'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(a()) -> g(h(a()))
, h(g(x)) -> g(h(f(x)))
, k(x, h(x), a()) -> h(x)
, k(f(x), y, x) -> f(x)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(a()) -> c_0(h^#(a()))
, h^#(g(x)) -> c_1(h^#(f(x)))
, k^#(x, h(x), a()) -> c_2(h^#(x))
, k^#(f(x), y, x) -> c_3(f^#(x))}
The usable rules are:
{ f(a()) -> g(h(a()))
, h(g(x)) -> g(h(f(x)))}
The estimated dependency graph contains the following edges:
{h^#(g(x)) -> c_1(h^#(f(x)))}
==> {h^#(g(x)) -> c_1(h^#(f(x)))}
{k^#(x, h(x), a()) -> c_2(h^#(x))}
==> {h^#(g(x)) -> c_1(h^#(f(x)))}
{k^#(f(x), y, x) -> c_3(f^#(x))}
==> {f^#(a()) -> c_0(h^#(a()))}
We consider the following path(s):
1) { k^#(x, h(x), a()) -> c_2(h^#(x))
, h^#(g(x)) -> c_1(h^#(f(x)))}
The usable rules for this path are the following:
{ f(a()) -> g(h(a()))
, h(g(x)) -> g(h(f(x)))}
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:
{ f(a()) -> g(h(a()))
, h(g(x)) -> g(h(f(x)))
, k^#(x, h(x), a()) -> c_2(h^#(x))
, h^#(g(x)) -> c_1(h^#(f(x)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{k^#(x, h(x), a()) -> c_2(h^#(x))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{k^#(x, h(x), a()) -> c_2(h^#(x))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [1]
a() = [0]
g(x1) = [1] x1 + [1]
h(x1) = [1] x1 + [0]
k(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
k^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{h^#(g(x)) -> c_1(h^#(f(x)))}
and weakly orienting the rules
{k^#(x, h(x), a()) -> c_2(h^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h^#(g(x)) -> c_1(h^#(f(x)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [1]
a() = [0]
g(x1) = [1] x1 + [9]
h(x1) = [1] x1 + [0]
k(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [4]
k^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [2]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] 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:
{ f(a()) -> g(h(a()))
, h(g(x)) -> g(h(f(x)))}
Weak Rules:
{ h^#(g(x)) -> c_1(h^#(f(x)))
, k^#(x, h(x), a()) -> c_2(h^#(x))}
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:
{ f(a()) -> g(h(a()))
, h(g(x)) -> g(h(f(x)))}
Weak Rules:
{ h^#(g(x)) -> c_1(h^#(f(x)))
, k^#(x, h(x), a()) -> c_2(h^#(x))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_0(2) -> 12
, f_0(3) -> 12
, f_1(2) -> 16
, f_1(3) -> 16
, f_1(13) -> 18
, a_0() -> 2
, a_1() -> 14
, g_0(2) -> 3
, g_0(3) -> 3
, g_1(13) -> 12
, g_1(13) -> 16
, h_1(14) -> 13
, h^#_0(2) -> 8
, h^#_0(3) -> 8
, h^#_0(12) -> 11
, h^#_1(16) -> 15
, h^#_1(18) -> 17
, c_1_0(11) -> 8
, c_1_1(15) -> 8
, c_1_1(17) -> 11
, c_1_1(17) -> 15
, k^#_0(2, 2, 2) -> 10
, k^#_0(2, 2, 3) -> 10
, k^#_0(2, 3, 2) -> 10
, k^#_0(2, 3, 3) -> 10
, k^#_0(3, 2, 2) -> 10
, k^#_0(3, 2, 3) -> 10
, k^#_0(3, 3, 2) -> 10
, k^#_0(3, 3, 3) -> 10}
2) { k^#(f(x), y, x) -> c_3(f^#(x))
, f^#(a()) -> c_0(h^#(a()))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
k(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
k^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [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: {f^#(a()) -> c_0(h^#(a()))}
Weak Rules: {k^#(f(x), y, x) -> c_3(f^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(a()) -> c_0(h^#(a()))}
and weakly orienting the rules
{k^#(f(x), y, x) -> c_3(f^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(a()) -> c_0(h^#(a()))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
k(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
k^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_2(x1) = [0] x1 + [0]
c_3(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:
{ f^#(a()) -> c_0(h^#(a()))
, k^#(f(x), y, x) -> c_3(f^#(x))}
Details:
The given problem does not contain any strict rules
3) {k^#(x, h(x), a()) -> c_2(h^#(x))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
k(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
k^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [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: {k^#(x, h(x), a()) -> c_2(h^#(x))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{k^#(x, h(x), a()) -> c_2(h^#(x))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{k^#(x, h(x), a()) -> c_2(h^#(x))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
k(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
k^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_2(x1) = [1] x1 + [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: {k^#(x, h(x), a()) -> c_2(h^#(x))}
Details:
The given problem does not contain any strict rules
4) {k^#(f(x), y, x) -> c_3(f^#(x))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
k(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
k^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [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: {k^#(f(x), y, x) -> c_3(f^#(x))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{k^#(f(x), y, x) -> c_3(f^#(x))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{k^#(f(x), y, x) -> c_3(f^#(x))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
k(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
k^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_2(x1) = [0] x1 + [0]
c_3(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: {k^#(f(x), y, x) -> c_3(f^#(x))}
Details:
The given problem does not contain any strict rules