'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(b(x1)) -> d(x1)
, b(a(x1)) -> a(b(x1))
, d(c(x1)) -> f(a(b(b(c(x1)))))
, d(f(x1)) -> f(a(b(x1)))
, a(f(x1)) -> a(x1)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(b(x1)) -> c_0(d^#(x1))
, b^#(a(x1)) -> c_1(a^#(b(x1)))
, d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))
, d^#(f(x1)) -> c_3(a^#(b(x1)))
, a^#(f(x1)) -> c_4(a^#(x1))}
The usable rules are:
{ b(a(x1)) -> a(b(x1))
, a(b(x1)) -> d(x1)
, a(f(x1)) -> a(x1)
, d(c(x1)) -> f(a(b(b(c(x1)))))
, d(f(x1)) -> f(a(b(x1)))}
The estimated dependency graph contains the following edges:
{a^#(b(x1)) -> c_0(d^#(x1))}
==> {d^#(f(x1)) -> c_3(a^#(b(x1)))}
{a^#(b(x1)) -> c_0(d^#(x1))}
==> {d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
{b^#(a(x1)) -> c_1(a^#(b(x1)))}
==> {a^#(f(x1)) -> c_4(a^#(x1))}
{b^#(a(x1)) -> c_1(a^#(b(x1)))}
==> {a^#(b(x1)) -> c_0(d^#(x1))}
{d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
==> {a^#(b(x1)) -> c_0(d^#(x1))}
{d^#(f(x1)) -> c_3(a^#(b(x1)))}
==> {a^#(f(x1)) -> c_4(a^#(x1))}
{d^#(f(x1)) -> c_3(a^#(b(x1)))}
==> {a^#(b(x1)) -> c_0(d^#(x1))}
{a^#(f(x1)) -> c_4(a^#(x1))}
==> {a^#(f(x1)) -> c_4(a^#(x1))}
{a^#(f(x1)) -> c_4(a^#(x1))}
==> {a^#(b(x1)) -> c_0(d^#(x1))}
We consider the following path(s):
1) { b^#(a(x1)) -> c_1(a^#(b(x1)))
, a^#(b(x1)) -> c_0(d^#(x1))
, a^#(f(x1)) -> c_4(a^#(x1))
, d^#(f(x1)) -> c_3(a^#(b(x1)))
, d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
The usable rules for this path are the following:
{ b(a(x1)) -> a(b(x1))
, a(b(x1)) -> d(x1)
, a(f(x1)) -> a(x1)
, d(c(x1)) -> f(a(b(b(c(x1)))))
, d(f(x1)) -> f(a(b(x1)))}
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:
{ b(a(x1)) -> a(b(x1))
, a(b(x1)) -> d(x1)
, a(f(x1)) -> a(x1)
, d(c(x1)) -> f(a(b(b(c(x1)))))
, d(f(x1)) -> f(a(b(x1)))
, b^#(a(x1)) -> c_1(a^#(b(x1)))
, a^#(b(x1)) -> c_0(d^#(x1))
, a^#(f(x1)) -> c_4(a^#(x1))
, d^#(f(x1)) -> c_3(a^#(b(x1)))
, d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ d^#(f(x1)) -> c_3(a^#(b(x1)))
, d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ d^#(f(x1)) -> c_3(a^#(b(x1)))
, d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(b(x1)) -> d(x1)}
and weakly orienting the rules
{ d^#(f(x1)) -> c_3(a^#(b(x1)))
, d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(b(x1)) -> d(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [4]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [2]
c_2(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [12]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(a(x1)) -> c_1(a^#(b(x1)))}
and weakly orienting the rules
{ a(b(x1)) -> d(x1)
, d^#(f(x1)) -> c_3(a^#(b(x1)))
, d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(a(x1)) -> c_1(a^#(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [3]
b^#(x1) = [1] x1 + [5]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [4]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(f(x1)) -> a(x1)
, a^#(f(x1)) -> c_4(a^#(x1))}
and weakly orienting the rules
{ b^#(a(x1)) -> c_1(a^#(b(x1)))
, a(b(x1)) -> d(x1)
, d^#(f(x1)) -> c_3(a^#(b(x1)))
, d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(f(x1)) -> a(x1)
