'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(x1) -> x1
, a(a(b(x1))) -> b(a(b(c(a(x1)))))
, c(b(x1)) -> a(x1)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(x1) -> c_0()
, a^#(a(b(x1))) -> c_1(a^#(b(c(a(x1)))))
, c^#(b(x1)) -> c_2(a^#(x1))}
The usable rules are:
{ a(x1) -> x1
, a(a(b(x1))) -> b(a(b(c(a(x1)))))
, c(b(x1)) -> a(x1)}
The estimated dependency graph contains the following edges:
{a^#(a(b(x1))) -> c_1(a^#(b(c(a(x1)))))}
==> {a^#(x1) -> c_0()}
{c^#(b(x1)) -> c_2(a^#(x1))}
==> {a^#(a(b(x1))) -> c_1(a^#(b(c(a(x1)))))}
{c^#(b(x1)) -> c_2(a^#(x1))}
==> {a^#(x1) -> c_0()}
We consider the following path(s):
1) { c^#(b(x1)) -> c_2(a^#(x1))
, a^#(a(b(x1))) -> c_1(a^#(b(c(a(x1)))))
, a^#(x1) -> c_0()}
The usable rules for this path are the following:
{ a(x1) -> x1
, a(a(b(x1))) -> b(a(b(c(a(x1)))))
, c(b(x1)) -> 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(b(x1))) -> b(a(b(c(a(x1)))))
, c(b(x1)) -> a(x1)
, a^#(a(b(x1))) -> c_1(a^#(b(c(a(x1)))))
, c^#(b(x1)) -> c_2(a^#(x1))
, a^#(x1) -> c_0()}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_0()}
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() = [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
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^#(x1))}
and weakly orienting the rules
{a^#(x1) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(b(x1)) -> c_2(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() = [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [9]
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(x1)}
and weakly orienting the rules
{ c^#(b(x1)) -> c_2(a^#(x1))
, a^#(x1) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(b(x1)) -> 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() = [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [13]
c_2(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(x1) -> x1}
and weakly orienting the rules
{ c(b(x1)) -> a(x1)
, c^#(b(x1)) -> c_2(a^#(x1))
, a^#(x1) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(x1) -> x1}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [12]
a^#(x1) = [1] x1 + [4]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [13]
c_2(x1) = [1] x1 + [5]
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))) -> b(a(b(c(a(x1)))))
, a^#(a(b(x1))) -> c_1(a^#(b(c(a(x1)))))}
Weak Rules:
{ a(x1) -> x1
, c(b(x1)) -> a(x1)
, c^#(b(x1)) -> c_2(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:
{ a(a(b(x1))) -> b(a(b(c(a(x1)))))
, a^#(a(b(x1))) -> c_1(a^#(b(c(a(x1)))))}
Weak Rules:
{ a(x1) -> x1
, c(b(x1)) -> a(x1)
, c^#(b(x1)) -> c_2(a^#(x1))
, a^#(x1) -> c_0()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ b_0(2) -> 2
, a^#_0(2) -> 4
, c_0_0() -> 4
, c^#_0(2) -> 7
, c_2_0(4) -> 7}
2) { c^#(b(x1)) -> c_2(a^#(x1))
, a^#(a(b(x1))) -> c_1(a^#(b(c(a(x1)))))}
The usable rules for this path are the following:
{ a(x1) -> x1
, a(a(b(x1))) -> b(a(b(c(a(x1)))))
, c(b(x1)) -> 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(b(x1))) -> b(a(b(c(a(x1)))))
, c(b(x1)) -> a(x1)
, c^#(b(x1)) -> c_2(a^#(x1))
, a^#(a(b(x1))) -> c_1(a^#(b(c(a(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c^#(b(x1)) -> c_2(a^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(b(x1)) -> c_2(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() = [0]
c_1(x1) = [1] x1 + [4]
c^#(x1) = [1] x1 + [4]
c_2(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(b(x1)) -> a(x1)}
and weakly orienting the rules
{c^#(b(x1)) -> c_2(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(b(x1)) -> 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() = [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [12]
c_2(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(x1) -> x1}
and weakly orienting the rules
{ c(b(x1)) -> a(x1)
, c^#(b(x1)) -> c_2(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 + [8]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [13]
a^#(x1) = [1] x1 + [13]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [13]
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(a(b(x1))) -> b(a(b(c(a(x1)))))
, a^#(a(b(x1))) -> c_1(a^#(b(c(a(x1)))))}
Weak Rules:
{ a(x1) -> x1
, c(b(x1)) -> a(x1)
, c^#(b(x1)) -> c_2(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))) -> b(a(b(c(a(x1)))))
, a^#(a(b(x1))) -> c_1(a^#(b(c(a(x1)))))}
Weak Rules:
{ a(x1) -> x1
, c(b(x1)) -> a(x1)
, c^#(b(x1)) -> c_2(a^#(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ b_0(2) -> 2
, a^#_0(2) -> 4
, c^#_0(2) -> 7
, c_2_0(4) -> 7}
3) {c^#(b(x1)) -> c_2(a^#(x1))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a(x1) = [0] x1 + [0]
b(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {c^#(b(x1)) -> c_2(a^#(x1))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{c^#(b(x1)) -> c_2(a^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(b(x1)) -> c_2(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [0] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {c^#(b(x1)) -> c_2(a^#(x1))}
Details:
The given problem does not contain any strict rules
4) { c^#(b(x1)) -> c_2(a^#(x1))
, a^#(x1) -> c_0()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a(x1) = [0] x1 + [0]
b(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {a^#(x1) -> c_0()}
Weak Rules: {c^#(b(x1)) -> c_2(a^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_0()}
and weakly orienting the rules
{c^#(b(x1)) -> c_2(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_0()}
Details:
Interpretation Functions:
a(x1) = [0] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules:
{ a^#(x1) -> c_0()
, c^#(b(x1)) -> c_2(a^#(x1))}
Details:
The given problem does not contain any strict rules