'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(b(c(x1)))) -> b(b(a(a(x1))))
, b(x1) -> c(c(a(a(x1))))
, b(c(x1)) -> a(x1)
, a(a(c(x1))) -> x1}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))
, b^#(x1) -> c_1(a^#(a(x1)))
, b^#(c(x1)) -> c_2(a^#(x1))
, a^#(a(c(x1))) -> c_3()}
The usable rules are:
{ a(a(b(c(x1)))) -> b(b(a(a(x1))))
, b(x1) -> c(c(a(a(x1))))
, b(c(x1)) -> a(x1)
, a(a(c(x1))) -> x1}
The estimated dependency graph contains the following edges:
{a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))}
==> {b^#(c(x1)) -> c_2(a^#(x1))}
{a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))}
==> {b^#(x1) -> c_1(a^#(a(x1)))}
{b^#(x1) -> c_1(a^#(a(x1)))}
==> {a^#(a(c(x1))) -> c_3()}
{b^#(x1) -> c_1(a^#(a(x1)))}
==> {a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))}
{b^#(c(x1)) -> c_2(a^#(x1))}
==> {a^#(a(c(x1))) -> c_3()}
{b^#(c(x1)) -> c_2(a^#(x1))}
==> {a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))}
We consider the following path(s):
1) { a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))
, b^#(c(x1)) -> c_2(a^#(x1))
, b^#(x1) -> c_1(a^#(a(x1)))
, a^#(a(c(x1))) -> c_3()}
The usable rules for this path are the following:
{ a(a(b(c(x1)))) -> b(b(a(a(x1))))
, b(x1) -> c(c(a(a(x1))))
, b(c(x1)) -> a(x1)
, a(a(c(x1))) -> 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(c(x1)))) -> b(b(a(a(x1))))
, b(x1) -> c(c(a(a(x1))))
, b(c(x1)) -> a(x1)
, a(a(c(x1))) -> x1
, a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))
, b^#(c(x1)) -> c_2(a^#(x1))
, b^#(x1) -> c_1(a^#(a(x1)))
, a^#(a(c(x1))) -> c_3()}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#(a(c(x1))) -> c_3()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(c(x1))) -> c_3()}
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 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))}
and weakly orienting the rules
{a^#(a(c(x1))) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(b(c(x1)))) -> c_0(b^#(b(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) = [1] x1 + [0]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(c(x1))) -> x1}
and weakly orienting the rules
{ a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))
, a^#(a(c(x1))) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(c(x1))) -> x1}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(a(b(c(x1)))) -> b(b(a(a(x1))))
, b(c(x1)) -> a(x1)
, b^#(c(x1)) -> c_2(a^#(x1))}
and weakly orienting the rules
{ a(a(c(x1))) -> x1
, a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))
, a^#(a(c(x1))) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(a(b(c(x1)))) -> b(b(a(a(x1))))
, b(c(x1)) -> a(x1)
, b^#(c(x1)) -> c_2(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [2]
b(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [10]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(x1) -> c_1(a^#(a(x1)))}
and weakly orienting the rules
{ a(a(b(c(x1)))) -> b(b(a(a(x1))))
, b(c(x1)) -> a(x1)
, b^#(c(x1)) -> c_2(a^#(x1))
, a(a(c(x1))) -> x1
, a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))
, a^#(a(c(x1))) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(x1) -> c_1(a^#(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [4]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3() = [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(x1) -> c(c(a(a(x1))))}
Weak Rules:
{ b^#(x1) -> c_1(a^#(a(x1)))
, a(a(b(c(x1)))) -> b(b(a(a(x1))))
, b(c(x1)) -> a(x1)
, b^#(c(x1)) -> c_2(a^#(x1))
, a(a(c(x1))) -> x1
, a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))
, a^#(a(c(x1))) -> c_3()}
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(x1) -> c(c(a(a(x1))))}
Weak Rules:
{ b^#(x1) -> c_1(a^#(a(x1)))
, a(a(b(c(x1)))) -> b(b(a(a(x1))))
, b(c(x1)) -> a(x1)
, b^#(c(x1)) -> c_2(a^#(x1))
, a(a(c(x1))) -> x1
, a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))
, a^#(a(c(x1))) -> c_3()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 8
, c_0(3) -> 3
, a^#_0(3) -> 4
, a^#_0(8) -> 7
, b^#_0(3) -> 6
, c_1_0(7) -> 6
, c_2_0(4) -> 6
, c_3_0() -> 7}
2) { a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))
, b^#(c(x1)) -> c_2(a^#(x1))
, b^#(x1) -> c_1(a^#(a(x1)))}
The usable rules for this path are the following:
{ a(a(b(c(x1)))) -> b(b(a(a(x1))))
, b(x1) -> c(c(a(a(x1))))
, b(c(x1)) -> a(x1)
, a(a(c(x1))) -> 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(c(x1)))) -> b(b(a(a(x1))))
, b(x1) -> c(c(a(a(x1))))
, b(c(x1)) -> a(x1)
, a(a(c(x1))) -> x1
, a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))
, b^#(c(x1)) -> c_2(a^#(x1))
, b^#(x1) -> c_1(a^#(a(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#(a(b(c(x1)))) -> c_0(b^#(b(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(c(x1)))) -> c_0(b^#(b(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) = [1] x1 + [0]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(c(x1))) -> x1}
and weakly orienting the rules
{a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(c(x1))) -> x1}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(a(b(c(x1)))) -> b(b(a(a(x1))))
, b(c(x1)) -> a(x1)
, b^#(c(x1)) -> c_2(a^#(x1))}
and weakly orienting the rules
{ a(a(c(x1))) -> x1
, a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(a(b(c(x1)))) -> b(b(a(a(x1))))
, b(c(x1)) -> a(x1)
, b^#(c(x1)) -> c_2(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [13]
b(x1) = [1] x1 + [6]
c(x1) = [1] x1 + [14]
a^#(x1) = [1] x1 + [7]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(x1) -> c_1(a^#(a(x1)))}
and weakly orienting the rules
{ a(a(b(c(x1)))) -> b(b(a(a(x1))))
, b(c(x1)) -> a(x1)
, b^#(c(x1)) -> c_2(a^#(x1))
, a(a(c(x1))) -> x1
, a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(x1) -> c_1(a^#(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [6]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3() = [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(x1) -> c(c(a(a(x1))))}
Weak Rules:
{ b^#(x1) -> c_1(a^#(a(x1)))
, a(a(b(c(x1)))) -> b(b(a(a(x1))))
, b(c(x1)) -> a(x1)
, b^#(c(x1)) -> c_2(a^#(x1))
, a(a(c(x1))) -> x1
, a^#(a(b(c(x1)))) -> c_0(b^#(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: {b(x1) -> c(c(a(a(x1))))}
Weak Rules:
{ b^#(x1) -> c_1(a^#(a(x1)))
, a(a(b(c(x1)))) -> b(b(a(a(x1))))
, b(c(x1)) -> a(x1)
, b^#(c(x1)) -> c_2(a^#(x1))
, a(a(c(x1))) -> x1
, a^#(a(b(c(x1)))) -> c_0(b^#(b(a(a(x1)))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 8
, c_0(3) -> 3
, a^#_0(3) -> 4
, a^#_0(8) -> 7
, b^#_0(3) -> 6
, c_1_0(7) -> 6
, c_2_0(4) -> 6}