'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(b(x1)) -> x1
, a(c(x1)) -> b(c(a(a(x1))))
, c(b(x1)) -> a(c(x1))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(b(x1)) -> c_0()
, a^#(c(x1)) -> c_1(c^#(a(a(x1))))
, c^#(b(x1)) -> c_2(a^#(c(x1)))}
The usable rules are:
{ a(b(x1)) -> x1
, a(c(x1)) -> b(c(a(a(x1))))
, c(b(x1)) -> a(c(x1))}
The estimated dependency graph contains the following edges:
{a^#(c(x1)) -> c_1(c^#(a(a(x1))))}
==> {c^#(b(x1)) -> c_2(a^#(c(x1)))}
{c^#(b(x1)) -> c_2(a^#(c(x1)))}
==> {a^#(c(x1)) -> c_1(c^#(a(a(x1))))}
{c^#(b(x1)) -> c_2(a^#(c(x1)))}
==> {a^#(b(x1)) -> c_0()}
We consider the following path(s):
1) { a^#(c(x1)) -> c_1(c^#(a(a(x1))))
, c^#(b(x1)) -> c_2(a^#(c(x1)))}
The usable rules for this path are the following:
{ a(b(x1)) -> x1
, a(c(x1)) -> b(c(a(a(x1))))
, c(b(x1)) -> a(c(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(b(x1)) -> x1
, a(c(x1)) -> b(c(a(a(x1))))
, c(b(x1)) -> a(c(x1))
, a^#(c(x1)) -> c_1(c^#(a(a(x1))))
, c^#(b(x1)) -> c_2(a^#(c(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a(b(x1)) -> x1}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(b(x1)) -> x1}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(b(x1)) -> c_2(a^#(c(x1)))}
and weakly orienting the rules
{a(b(x1)) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(b(x1)) -> c_2(a^#(c(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [9]
c^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(b(x1)) -> a(c(x1))}
and weakly orienting the rules
{ c^#(b(x1)) -> c_2(a^#(c(x1)))
, a(b(x1)) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(b(x1)) -> a(c(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [9]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(c(x1)) -> c_1(c^#(a(a(x1))))}
and weakly orienting the rules
{ c(b(x1)) -> a(c(x1))
, c^#(b(x1)) -> c_2(a^#(c(x1)))
, a(b(x1)) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(c(x1)) -> c_1(c^#(a(a(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [7]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [3]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(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(c(x1)) -> b(c(a(a(x1))))}
Weak Rules:
{ a^#(c(x1)) -> c_1(c^#(a(a(x1))))
, c(b(x1)) -> a(c(x1))
, c^#(b(x1)) -> c_2(a^#(c(x1)))
, a(b(x1)) -> 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(c(x1)) -> b(c(a(a(x1))))}
Weak Rules:
{ a^#(c(x1)) -> c_1(c^#(a(a(x1))))
, c(b(x1)) -> a(c(x1))
, c^#(b(x1)) -> c_2(a^#(c(x1)))
, a(b(x1)) -> x1}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 12
, a_0(9) -> 9
, a_0(12) -> 11
, a_1(2) -> 15
, a_1(14) -> 15
, a_1(15) -> 14
, a_1(18) -> 9
, a_1(18) -> 13
, a_1(18) -> 18
, a_1(18) -> 19
, a_1(20) -> 15
, a_2(2) -> 21
, a_2(20) -> 21
, a_2(21) -> 20
, b_0(2) -> 2
, b_0(2) -> 11
, b_0(2) -> 12
, b_0(2) -> 14
, b_0(2) -> 15
, b_0(2) -> 20
, b_0(2) -> 21
, b_1(13) -> 9
, b_2(19) -> 9
, b_2(19) -> 13
, b_2(19) -> 18
, b_2(19) -> 19
, c_0(2) -> 9
, c_1(2) -> 18
, c_1(14) -> 9
, c_1(14) -> 13
, c_2(20) -> 9
, c_2(20) -> 13
, c_2(20) -> 18
, c_2(20) -> 19
, a^#_0(2) -> 4
, a^#_0(9) -> 8
, a^#_1(18) -> 17
, c_1_0(10) -> 8
