'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(f(X)) -> c(f(g(f(X))))
, c(X) -> d(X)
, h(X) -> c(d(X))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(f(X)) -> c_0(c^#(f(g(f(X)))))
, c^#(X) -> c_1()
, h^#(X) -> c_2(c^#(d(X)))}
The usable rules are:
{ f(f(X)) -> c(f(g(f(X))))
, c(X) -> d(X)}
The estimated dependency graph contains the following edges:
{f^#(f(X)) -> c_0(c^#(f(g(f(X)))))}
==> {c^#(X) -> c_1()}
{h^#(X) -> c_2(c^#(d(X)))}
==> {c^#(X) -> c_1()}
We consider the following path(s):
1) { f^#(f(X)) -> c_0(c^#(f(g(f(X)))))
, c^#(X) -> c_1()}
The usable rules for this path are the following:
{ f(f(X)) -> c(f(g(f(X))))
, c(X) -> d(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(f(X)) -> c(f(g(f(X))))
, c(X) -> d(X)
, f^#(f(X)) -> c_0(c^#(f(g(f(X)))))
, c^#(X) -> c_1()}
Details:
We apply the weight gap principle, strictly orienting the rules
{c(X) -> d(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(X) -> d(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
h(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_1() = [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(f(X)) -> c_0(c^#(f(g(f(X)))))}
and weakly orienting the rules
{c(X) -> d(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(f(X)) -> c_0(c^#(f(g(f(X)))))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
h(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1() = [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(X) -> c_1()}
and weakly orienting the rules
{ f^#(f(X)) -> c_0(c^#(f(g(f(X)))))
, c(X) -> d(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(X) -> c_1()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
h(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1() = [0]
h^#(x1) = [0] x1 + [0]
c_2(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(f(X)) -> c(f(g(f(X))))}
Weak Rules:
{ c^#(X) -> c_1()
, f^#(f(X)) -> c_0(c^#(f(g(f(X)))))
, c(X) -> d(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(f(X)) -> c(f(g(f(X))))}
Weak Rules:
{ c^#(X) -> c_1()
, f^#(f(X)) -> c_0(c^#(f(g(f(X)))))
, c(X) -> d(X)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ g_0(3) -> 3
, g_0(4) -> 3
, d_0(3) -> 4
, d_0(4) -> 4
, f^#_0(3) -> 6
, f^#_0(4) -> 6
, c^#_0(3) -> 8
, c^#_0(4) -> 8
, c_1_0() -> 8}
2) {f^#(f(X)) -> c_0(c^#(f(g(f(X)))))}
The usable rules for this path are the following:
{ f(f(X)) -> c(f(g(f(X))))
, c(X) -> d(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(f(X)) -> c(f(g(f(X))))
, c(X) -> d(X)
, f^#(f(X)) -> c_0(c^#(f(g(f(X)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c(X) -> d(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(X) -> d(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
h(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_1() = [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(f(X)) -> c_0(c^#(f(g(f(X)))))}
and weakly orienting the rules
{c(X) -> d(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(f(X)) -> c_0(c^#(f(g(f(X)))))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
h(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1() = [0]
h^#(x1) = [0] x1 + [0]
c_2(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(f(X)) -> c(f(g(f(X))))}
Weak Rules:
{ f^#(f(X)) -> c_0(c^#(f(g(f(X)))))
, c(X) -> d(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(f(X)) -> c(f(g(f(X))))}
Weak Rules:
{ f^#(f(X)) -> c_0(c^#(f(g(f(X)))))
, c(X) -> d(X)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ g_0(3) -> 3
, g_0(4) -> 3
, d_0(3) -> 4
, d_0(4) -> 4
, f^#_0(3) -> 6
, f^#_0(4) -> 6
, c^#_0(3) -> 8
, c^#_0(4) -> 8}
3) {h^#(X) -> c_2(c^#(d(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]
c(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1() = [0]
h^#(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: {h^#(X) -> c_2(c^#(d(X)))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{h^#(X) -> c_2(c^#(d(X)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h^#(X) -> c_2(c^#(d(X)))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
d(x1) = [1] x1 + [0]
h(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1() = [0]
h^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [1]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {h^#(X) -> c_2(c^#(d(X)))}
Details:
The given problem does not contain any strict rules
4) { h^#(X) -> c_2(c^#(d(X)))
, c^#(X) -> c_1()}
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]
c(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1() = [0]
h^#(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^#(X) -> c_1()}
Weak Rules: {h^#(X) -> c_2(c^#(d(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c^#(X) -> c_1()}
and weakly orienting the rules
{h^#(X) -> c_2(c^#(d(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(X) -> c_1()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
d(x1) = [1] x1 + [0]
h(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_1() = [0]
h^#(x1) = [1] x1 + [12]
c_2(x1) = [1] x1 + [1]
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^#(X) -> c_1()
, h^#(X) -> c_2(c^#(d(X)))}
Details:
The given problem does not contain any strict rules