'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(x1) -> x1
, a(a(x1)) -> b(x1)
, b(x1) -> a(x1)
, b(c(x1)) -> c(c(b(a(x1))))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(x1) -> c_0()
, a^#(a(x1)) -> c_1(b^#(x1))
, b^#(x1) -> c_2(a^#(x1))
, b^#(c(x1)) -> c_3(b^#(a(x1)))}
The usable rules are:
{ a(x1) -> x1
, a(a(x1)) -> b(x1)
, b(x1) -> a(x1)
, b(c(x1)) -> c(c(b(a(x1))))}
The estimated dependency graph contains the following edges:
{a^#(a(x1)) -> c_1(b^#(x1))}
==> {b^#(c(x1)) -> c_3(b^#(a(x1)))}
{a^#(a(x1)) -> c_1(b^#(x1))}
==> {b^#(x1) -> c_2(a^#(x1))}
{b^#(x1) -> c_2(a^#(x1))}
==> {a^#(a(x1)) -> c_1(b^#(x1))}
{b^#(x1) -> c_2(a^#(x1))}
==> {a^#(x1) -> c_0()}
{b^#(c(x1)) -> c_3(b^#(a(x1)))}
==> {b^#(c(x1)) -> c_3(b^#(a(x1)))}
{b^#(c(x1)) -> c_3(b^#(a(x1)))}
==> {b^#(x1) -> c_2(a^#(x1))}
We consider the following path(s):
1) { a^#(a(x1)) -> c_1(b^#(x1))
, b^#(x1) -> c_2(a^#(x1))
, b^#(c(x1)) -> c_3(b^#(a(x1)))}
The usable rules for this path are the following:
{ a(x1) -> x1
, a(a(x1)) -> b(x1)
, b(x1) -> a(x1)
, b(c(x1)) -> c(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:
{ a(x1) -> x1
, a(a(x1)) -> b(x1)
, b(x1) -> a(x1)
, b(c(x1)) -> c(c(b(a(x1))))
, a^#(a(x1)) -> c_1(b^#(x1))
, b^#(x1) -> c_2(a^#(x1))
, b^#(c(x1)) -> c_3(b^#(a(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b(x1) -> a(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(x1) -> a(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(x1) -> c_2(a^#(x1))}
and weakly orienting the rules
{b(x1) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(x1) -> c_2(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(c(x1)) -> c_3(b^#(a(x1)))}
and weakly orienting the rules
{ b^#(x1) -> c_2(a^#(x1))
, b(x1) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(x1)) -> c_3(b^#(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [9]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [4]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(x1) -> x1}
and weakly orienting the rules
{ b^#(c(x1)) -> c_3(b^#(a(x1)))
, b^#(x1) -> c_2(a^#(x1))
, b(x1) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(x1) -> x1}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [7]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(a(x1)) -> c_1(b^#(x1))}
and weakly orienting the rules
{ a(x1) -> x1
, b^#(c(x1)) -> c_3(b^#(a(x1)))
, b^#(x1) -> c_2(a^#(x1))
, b(x1) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(x1)) -> c_1(b^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [2]
b(x1) = [1] x1 + [7]
c(x1) = [1] x1 + [3]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(x1)) -> b(x1)}
and weakly orienting the rules
{ a^#(a(x1)) -> c_1(b^#(x1))
, a(x1) -> x1
, b^#(c(x1)) -> c_3(b^#(a(x1)))
, b^#(x1) -> c_2(a^#(x1))
, b(x1) -> a(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(x1)) -> b(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [2]
c_2(x1) = [1] x1 + [0]
c_3(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(x1)) -> c(c(b(a(x1))))}
Weak Rules:
{ a(a(x1)) -> b(x1)