, c_1_1(16) -> 8
, c_1_1(16) -> 17
, c_1_2(22) -> 17
, c^#_0(2) -> 7
, c^#_0(11) -> 10
, c^#_1(14) -> 16
, c^#_2(20) -> 22
, c_2_0(8) -> 7
, c_2_1(17) -> 7
, c_2_1(17) -> 10
, c_2_1(17) -> 16
, c_2_1(17) -> 22}
2) { a^#(c(x1)) -> c_1(c^#(a(a(x1))))
, c^#(b(x1)) -> c_2(a^#(c(x1)))
, a^#(b(x1)) -> c_0()}
The usable rules for this path are the following:
{ a(b(x1)) -> x1
, a(c(x1)) -> b(c(a(a(x1))))
, c(b(x1)) -> a(c(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(b(x1)) -> x1
, a(c(x1)) -> b(c(a(a(x1))))
, c(b(x1)) -> a(c(x1))
, a^#(c(x1)) -> c_1(c^#(a(a(x1))))
, c^#(b(x1)) -> c_2(a^#(c(x1)))
, a^#(b(x1)) -> c_0()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(b(x1)) -> x1
, a^#(b(x1)) -> c_0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(b(x1)) -> x1
, a^#(b(x1)) -> c_0()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(b(x1)) -> c_2(a^#(c(x1)))}
and weakly orienting the rules
{ a(b(x1)) -> x1
, a^#(b(x1)) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(b(x1)) -> c_2(a^#(c(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [7]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(b(x1)) -> a(c(x1))}
and weakly orienting the rules
{ c^#(b(x1)) -> c_2(a^#(c(x1)))
, a(b(x1)) -> x1
, a^#(b(x1)) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(b(x1)) -> a(c(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [9]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(c(x1)) -> c_1(c^#(a(a(x1))))}
and weakly orienting the rules
{ c(b(x1)) -> a(c(x1))
, c^#(b(x1)) -> c_2(a^#(c(x1)))
, a(b(x1)) -> x1
, a^#(b(x1)) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(c(x1)) -> c_1(c^#(a(a(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [9]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [3]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [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: {a(c(x1)) -> b(c(a(a(x1))))}
Weak Rules:
{ a^#(c(x1)) -> c_1(c^#(a(a(x1))))
, c(b(x1)) -> a(c(x1))
, c^#(b(x1)) -> c_2(a^#(c(x1)))
, a(b(x1)) -> x1
, a^#(b(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: {a(c(x1)) -> b(c(a(a(x1))))}
Weak Rules:
{ a^#(c(x1)) -> c_1(c^#(a(a(x1))))
, c(b(x1)) -> a(c(x1))
, c^#(b(x1)) -> c_2(a^#(c(x1)))
, a(b(x1)) -> x1
, a^#(b(x1)) -> c_0()}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 12
, a_0(9) -> 9
, a_0(12) -> 11
, a_1(2) -> 15
, a_1(14) -> 15
, a_1(15) -> 14
, a_1(18) -> 9
, a_1(18) -> 13
, a_1(18) -> 18
, a_1(18) -> 19
, a_1(20) -> 15
, a_2(2) -> 21
, a_2(20) -> 21
, a_2(21) -> 20
, b_0(2) -> 2
, b_0(2) -> 11
, b_0(2) -> 12
, b_0(2) -> 14
, b_0(2) -> 15
, b_0(2) -> 20
, b_0(2) -> 21
, b_1(13) -> 9
, b_2(19) -> 9
, b_2(19) -> 13
, b_2(19) -> 18
, b_2(19) -> 19
, c_0(2) -> 9
, c_1(2) -> 18
, c_1(14) -> 9
, c_1(14) -> 13
, c_2(20) -> 9
, c_2(20) -> 13
, c_2(20) -> 18
, c_2(20) -> 19
, a^#_0(2) -> 4
, a^#_0(9) -> 8
, a^#_1(18) -> 17
, c_0_0() -> 4
, c_0_1() -> 8
, c_0_2() -> 17
, c_1_0(10) -> 8
, c_1_1(16) -> 8
, c_1_1(16) -> 17
, c_1_2(22) -> 17
, c^#_0(2) -> 7
, c^#_0(11) -> 10
, c^#_1(14) -> 16
, c^#_2(20) -> 22
, c_2_0(8) -> 7
, c_2_1(17) -> 7
, c_2_1(17) -> 10
, c_2_1(17) -> 16
, c_2_1(17) -> 22}