'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(b(c(x1)))
, c(x1) -> x1
, c(b(x1)) -> b(c(a(x1)))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(x1) -> c_0()
, a^#(a(x1)) -> c_1(c^#(x1))
, c^#(x1) -> c_2()
, c^#(b(x1)) -> c_3(c^#(a(x1)))}
The usable rules are:
{ a(x1) -> x1
, a(a(x1)) -> b(b(c(x1)))
, c(x1) -> x1
, c(b(x1)) -> b(c(a(x1)))}
The estimated dependency graph contains the following edges:
{a^#(a(x1)) -> c_1(c^#(x1))}
==> {c^#(b(x1)) -> c_3(c^#(a(x1)))}
{a^#(a(x1)) -> c_1(c^#(x1))}
==> {c^#(x1) -> c_2()}
{c^#(b(x1)) -> c_3(c^#(a(x1)))}
==> {c^#(b(x1)) -> c_3(c^#(a(x1)))}
{c^#(b(x1)) -> c_3(c^#(a(x1)))}
==> {c^#(x1) -> c_2()}
We consider the following path(s):
1) { a^#(a(x1)) -> c_1(c^#(x1))
, c^#(b(x1)) -> c_3(c^#(a(x1)))
, c^#(x1) -> c_2()}
The usable rules for this path are the following:
{ a(x1) -> x1
, a(a(x1)) -> b(b(c(x1)))
, c(x1) -> x1
, c(b(x1)) -> b(c(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(b(c(x1)))
, c(x1) -> x1
, c(b(x1)) -> b(c(a(x1)))
, c^#(b(x1)) -> c_3(c^#(a(x1)))
, a^#(a(x1)) -> c_1(c^#(x1))
, c^#(x1) -> c_2()}
Details:
We apply the weight gap principle, strictly orienting the rules
{c^#(x1) -> c_2()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(x1) -> c_2()}
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() = [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(c^#(x1))}
and weakly orienting the rules
{c^#(x1) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(x1)) -> c_1(c^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(b(x1)) -> c_3(c^#(a(x1)))}
and weakly orienting the rules
{ a^#(a(x1)) -> c_1(c^#(x1))
, c^#(x1) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(b(x1)) -> c_3(c^#(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [13]
c_0() = [0]
c_1(x1) = [1] x1 + [4]
c^#(x1) = [1] x1 + [8]
c_2() = [0]
c_3(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(x1) -> x1}
and weakly orienting the rules
{ c^#(b(x1)) -> c_3(c^#(a(x1)))
, a^#(a(x1)) -> c_1(c^#(x1))
, c^#(x1) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(x1) -> x1}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2() = [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}
and weakly orienting the rules
{ c(x1) -> x1
, c^#(b(x1)) -> c_3(c^#(a(x1)))
, a^#(a(x1)) -> c_1(c^#(x1))
, c^#(x1) -> c_2()}
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 + [8]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1(x1) = [1] x1 + [9]
c^#(x1) = [1] x1 + [0]
c_2() = [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:
{ a(a(x1)) -> b(b(c(x1)))
, c(b(x1)) -> b(c(a(x1)))}
Weak Rules:
{ a(x1) -> x1
, c(x1) -> x1
, c^#(b(x1)) -> c_3(c^#(a(x1)))
, a^#(a(x1)) -> c_1(c^#(x1))
, c^#(x1) -> c_2()}
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(x1)) -> b(b(c(x1)))
, c(b(x1)) -> b(c(a(x1)))}
Weak Rules:
{ a(x1) -> x1
, c(x1) -> x1
, c^#(b(x1)) -> c_3(c^#(a(x1)))
, a^#(a(x1)) -> c_1(c^#(x1))
, c^#(x1) -> c_2()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 9
, b_0(2) -> 2
, b_0(2) -> 9
, a^#_0(2) -> 4
, c^#_0(2) -> 7
, c^#_0(9) -> 8
, c_2_0() -> 7
, c_2_0() -> 8
, c_3_0(8) -> 7
, c_3_0(8) -> 8}
2) { a^#(a(x1)) -> c_1(c^#(x1))
, c^#(b(x1)) -> c_3(c^#(a(x1)))}
The usable rules for this path are the following:
{ a(x1) -> x1
, a(a(x1)) -> b(b(c(x1)))
, c(x1) -> x1
, c(b(x1)) -> b(c(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(b(c(x1)))
, c(x1) -> x1
, c(b(x1)) -> b(c(a(x1)))
, a^#(a(x1)) -> c_1(c^#(x1))
, c^#(b(x1)) -> c_3(c^#(a(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#(a(x1)) -> c_1(c^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(x1)) -> c_1(c^#(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 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(b(x1)) -> c_3(c^#(a(x1)))}
and weakly orienting the rules
{a^#(a(x1)) -> c_1(c^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(b(x1)) -> c_3(c^#(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [5]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(x1) -> x1}
and weakly orienting the rules
{ c^#(b(x1)) -> c_3(c^#(a(x1)))
, a^#(a(x1)) -> c_1(c^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(x1) -> x1}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1(x1) = [1] x1 + [2]
c^#(x1) = [1] x1 + [3]
c_2() = [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}
and weakly orienting the rules
{ c(x1) -> x1
, c^#(b(x1)) -> c_3(c^#(a(x1)))
, a^#(a(x1)) -> c_1(c^#(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 + [8]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1(x1) = [1] x1 + [7]
c^#(x1) = [1] x1 + [1]
c_2() = [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:
{ a(a(x1)) -> b(b(c(x1)))
, c(b(x1)) -> b(c(a(x1)))}
Weak Rules:
{ a(x1) -> x1
, c(x1) -> x1
, c^#(b(x1)) -> c_3(c^#(a(x1)))
, a^#(a(x1)) -> c_1(c^#(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(x1)) -> b(b(c(x1)))
, c(b(x1)) -> b(c(a(x1)))}
Weak Rules:
{ a(x1) -> x1
, c(x1) -> x1
, c^#(b(x1)) -> c_3(c^#(a(x1)))
, a^#(a(x1)) -> c_1(c^#(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 4
, b_0(2) -> 2
, b_0(2) -> 4
, a^#_0(2) -> 1
, c^#_0(2) -> 1
, c^#_0(4) -> 3
, c_3_0(3) -> 1
, c_3_0(3) -> 3}
3) {a^#(a(x1)) -> c_1(c^#(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() = [0]
c_3(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^#(a(x1)) -> c_1(c^#(x1))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#(a(x1)) -> c_1(c^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(x1)) -> c_1(c^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] 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^#(a(x1)) -> c_1(c^#(x1))}
Details:
The given problem does not contain any strict rules
4) { a^#(a(x1)) -> c_1(c^#(x1))
, c^#(x1) -> c_2()}
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() = [0]
c_3(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^#(x1) -> c_2()}
Weak Rules: {a^#(a(x1)) -> c_1(c^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c^#(x1) -> c_2()}
and weakly orienting the rules
{a^#(a(x1)) -> c_1(c^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(x1) -> c_2()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3(x1) = [0] 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^#(x1) -> c_2()
, a^#(a(x1)) -> c_1(c^#(x1))}
Details:
The given problem does not contain any strict rules
5) {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() = [0]
c_3(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: {}
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) = [0] x1 + [0]
b(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [4]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] 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()}
Details:
The given problem does not contain any strict rules