, a^#(f(x1)) -> c_4(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [12]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [3]
b^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [4]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(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:
{ b(a(x1)) -> a(b(x1))
, d(c(x1)) -> f(a(b(b(c(x1)))))
, d(f(x1)) -> f(a(b(x1)))
, a^#(b(x1)) -> c_0(d^#(x1))}
Weak Rules:
{ a(f(x1)) -> a(x1)
, a^#(f(x1)) -> c_4(a^#(x1))
, b^#(a(x1)) -> c_1(a^#(b(x1)))
, a(b(x1)) -> d(x1)
, d^#(f(x1)) -> c_3(a^#(b(x1)))
, d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
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:
{ b(a(x1)) -> a(b(x1))
, d(c(x1)) -> f(a(b(b(c(x1)))))
, d(f(x1)) -> f(a(b(x1)))
, a^#(b(x1)) -> c_0(d^#(x1))}
Weak Rules:
{ a(f(x1)) -> a(x1)
, a^#(f(x1)) -> c_4(a^#(x1))
, b^#(a(x1)) -> c_1(a^#(b(x1)))
, a(b(x1)) -> d(x1)
, d^#(f(x1)) -> c_3(a^#(b(x1)))
, d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ b_0(2) -> 4
, b_0(4) -> 6
, b_1(2) -> 10
, b_1(13) -> 12
, b_1(14) -> 13
, c_0(2) -> 2
, c_1(2) -> 14
, f_0(2) -> 2
, a^#_0(2) -> 1
, a^#_0(4) -> 3
, a^#_0(6) -> 5
, a^#_1(10) -> 9
, a^#_1(12) -> 11
, c_0_1(7) -> 3
, c_0_1(8) -> 5
, c_0_2(15) -> 9
, c_0_2(16) -> 11
, d^#_0(2) -> 1
, d^#_1(2) -> 7
, d^#_1(4) -> 8
, d^#_2(2) -> 15
, d^#_2(13) -> 16
, b^#_0(2) -> 1
, c_2_0(5) -> 1
, c_2_1(11) -> 1
, c_2_1(11) -> 7
, c_2_1(11) -> 15
, c_3_0(3) -> 1
, c_3_1(9) -> 1
, c_3_1(9) -> 7
, c_3_1(9) -> 15
, c_4_0(1) -> 1}
2) {b^#(a(x1)) -> c_1(a^#(b(x1)))}
The usable rules for this path are the following:
{ b(a(x1)) -> a(b(x1))
, a(b(x1)) -> d(x1)
, a(f(x1)) -> a(x1)
, d(c(x1)) -> f(a(b(b(c(x1)))))
, d(f(x1)) -> f(a(b(x1)))}
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:
{ b(a(x1)) -> a(b(x1))
, a(b(x1)) -> d(x1)
, a(f(x1)) -> a(x1)
, d(c(x1)) -> f(a(b(b(c(x1)))))
, d(f(x1)) -> f(a(b(x1)))
, b^#(a(x1)) -> c_1(a^#(b(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b^#(a(x1)) -> c_1(a^#(b(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(a(x1)) -> c_1(a^#(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [5]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(b(x1)) -> d(x1)}
and weakly orienting the rules
{b^#(a(x1)) -> c_1(a^#(b(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(b(x1)) -> d(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [3]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(f(x1)) -> a(x1)}
and weakly orienting the rules
{ a(b(x1)) -> d(x1)
, b^#(a(x1)) -> c_1(a^#(b(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(f(x1)) -> a(x1)}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(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:
{ b(a(x1)) -> a(b(x1))
, d(c(x1)) -> f(a(b(b(c(x1)))))
, d(f(x1)) -> f(a(b(x1)))}
Weak Rules:
{ a(f(x1)) -> a(x1)
, a(b(x1)) -> d(x1)
, b^#(a(x1)) -> c_1(a^#(b(x1)))}
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:
{ b(a(x1)) -> a(b(x1))
, d(c(x1)) -> f(a(b(b(c(x1)))))
, d(f(x1)) -> f(a(b(x1)))}
Weak Rules:
{ a(f(x1)) -> a(x1)
, a(b(x1)) -> d(x1)
, b^#(a(x1)) -> c_1(a^#(b(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(4) -> 4
, c_0(5) -> 4
, f_0(4) -> 5
, f_0(5) -> 5
, a^#_0(4) -> 6
, a^#_0(5) -> 6
, b^#_0(4) -> 9
, b^#_0(5) -> 9}