'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 0(*(x1)) -> *(1(x1))
, 1(*(x1)) -> 0(#(x1))
, #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, #($(x1)) -> *($(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ 0^#(*(x1)) -> c_0(1^#(x1))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))
, #^#(0(x1)) -> c_2(0^#(#(x1)))
, #^#(1(x1)) -> c_3(1^#(#(x1)))
, #^#($(x1)) -> c_4()
, #^#(#(x1)) -> c_5(#^#(x1))
, #^#(*(x1)) -> c_6()}
The usable rules are:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, #($(x1)) -> *($(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 0(*(x1)) -> *(1(x1))
, 1(*(x1)) -> 0(#(x1))}
The estimated dependency graph contains the following edges:
{0^#(*(x1)) -> c_0(1^#(x1))}
==> {1^#(*(x1)) -> c_1(0^#(#(x1)))}
{1^#(*(x1)) -> c_1(0^#(#(x1)))}
==> {0^#(*(x1)) -> c_0(1^#(x1))}
{#^#(0(x1)) -> c_2(0^#(#(x1)))}
==> {0^#(*(x1)) -> c_0(1^#(x1))}
{#^#(1(x1)) -> c_3(1^#(#(x1)))}
==> {1^#(*(x1)) -> c_1(0^#(#(x1)))}
{#^#(#(x1)) -> c_5(#^#(x1))}
==> {#^#(*(x1)) -> c_6()}
{#^#(#(x1)) -> c_5(#^#(x1))}
==> {#^#(#(x1)) -> c_5(#^#(x1))}
{#^#(#(x1)) -> c_5(#^#(x1))}
==> {#^#($(x1)) -> c_4()}
{#^#(#(x1)) -> c_5(#^#(x1))}
==> {#^#(1(x1)) -> c_3(1^#(#(x1)))}
{#^#(#(x1)) -> c_5(#^#(x1))}
==> {#^#(0(x1)) -> c_2(0^#(#(x1)))}
We consider the following path(s):
1) { #^#(#(x1)) -> c_5(#^#(x1))
, #^#(1(x1)) -> c_3(1^#(#(x1)))
, 0^#(*(x1)) -> c_0(1^#(x1))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))}
The usable rules for this path are the following:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, #($(x1)) -> *($(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 0(*(x1)) -> *(1(x1))
, 1(*(x1)) -> 0(#(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:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, #($(x1)) -> *($(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 0(*(x1)) -> *(1(x1))
, 1(*(x1)) -> 0(#(x1))
, #^#(1(x1)) -> c_3(1^#(#(x1)))
, #^#(#(x1)) -> c_5(#^#(x1))
, 0^#(*(x1)) -> c_0(1^#(x1))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, #^#(#(x1)) -> c_5(#^#(x1))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, #^#(#(x1)) -> c_5(#^#(x1))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [0]
*(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
#(x1) = [1] x1 + [1]
$(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [7]
c_1(x1) = [1] x1 + [0]
#^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{#^#(1(x1)) -> c_3(1^#(#(x1)))}
and weakly orienting the rules
{ #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, #^#(#(x1)) -> c_5(#^#(x1))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{#^#(1(x1)) -> c_3(1^#(#(x1)))}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [0]
*(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
#(x1) = [1] x1 + [1]
$(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
#^#(x1) = [1] x1 + [7]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(*(x1)) -> c_0(1^#(x1))}
and weakly orienting the rules
{ #^#(1(x1)) -> c_3(1^#(#(x1)))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, #^#(#(x1)) -> c_5(#^#(x1))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(*(x1)) -> c_0(1^#(x1))}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [0]
*(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
#(x1) = [1] x1 + [1]
$(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
#^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{1(*(x1)) -> 0(#(x1))}
and weakly orienting the rules
{ 0^#(*(x1)) -> c_0(1^#(x1))
, #^#(1(x1)) -> c_3(1^#(#(x1)))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, #^#(#(x1)) -> c_5(#^#(x1))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1(*(x1)) -> 0(#(x1))}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [0]
*(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [4]
#(x1) = [1] x1 + [1]
$(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
#^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4() = [0]
c_5(x1) = [1] x1 + [1]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(*(x1)) -> *(1(x1))}
and weakly orienting the rules
{ 1(*(x1)) -> 0(#(x1))
, 0^#(*(x1)) -> c_0(1^#(x1))
, #^#(1(x1)) -> c_3(1^#(#(x1)))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, #^#(#(x1)) -> c_5(#^#(x1))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(*(x1)) -> *(1(x1))}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [15]
