'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(s(x1)) -> s(a(x1))
, b(a(b(s(x1)))) -> a(b(s(a(x1))))
, b(a(b(b(x1)))) -> c(s(x1))
, c(s(x1)) -> a(b(a(b(x1))))
, a(b(a(a(x1)))) -> b(a(b(a(x1))))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(s(x1)) -> c_0(a^#(x1))
, b^#(a(b(s(x1)))) -> c_1(a^#(b(s(a(x1)))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))
, c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))
, a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))}
The usable rules are:
{ a(s(x1)) -> s(a(x1))
, b(a(b(s(x1)))) -> a(b(s(a(x1))))
, b(a(b(b(x1)))) -> c(s(x1))
, a(b(a(a(x1)))) -> b(a(b(a(x1))))
, c(s(x1)) -> a(b(a(b(x1))))}
The estimated dependency graph contains the following edges:
{a^#(s(x1)) -> c_0(a^#(x1))}
==> {a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))}
{a^#(s(x1)) -> c_0(a^#(x1))}
==> {a^#(s(x1)) -> c_0(a^#(x1))}
{b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))}
==> {c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))}
{c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))}
==> {a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))}
{c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))}
==> {a^#(s(x1)) -> c_0(a^#(x1))}
{a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))}
==> {b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))}
{a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))}
==> {b^#(a(b(s(x1)))) -> c_1(a^#(b(s(a(x1)))))}
We consider the following path(s):
1) { a^#(s(x1)) -> c_0(a^#(x1))
, c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))
, a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))
, b^#(a(b(s(x1)))) -> c_1(a^#(b(s(a(x1)))))}
The usable rules for this path are the following:
{ a(s(x1)) -> s(a(x1))
, b(a(b(s(x1)))) -> a(b(s(a(x1))))
, b(a(b(b(x1)))) -> c(s(x1))
, a(b(a(a(x1)))) -> b(a(b(a(x1))))
, c(s(x1)) -> a(b(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:
{ a(s(x1)) -> s(a(x1))
, b(a(b(s(x1)))) -> a(b(s(a(x1))))
, b(a(b(b(x1)))) -> c(s(x1))
, a(b(a(a(x1)))) -> b(a(b(a(x1))))
, c(s(x1)) -> a(b(a(b(x1))))
, a^#(s(x1)) -> c_0(a^#(x1))
, c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))
, a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))
, b^#(a(b(s(x1)))) -> c_1(a^#(b(s(a(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(b(a(a(x1)))) -> b(a(b(a(x1))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(b(a(a(x1)))) -> b(a(b(a(x1))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(a(b(s(x1)))) -> c_1(a^#(b(s(a(x1)))))}
and weakly orienting the rules
{ a(b(a(a(x1)))) -> b(a(b(a(x1))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(a(b(s(x1)))) -> c_1(a^#(b(s(a(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [6]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))}
and weakly orienting the rules
{ b^#(a(b(s(x1)))) -> c_1(a^#(b(s(a(x1)))))
, a(b(a(a(x1)))) -> b(a(b(a(x1))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [7]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [4]
c^#(x1) = [1] x1 + [3]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [8]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(a(b(b(x1)))) -> c(s(x1))}
and weakly orienting the rules
{ c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))
, b^#(a(b(s(x1)))) -> c_1(a^#(b(s(a(x1)))))
, a(b(a(a(x1)))) -> b(a(b(a(x1))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(a(b(b(x1)))) -> c(s(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [14]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(s(x1)) -> c_0(a^#(x1))}
and weakly orienting the rules
{ b(a(b(b(x1)))) -> c(s(x1))
, c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))
, b^#(a(b(s(x1)))) -> c_1(a^#(b(s(a(x1)))))
, a(b(a(a(x1)))) -> b(a(b(a(x1))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(s(x1)) -> c_0(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [4]
s(x1) = [1] x1 + [2]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [4]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [8]
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(s(x1)) -> s(a(x1))
, b(a(b(s(x1)))) -> a(b(s(a(x1))))