, a^#(a(x1)) -> c_1(b^#(x1))
, a(x1) -> x1
, b^#(c(x1)) -> c_3(b^#(a(x1)))
, b^#(x1) -> c_2(a^#(x1))
, b(x1) -> 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(c(x1)) -> c(c(b(a(x1))))}
Weak Rules:
{ a(a(x1)) -> b(x1)
, a^#(a(x1)) -> c_1(b^#(x1))
, a(x1) -> x1
, b^#(c(x1)) -> c_3(b^#(a(x1)))
, b^#(x1) -> c_2(a^#(x1))
, b(x1) -> a(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 9
, c_0(3) -> 3
, c_0(3) -> 9
, a^#_0(3) -> 4
, a^#_0(9) -> 10
, c_1_0(7) -> 10
, b^#_0(3) -> 7
, b^#_0(9) -> 8
, c_2_0(4) -> 7
, c_2_0(10) -> 8
, c_3_0(8) -> 7
, c_3_0(8) -> 8}
2) { a^#(a(x1)) -> c_1(b^#(x1))
, b^#(x1) -> c_2(a^#(x1))
, b^#(c(x1)) -> c_3(b^#(a(x1)))
, a^#(x1) -> c_0()}
The usable rules for this path are the following:
{ a(x1) -> x1
, a(a(x1)) -> b(x1)
, b(x1) -> a(x1)
, b(c(x1)) -> c(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:
{ a(x1) -> x1
, a(a(x1)) -> b(x1)
, b(x1) -> a(x1)
, b(c(x1)) -> c(c(b(a(x1))))
, a^#(a(x1)) -> c_1(b^#(x1))
, b^#(x1) -> c_2(a^#(x1))
, b^#(c(x1)) -> c_3(b^#(a(x1)))
, a^#(x1) -> c_0()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(x1) -> a(x1)
, a^#(x1) -> c_0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(x1) -> a(x1)
, a^#(x1) -> c_0()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(x1) -> c_2(a^#(x1))}
and weakly orienting the rules
{ b(x1) -> a(x1)
, a^#(x1) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(x1) -> c_2(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(c(x1)) -> c_3(b^#(a(x1)))}
and weakly orienting the rules
{ b^#(x1) -> c_2(a^#(x1))
, b(x1) -> a(x1)
, a^#(x1) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(x1)) -> c_3(b^#(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [3]
b^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(x1) -> x1
, a(a(x1)) -> b(x1)
, a^#(a(x1)) -> c_1(b^#(x1))}
and weakly orienting the rules
{ b^#(c(x1)) -> c_3(b^#(a(x1)))
, b^#(x1) -> c_2(a^#(x1))
, b(x1) -> a(x1)
, a^#(x1) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(x1) -> x1
, a(a(x1)) -> b(x1)
, a^#(a(x1)) -> c_1(b^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [7]
b(x1) = [1] x1 + [9]
c(x1) = [1] x1 + [12]
a^#(x1) = [1] x1 + [4]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [4]
c_2(x1) = [1] x1 + [0]
c_3(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(x1)) -> c(c(b(a(x1))))}
Weak Rules:
{ a(x1) -> x1
, a(a(x1)) -> b(x1)
, a^#(a(x1)) -> c_1(b^#(x1))
, b^#(c(x1)) -> c_3(b^#(a(x1)))
, b^#(x1) -> c_2(a^#(x1))
, b(x1) -> a(x1)
, a^#(x1) -> c_0()}
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(x1)) -> c(c(b(a(x1))))}
Weak Rules:
{ a(x1) -> x1
, a(a(x1)) -> b(x1)
, a^#(a(x1)) -> c_1(b^#(x1))
, b^#(c(x1)) -> c_3(b^#(a(x1)))
, b^#(x1) -> c_2(a^#(x1))
, b(x1) -> a(x1)
, a^#(x1) -> c_0()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 9
, c_0(3) -> 3
, c_0(3) -> 9
, a^#_0(3) -> 4
, a^#_0(9) -> 10
, c_0_0() -> 4
, c_0_0() -> 10
, c_1_0(7) -> 10
, b^#_0(3) -> 7
, b^#_0(9) -> 8
, c_2_0(4) -> 7
, c_2_0(10) -> 8
, c_3_0(8) -> 7
, c_3_0(8) -> 8}