'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ c(c(c(y))) -> c(c(a(y, 0())))
, c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c(y) -> y}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ c^#(c(c(y))) -> c_0(c^#(c(a(y, 0()))))
, c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))
, c^#(y) -> c_2()}
The usable rules are:
{ c(c(c(y))) -> c(c(a(y, 0())))
, c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c(y) -> y}
The estimated dependency graph contains the following edges:
{c^#(c(c(y))) -> c_0(c^#(c(a(y, 0()))))}
==> {c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))}
{c^#(c(c(y))) -> c_0(c^#(c(a(y, 0()))))}
==> {c^#(y) -> c_2()}
{c^#(c(c(y))) -> c_0(c^#(c(a(y, 0()))))}
==> {c^#(c(c(y))) -> c_0(c^#(c(a(y, 0()))))}
{c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))}
==> {c^#(y) -> c_2()}
{c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))}
==> {c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))}
{c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))}
==> {c^#(c(c(y))) -> c_0(c^#(c(a(y, 0()))))}
We consider the following path(s):
1) { c^#(c(c(y))) -> c_0(c^#(c(a(y, 0()))))
, c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))
, c^#(y) -> c_2()}
The usable rules for this path are the following:
{ c(c(c(y))) -> c(c(a(y, 0())))
, c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c(y) -> y}
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:
{ c(c(c(y))) -> c(c(a(y, 0())))
, c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c(y) -> y
, c^#(c(c(y))) -> c_0(c^#(c(a(y, 0()))))
, c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))
, c^#(y) -> c_2()}
Details:
We apply the weight gap principle, strictly orienting the rules
{c^#(y) -> c_2()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(y) -> c_2()}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
c^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))}
and weakly orienting the rules
{c^#(y) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
c^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [9]
c_2() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(y) -> y}
and weakly orienting the rules
{ c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))
, c^#(y) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(y) -> y}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [1]
a(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [0]
c^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [0]
c_2() = [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:
{ c(c(c(y))) -> c(c(a(y, 0())))
, c^#(c(c(y))) -> c_0(c^#(c(a(y, 0()))))}
Weak Rules:
{ c(y) -> y
, c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))
, c^#(y) -> 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:
{ c(c(c(y))) -> c(c(a(y, 0())))
, c^#(c(c(y))) -> c_0(c^#(c(a(y, 0()))))}
Weak Rules:
{ c(y) -> y
, c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))
, c^#(y) -> c_2()}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ c_0(3) -> 6
, c_0(3) -> 7
, c_0(7) -> 6
, c_1(10) -> 9
, a_0(2, 2) -> 2
, a_0(2, 3) -> 2
, a_0(3, 2) -> 2
, a_0(3, 3) -> 2
, a_1(3, 11) -> 9
, a_1(3, 11) -> 10
, 0_0() -> 3
, 0_0() -> 6
, 0_0() -> 7
, 0_1() -> 11
, c^#_0(2) -> 4
, c^#_0(3) -> 4
, c^#_0(6) -> 5
, c^#_1(9) -> 8
, c_0_1(8) -> 5
, c_1_0(5) -> 4
, c_2_0() -> 4
, c_2_0() -> 5
, c_2_1() -> 8}
2) { c^#(c(c(y))) -> c_0(c^#(c(a(y, 0()))))
, c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))}
The usable rules for this path are the following:
{ c(c(c(y))) -> c(c(a(y, 0())))
, c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c(y) -> y}
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:
{ c(c(c(y))) -> c(c(a(y, 0())))
, c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c(y) -> y
, c^#(c(c(y))) -> c_0(c^#(c(a(y, 0()))))
, c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [0]
a(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
c^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(y) -> y}
and weakly orienting the rules
{ c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(y) -> y}
Details:
Interpretation Functions:
c(x1) = [1] x1 + [1]
a(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [0]
c^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [2]
c_2() = [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:
{ c(c(c(y))) -> c(c(a(y, 0())))
, c^#(c(c(y))) -> c_0(c^#(c(a(y, 0()))))}
Weak Rules:
{ c(y) -> y
, c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c^#(a(a(0(), x), y)) -> c_1(c^#(c(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:
{ c(c(c(y))) -> c(c(a(y, 0())))
, c^#(c(c(y))) -> c_0(c^#(c(a(y, 0()))))}
Weak Rules:
{ c(y) -> y
, c(a(a(0(), x), y)) -> a(c(c(c(0()))), y)
, c^#(a(a(0(), x), y)) -> c_1(c^#(c(c(0()))))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ c_0(3) -> 6
, c_0(3) -> 7
, c_0(7) -> 6
, c_1(10) -> 9
, a_0(2, 2) -> 2
, a_0(2, 3) -> 2
, a_0(3, 2) -> 2
, a_0(3, 3) -> 2
, a_1(3, 11) -> 9
, a_1(3, 11) -> 10
, 0_0() -> 3
, 0_0() -> 6
, 0_0() -> 7
, 0_1() -> 11
, c^#_0(2) -> 4
, c^#_0(3) -> 4
, c^#_0(6) -> 5
, c^#_1(9) -> 8
, c_0_1(8) -> 5
, c_1_0(5) -> 4}