'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ b(x, y) -> c(a(c(y), a(0(), x)))
, a(y, x) -> y
, a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
, a^#(y, x) -> c_1()
, a^#(y, c(b(a(0(), x), 0()))) ->
c_2(b^#(a(c(b(0(), y)), x), 0()))}
The usable rules are:
{ b(x, y) -> c(a(c(y), a(0(), x)))
, a(y, x) -> y
, a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())}
The estimated dependency graph contains the following edges:
{b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
==> {a^#(y, c(b(a(0(), x), 0()))) ->
c_2(b^#(a(c(b(0(), y)), x), 0()))}
{b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
==> {a^#(y, x) -> c_1()}
{a^#(y, c(b(a(0(), x), 0()))) -> c_2(b^#(a(c(b(0(), y)), x), 0()))}
==> {b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
We consider the following path(s):
1) { b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
, a^#(y, c(b(a(0(), x), 0()))) ->
c_2(b^#(a(c(b(0(), y)), x), 0()))}
The usable rules for this path are the following:
{ b(x, y) -> c(a(c(y), a(0(), x)))
, a(y, x) -> y
, a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())}
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(x, y) -> c(a(c(y), a(0(), x)))
, a(y, x) -> y
, a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())
, b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
, a^#(y, c(b(a(0(), x), 0()))) ->
c_2(b^#(a(c(b(0(), y)), x), 0()))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
Details:
Interpretation Functions:
b(x1, x2) = [1] x1 + [1] x2 + [0]
c(x1) = [1] x1 + [0]
a(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
b^#(x1, x2) = [1] x1 + [1] x2 + [8]
c_0(x1) = [1] x1 + [1]
a^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(x, y) -> c(a(c(y), a(0(), x)))}
and weakly orienting the rules
{b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(x, y) -> c(a(c(y), a(0(), x)))}
Details:
Interpretation Functions:
b(x1, x2) = [1] x1 + [1] x2 + [8]
c(x1) = [1] x1 + [0]
a(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
b^#(x1, x2) = [1] x1 + [1] x2 + [8]
c_0(x1) = [1] x1 + [0]
a^#(x1, x2) = [1] x1 + [1] x2 + [8]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(y, x) -> y}
and weakly orienting the rules
{ b(x, y) -> c(a(c(y), a(0(), x)))
, b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(y, x) -> y}
Details:
Interpretation Functions:
b(x1, x2) = [1] x1 + [1] x2 + [4]
c(x1) = [1] x1 + [0]
a(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [1]
b^#(x1, x2) = [1] x1 + [1] x2 + [8]
c_0(x1) = [1] x1 + [0]
a^#(x1, x2) = [1] x1 + [1] x2 + [6]
c_1() = [0]
c_2(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(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())
, a^#(y, c(b(a(0(), x), 0()))) ->
c_2(b^#(a(c(b(0(), y)), x), 0()))}
Weak Rules:
{ a(y, x) -> y
, b(x, y) -> c(a(c(y), a(0(), x)))
, b^#(x, y) -> c_0(a^#(c(y), a(0(), 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:
{ a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())
, a^#(y, c(b(a(0(), x), 0()))) ->
c_2(b^#(a(c(b(0(), y)), x), 0()))}
Weak Rules:
{ a(y, x) -> y
, b(x, y) -> c(a(c(y), a(0(), x)))
, b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 2
, c_0(4) -> 2
, a_0(4, 2) -> 9
, a_0(4, 4) -> 9
, 0_0() -> 4
, 0_0() -> 9
, b^#_0(2, 2) -> 5
, b^#_0(2, 4) -> 5
, b^#_0(4, 2) -> 5
, b^#_0(4, 4) -> 5
, c_0_0(8) -> 5
, a^#_0(2, 2) -> 7
, a^#_0(2, 4) -> 7
, a^#_0(2, 9) -> 8
, a^#_0(4, 2) -> 7
, a^#_0(4, 4) -> 7}
2) { b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
, a^#(y, c(b(a(0(), x), 0()))) -> c_2(b^#(a(c(b(0(), y)), x), 0()))
, a^#(y, x) -> c_1()}
The usable rules for this path are the following:
{ b(x, y) -> c(a(c(y), a(0(), x)))
, a(y, x) -> y
, a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())}
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(x, y) -> c(a(c(y), a(0(), x)))
, a(y, x) -> y
, a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())
, b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
, a^#(y, c(b(a(0(), x), 0()))) -> c_2(b^#(a(c(b(0(), y)), x), 0()))
, a^#(y, x) -> c_1()}
Details:
We apply the weight gap principle, strictly orienting the rules
{b(x, y) -> c(a(c(y), a(0(), x)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(x, y) -> c(a(c(y), a(0(), x)))}
Details:
Interpretation Functions:
b(x1, x2) = [1] x1 + [1] x2 + [1]
c(x1) = [1] x1 + [0]
a(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
b^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_0(x1) = [1] x1 + [1]
a^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
and weakly orienting the rules
{b(x, y) -> c(a(c(y), a(0(), x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
Details:
Interpretation Functions:
b(x1, x2) = [1] x1 + [1] x2 + [0]
c(x1) = [1] x1 + [0]
a(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
b^#(x1, x2) = [1] x1 + [1] x2 + [8]
c_0(x1) = [1] x1 + [1]
a^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(y, x) -> c_1()}
and weakly orienting the rules
{ b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
, b(x, y) -> c(a(c(y), a(0(), x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(y, x) -> c_1()}
Details:
Interpretation Functions:
b(x1, x2) = [1] x1 + [1] x2 + [0]
c(x1) = [1] x1 + [0]
a(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
b^#(x1, x2) = [1] x1 + [1] x2 + [8]
c_0(x1) = [1] x1 + [1]
a^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(y, x) -> y}
and weakly orienting the rules
{ a^#(y, x) -> c_1()
, b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
, b(x, y) -> c(a(c(y), a(0(), x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(y, x) -> y}
Details:
Interpretation Functions:
b(x1, x2) = [1] x1 + [1] x2 + [11]
c(x1) = [1] x1 + [0]
a(x1, x2) = [1] x1 + [1] x2 + [3]
0() = [1]
b^#(x1, x2) = [1] x1 + [1] x2 + [8]
c_0(x1) = [1] x1 + [2]
a^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2(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(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())
, a^#(y, c(b(a(0(), x), 0()))) ->
c_2(b^#(a(c(b(0(), y)), x), 0()))}
Weak Rules:
{ a(y, x) -> y
, a^#(y, x) -> c_1()
, b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
, b(x, y) -> c(a(c(y), a(0(), 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:
{ a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())
, a^#(y, c(b(a(0(), x), 0()))) ->
c_2(b^#(a(c(b(0(), y)), x), 0()))}
Weak Rules:
{ a(y, x) -> y
, a^#(y, x) -> c_1()
, b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
, b(x, y) -> c(a(c(y), a(0(), x)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 2
, c_0(4) -> 2
, a_0(4, 2) -> 9
, a_0(4, 4) -> 9
, 0_0() -> 4
, 0_0() -> 9
, b^#_0(2, 2) -> 5
, b^#_0(2, 4) -> 5
, b^#_0(4, 2) -> 5
, b^#_0(4, 4) -> 5
, c_0_0(8) -> 5
, a^#_0(2, 2) -> 7
, a^#_0(2, 4) -> 7
, a^#_0(2, 9) -> 8
, a^#_0(4, 2) -> 7
, a^#_0(4, 4) -> 7
, c_1_0() -> 7
, c_1_0() -> 8}