, c(s(x1)) -> a(b(a(b(x1))))
, a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))}
Weak Rules:
{ a^#(s(x1)) -> c_0(a^#(x1))
, b(a(b(b(x1)))) -> c(s(x1))
, c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))
, b^#(a(b(s(x1)))) -> c_1(a^#(b(s(a(x1)))))
, a(b(a(a(x1)))) -> b(a(b(a(x1))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(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:
{ a(s(x1)) -> s(a(x1))
, b(a(b(s(x1)))) -> a(b(s(a(x1))))
, c(s(x1)) -> a(b(a(b(x1))))
, a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))}
Weak Rules:
{ a^#(s(x1)) -> c_0(a^#(x1))
, b(a(b(b(x1)))) -> c(s(x1))
, c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))
, b^#(a(b(s(x1)))) -> c_1(a^#(b(s(a(x1)))))
, a(b(a(a(x1)))) -> b(a(b(a(x1))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a_0(6) -> 5
, a_1(2) -> 9
, a_1(7) -> 4
, a_1(7) -> 11
, a_1(13) -> 12
, s_0(2) -> 2
, s_1(9) -> 8
, s_1(9) -> 9
, b_0(2) -> 6
, b_0(5) -> 4
, b_1(2) -> 13
, b_1(8) -> 7
, b_1(12) -> 11
, a^#_0(2) -> 1
, a^#_0(4) -> 3
, a^#_1(11) -> 10
, c_0_0(1) -> 1
, b^#_0(2) -> 1
, c^#_0(2) -> 1
, c_3_0(3) -> 1
, c_3_1(10) -> 1}
2) { a^#(s(x1)) -> c_0(a^#(x1))
, c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))
, a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))}
The usable rules for this path are the following:
{ a(s(x1)) -> s(a(x1))
, b(a(b(s(x1)))) -> a(b(s(a(x1))))
, b(a(b(b(x1)))) -> c(s(x1))
, a(b(a(a(x1)))) -> b(a(b(a(x1))))
, c(s(x1)) -> a(b(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:
{ a(s(x1)) -> s(a(x1))
, b(a(b(s(x1)))) -> a(b(s(a(x1))))
, b(a(b(b(x1)))) -> c(s(x1))
, a(b(a(a(x1)))) -> b(a(b(a(x1))))
, c(s(x1)) -> a(b(a(b(x1))))
, a^#(s(x1)) -> c_0(a^#(x1))
, c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))
, a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(b(a(a(x1)))) -> b(a(b(a(x1))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(b(a(a(x1)))) -> b(a(b(a(x1))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [3]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [4]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))}
and weakly orienting the rules
{ a(b(a(a(x1)))) -> b(a(b(a(x1))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(a(b(b(x1)))) -> c(s(x1))}
and weakly orienting the rules
{ a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))
, a(b(a(a(x1)))) -> b(a(b(a(x1))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(a(b(b(x1)))) -> c(s(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [7]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(s(x1)) -> c_0(a^#(x1))}
and weakly orienting the rules
{ b(a(b(b(x1)))) -> c(s(x1))
, a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))
, a(b(a(a(x1)))) -> b(a(b(a(x1))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(s(x1)) -> c_0(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [9]
s(x1) = [1] x1 + [2]
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [3]
a^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [6]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
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(s(x1)) -> s(a(x1))
, b(a(b(s(x1)))) -> a(b(s(a(x1))))
, c(s(x1)) -> a(b(a(b(x1))))
, c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))}
Weak Rules:
{ a^#(s(x1)) -> c_0(a^#(x1))
, b(a(b(b(x1)))) -> c(s(x1))
, a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))
, a(b(a(a(x1)))) -> b(a(b(a(x1))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(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:
{ a(s(x1)) -> s(a(x1))
, b(a(b(s(x1)))) -> a(b(s(a(x1))))
, c(s(x1)) -> a(b(a(b(x1))))
, c^#(s(x1)) -> c_3(a^#(b(a(b(x1)))))}
Weak Rules:
{ a^#(s(x1)) -> c_0(a^#(x1))
, b(a(b(b(x1)))) -> c(s(x1))
, a^#(b(a(a(x1)))) -> c_4(b^#(a(b(a(x1)))))
, a(b(a(a(x1)))) -> b(a(b(a(x1))))
, b^#(a(b(b(x1)))) -> c_2(c^#(s(x1)))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a_1(2) -> 17
, a_1(14) -> 13
, a_1(15) -> 12
, s_0(2) -> 2
, s_1(17) -> 16
, s_1(17) -> 17
, b_1(2) -> 14
, b_1(13) -> 12
, b_1(16) -> 15
, a^#_0(2) -> 5
, a^#_1(12) -> 11
, c_0_0(5) -> 5
, b^#_0(2) -> 7
, c^#_0(2) -> 10
, c_3_1(11) -> 10}