'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(f(X)) -> f(a(b(f(X))))
, f(a(g(X))) -> b(X)
, b(X) -> a(X)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(f(X)) -> c_0(f^#(a(b(f(X)))))
, f^#(a(g(X))) -> c_1(b^#(X))
, b^#(X) -> c_2()}
The usable rules are:
{ f(f(X)) -> f(a(b(f(X))))
, f(a(g(X))) -> b(X)
, b(X) -> a(X)}
The estimated dependency graph contains the following edges:
{f^#(f(X)) -> c_0(f^#(a(b(f(X)))))}
==> {f^#(a(g(X))) -> c_1(b^#(X))}
{f^#(a(g(X))) -> c_1(b^#(X))}
==> {b^#(X) -> c_2()}
We consider the following path(s):
1) { f^#(f(X)) -> c_0(f^#(a(b(f(X)))))
, f^#(a(g(X))) -> c_1(b^#(X))
, b^#(X) -> c_2()}
The usable rules for this path are the following:
{ f(f(X)) -> f(a(b(f(X))))
, f(a(g(X))) -> b(X)
, b(X) -> a(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)) -> f(a(b(f(X))))
, f(a(g(X))) -> b(X)
, b(X) -> a(X)
, f^#(a(g(X))) -> c_1(b^#(X))
, f^#(f(X)) -> c_0(f^#(a(b(f(X)))))
, b^#(X) -> c_2()}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(a(g(X))) -> c_1(b^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(a(g(X))) -> c_1(b^#(X))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [7]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(X) -> c_2()}
and weakly orienting the rules
{f^#(a(g(X))) -> c_1(b^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(X) -> c_2()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(a(g(X))) -> b(X)}
and weakly orienting the rules
{ b^#(X) -> c_2()
, f^#(a(g(X))) -> c_1(b^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(a(g(X))) -> b(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [2]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [2]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(X) -> a(X)}
and weakly orienting the rules
{ f(a(g(X))) -> b(X)
, b^#(X) -> c_2()
, f^#(a(g(X))) -> c_1(b^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(X) -> a(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [4]
b(x1) = [1] x1 + [8]
g(x1) = [1] x1 + [7]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [3]
b^#(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:
{ f(f(X)) -> f(a(b(f(X))))
, f^#(f(X)) -> c_0(f^#(a(b(f(X)))))}
Weak Rules:
{ b(X) -> a(X)
, f(a(g(X))) -> b(X)
, b^#(X) -> c_2()
, f^#(a(g(X))) -> c_1(b^#(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)) -> f(a(b(f(X))))
, f^#(f(X)) -> c_0(f^#(a(b(f(X)))))}
Weak Rules:
{ b(X) -> a(X)
, f(a(g(X))) -> b(X)
, b^#(X) -> c_2()
, f^#(a(g(X))) -> c_1(b^#(X))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 2
, a_0(4) -> 2
, g_0(2) -> 4
, g_0(4) -> 4
, f^#_0(2) -> 5
, f^#_0(4) -> 5
, c_1_0(8) -> 5
, b^#_0(2) -> 8
, b^#_0(4) -> 8
, c_2_0() -> 8}
2) { f^#(f(X)) -> c_0(f^#(a(b(f(X)))))
, f^#(a(g(X))) -> c_1(b^#(X))}
The usable rules for this path are the following:
{ f(f(X)) -> f(a(b(f(X))))
, f(a(g(X))) -> b(X)
, b(X) -> a(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)) -> f(a(b(f(X))))
, f(a(g(X))) -> b(X)
, b(X) -> a(X)
, f^#(f(X)) -> c_0(f^#(a(b(f(X)))))
, f^#(a(g(X))) -> c_1(b^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(a(g(X))) -> c_1(b^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(a(g(X))) -> c_1(b^#(X))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(a(g(X))) -> b(X)}
and weakly orienting the rules
{f^#(a(g(X))) -> c_1(b^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(a(g(X))) -> b(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_2() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(X) -> a(X)}
and weakly orienting the rules
{ f(a(g(X))) -> b(X)
, f^#(a(g(X))) -> c_1(b^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(X) -> a(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [8]
g(x1) = [1] x1 + [12]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [3]
b^#(x1) = [1] x1 + [1]
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:
{ f(f(X)) -> f(a(b(f(X))))
, f^#(f(X)) -> c_0(f^#(a(b(f(X)))))}
Weak Rules:
{ b(X) -> a(X)
, f(a(g(X))) -> b(X)
, f^#(a(g(X))) -> c_1(b^#(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)) -> f(a(b(f(X))))
, f^#(f(X)) -> c_0(f^#(a(b(f(X)))))}
Weak Rules:
{ b(X) -> a(X)
, f(a(g(X))) -> b(X)
, f^#(a(g(X))) -> c_1(b^#(X))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 2
, a_0(4) -> 2
, g_0(2) -> 4
, g_0(4) -> 4
, f^#_0(2) -> 5
, f^#_0(4) -> 5
, c_1_0(8) -> 5
, b^#_0(2) -> 8
, b^#_0(4) -> 8}
3) {f^#(f(X)) -> c_0(f^#(a(b(f(X)))))}
The usable rules for this path are the following:
{ f(f(X)) -> f(a(b(f(X))))
, f(a(g(X))) -> b(X)
, b(X) -> a(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)) -> f(a(b(f(X))))
, f(a(g(X))) -> b(X)
, b(X) -> a(X)
, f^#(f(X)) -> c_0(f^#(a(b(f(X)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f(a(g(X))) -> b(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(a(g(X))) -> b(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [4]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [4]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_2() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(X) -> a(X)}
and weakly orienting the rules
{f(a(g(X))) -> b(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(X) -> a(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [8]
g(x1) = [1] x1 + [12]
f^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [0] 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:
{ f(f(X)) -> f(a(b(f(X))))
, f^#(f(X)) -> c_0(f^#(a(b(f(X)))))}
Weak Rules:
{ b(X) -> a(X)
, f(a(g(X))) -> b(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)) -> f(a(b(f(X))))
, f^#(f(X)) -> c_0(f^#(a(b(f(X)))))}
Weak Rules:
{ b(X) -> a(X)
, f(a(g(X))) -> b(X)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 2
, a_0(4) -> 2
, g_0(2) -> 4
, g_0(4) -> 4
, f^#_0(2) -> 5
, f^#_0(4) -> 5}