a(a(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(c(x1)) -> c(c(a(a(a(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [4]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [4]
b^#(x1) = [1] x1 + [1]
c_1(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:
{ a(a(a(x1))) -> a(b(x1))
, a^#(a(a(x1))) -> c_0(a^#(b(x1)))}
Weak Rules:
{ b(c(x1)) -> c(c(a(a(a(x1)))))
, b^#(c(x1)) -> c_1(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(a(x1))) -> a(b(x1))
, a^#(a(a(x1))) -> c_0(a^#(b(x1)))}
Weak Rules:
{ b(c(x1)) -> c(c(a(a(a(x1)))))
, b^#(c(x1)) -> c_1(a^#(a(a(x1))))}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 9
, a_0(9) -> 8
, a_1(3) -> 15
, a_1(14) -> 13
, a_1(15) -> 14
, a_2(17) -> 13
, b_1(3) -> 11
, b_2(3) -> 17
, c_0(3) -> 3
, c_1(12) -> 11
, c_1(12) -> 17
, c_1(13) -> 12
, a^#_0(3) -> 4
, a^#_0(8) -> 7
, a^#_1(11) -> 10
, a^#_1(14) -> 16
, a^#_2(17) -> 18
, c_0_1(10) -> 7
, c_0_2(18) -> 16
, b^#_0(3) -> 6
, c_1_0(7) -> 6
, c_1_1(16) -> 6}