'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
, f(a(x), a(y)) -> a(f(x, y))
, f(b(x), b(y)) -> b(f(x, y))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(a(f(x, y))) -> c_0(f^#(a(b(a(b(a(x))))), a(b(a(b(a(y)))))))
, f^#(a(x), a(y)) -> c_1(a^#(f(x, y)))
, f^#(b(x), b(y)) -> c_2(f^#(x, y))}
The usable rules are:
{ a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
, f(a(x), a(y)) -> a(f(x, y))
, f(b(x), b(y)) -> b(f(x, y))}
The estimated dependency graph contains the following edges:
{a^#(a(f(x, y))) -> c_0(f^#(a(b(a(b(a(x))))), a(b(a(b(a(y)))))))}
==> {f^#(a(x), a(y)) -> c_1(a^#(f(x, y)))}
{f^#(a(x), a(y)) -> c_1(a^#(f(x, y)))}
==> {a^#(a(f(x, y))) ->
c_0(f^#(a(b(a(b(a(x))))), a(b(a(b(a(y)))))))}
{f^#(b(x), b(y)) -> c_2(f^#(x, y))}
==> {f^#(b(x), b(y)) -> c_2(f^#(x, y))}
{f^#(b(x), b(y)) -> c_2(f^#(x, y))}
==> {f^#(a(x), a(y)) -> c_1(a^#(f(x, y)))}
We consider the following path(s):
1) { f^#(b(x), b(y)) -> c_2(f^#(x, y))
, a^#(a(f(x, y))) -> c_0(f^#(a(b(a(b(a(x))))), a(b(a(b(a(y)))))))
, f^#(a(x), a(y)) -> c_1(a^#(f(x, y)))}
The usable rules for this path are the following:
{ a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
, f(a(x), a(y)) -> a(f(x, y))
, f(b(x), b(y)) -> b(f(x, 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:
{ a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
, f(a(x), a(y)) -> a(f(x, y))
, f(b(x), b(y)) -> b(f(x, y))
, f^#(b(x), b(y)) -> c_2(f^#(x, y))
, a^#(a(f(x, y))) -> c_0(f^#(a(b(a(b(a(x))))), a(b(a(b(a(y)))))))
, f^#(a(x), a(y)) -> c_1(a^#(f(x, y)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#(a(f(x, y))) -> c_0(f^#(a(b(a(b(a(x))))), a(b(a(b(a(y)))))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(f(x, y))) -> c_0(f^#(a(b(a(b(a(x))))), a(b(a(b(a(y)))))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
f(x1, x2) = [1] x1 + [1] x2 + [0]
b(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [0]
f^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_1(x1) = [1] x1 + [11]
c_2(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(b(x), b(y)) -> b(f(x, y))
, f^#(b(x), b(y)) -> c_2(f^#(x, y))}
and weakly orienting the rules
{a^#(a(f(x, y))) -> c_0(f^#(a(b(a(b(a(x))))), a(b(a(b(a(y)))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(b(x), b(y)) -> b(f(x, y))
, f^#(b(x), b(y)) -> c_2(f^#(x, y))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
f(x1, x2) = [1] x1 + [1] x2 + [0]
b(x1) = [1] x1 + [2]
a^#(x1) = [1] x1 + [12]
c_0(x1) = [1] x1 + [0]
f^#(x1, x2) = [1] x1 + [1] x2 + [4]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(a(x), a(y)) -> a(f(x, y))}
and weakly orienting the rules
{ f(b(x), b(y)) -> b(f(x, y))
, f^#(b(x), b(y)) -> c_2(f^#(x, y))
, a^#(a(f(x, y))) -> c_0(f^#(a(b(a(b(a(x))))), a(b(a(b(a(y)))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(a(x), a(y)) -> a(f(x, y))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [4]
f(x1, x2) = [1] x1 + [1] x2 + [9]
b(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [11]
c_0(x1) = [1] x1 + [0]
f^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [1] x1 + [1]
c_2(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(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
, f^#(a(x), a(y)) -> c_1(a^#(f(x, y)))}
Weak Rules:
{ f(a(x), a(y)) -> a(f(x, y))
, f(b(x), b(y)) -> b(f(x, y))
, f^#(b(x), b(y)) -> c_2(f^#(x, y))
, a^#(a(f(x, y))) -> c_0(f^#(a(b(a(b(a(x))))), a(b(a(b(a(y)))))))}
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(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
, f^#(a(x), a(y)) -> c_1(a^#(f(x, y)))}
Weak Rules:
{ f(a(x), a(y)) -> a(f(x, y))
, f(b(x), b(y)) -> b(f(x, y))
, f^#(b(x), b(y)) -> c_2(f^#(x, y))
, a^#(a(f(x, y))) -> c_0(f^#(a(b(a(b(a(x))))), a(b(a(b(a(y)))))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ b_0(2) -> 2
, a^#_0(2) -> 1
, f^#_0(2, 2) -> 1
, c_2_0(1) -> 1}
2) {f^#(b(x), b(y)) -> c_2(f^#(x, y))}
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]
f(x1, x2) = [0] x1 + [0] x2 + [0]
b(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(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: {f^#(b(x), b(y)) -> c_2(f^#(x, y))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(b(x), b(y)) -> c_2(f^#(x, y))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(b(x), b(y)) -> c_2(f^#(x, y))}
Details:
Interpretation Functions:
a(x1) = [0] x1 + [0]
f(x1, x2) = [0] x1 + [0] x2 + [0]
b(x1) = [1] x1 + [8]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [11]
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^#(b(x), b(y)) -> c_2(f^#(x, y))}
Details:
The given problem does not contain any strict rules