*(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [14]
#(x1) = [1] x1 + [0]
$(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
#^#(x1) = [1] x1 + [9]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [4]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, #($(x1)) -> *($(x1))}
Weak Rules:
{ 0(*(x1)) -> *(1(x1))
, 1(*(x1)) -> 0(#(x1))
, 0^#(*(x1)) -> c_0(1^#(x1))
, #^#(1(x1)) -> c_3(1^#(#(x1)))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, #^#(#(x1)) -> c_5(#^#(x1))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, #($(x1)) -> *($(x1))}
Weak Rules:
{ 0(*(x1)) -> *(1(x1))
, 1(*(x1)) -> 0(#(x1))
, 0^#(*(x1)) -> c_0(1^#(x1))
, #^#(1(x1)) -> c_3(1^#(#(x1)))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, #^#(#(x1)) -> c_5(#^#(x1))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ *_0(2) -> 2
, *_0(2) -> 12
, *_0(5) -> 2
, *_0(5) -> 12
, *_1(2) -> 16
, *_1(5) -> 16
, *_1(13) -> 12
, *_1(13) -> 16
, #_0(2) -> 12
, #_0(5) -> 12
, #_1(2) -> 16
, #_1(5) -> 16
, $_0(2) -> 5
, $_0(5) -> 5
, $_1(2) -> 13
, $_1(5) -> 13
, 0^#_0(2) -> 6
, 0^#_0(5) -> 6
, 0^#_0(12) -> 11
, 0^#_1(16) -> 15
, c_0_0(8) -> 6
, c_0_0(8) -> 11
, c_0_1(14) -> 11
, c_0_1(14) -> 15
, 1^#_0(2) -> 8
, 1^#_0(5) -> 8
, 1^#_1(2) -> 14
, 1^#_1(5) -> 14
, 1^#_1(13) -> 14
, c_1_0(11) -> 8
, c_1_1(15) -> 8
, c_1_1(15) -> 14
, #^#_0(2) -> 10
, #^#_0(5) -> 10}
2) { #^#(#(x1)) -> c_5(#^#(x1))
, #^#(0(x1)) -> c_2(0^#(#(x1)))
, 0^#(*(x1)) -> c_0(1^#(x1))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))}
The usable rules for this path are the following:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, #($(x1)) -> *($(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 0(*(x1)) -> *(1(x1))
, 1(*(x1)) -> 0(#(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, #($(x1)) -> *($(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 0(*(x1)) -> *(1(x1))
, 1(*(x1)) -> 0(#(x1))
, #^#(0(x1)) -> c_2(0^#(#(x1)))
, #^#(#(x1)) -> c_5(#^#(x1))
, 0^#(*(x1)) -> c_0(1^#(x1))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 1(*(x1)) -> 0(#(x1))
, #^#(#(x1)) -> c_5(#^#(x1))
, 0^#(*(x1)) -> c_0(1^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 1(*(x1)) -> 0(#(x1))
, #^#(#(x1)) -> c_5(#^#(x1))
, 0^#(*(x1)) -> c_0(1^#(x1))}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [0]
*(x1) = [1] x1 + [5]
1(x1) = [1] x1 + [0]
#(x1) = [1] x1 + [1]
$(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [15]
c_0(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
#^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{1^#(*(x1)) -> c_1(0^#(#(x1)))}
and weakly orienting the rules
{ #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 1(*(x1)) -> 0(#(x1))
, #^#(#(x1)) -> c_5(#^#(x1))
, 0^#(*(x1)) -> c_0(1^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(*(x1)) -> c_1(0^#(#(x1)))}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [0]
*(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
#(x1) = [1] x1 + [1]
$(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
#^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [7]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [1]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{#^#(0(x1)) -> c_2(0^#(#(x1)))}
and weakly orienting the rules
{ 1^#(*(x1)) -> c_1(0^#(#(x1)))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 1(*(x1)) -> 0(#(x1))
, #^#(#(x1)) -> c_5(#^#(x1))
, 0^#(*(x1)) -> c_0(1^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{#^#(0(x1)) -> c_2(0^#(#(x1)))}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [0]
*(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
#(x1) = [1] x1 + [1]
$(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
#^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(*(x1)) -> *(1(x1))}
and weakly orienting the rules
{ #^#(0(x1)) -> c_2(0^#(#(x1)))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 1(*(x1)) -> 0(#(x1))
, #^#(#(x1)) -> c_5(#^#(x1))
, 0^#(*(x1)) -> c_0(1^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(*(x1)) -> *(1(x1))}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [10]
*(x1) = [1] x1 + [10]
1(x1) = [1] x1 + [8]
#(x1) = [1] x1 + [0]
$(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [10]
c_0(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [10]
c_1(x1) = [1] x1 + [1]
#^#(x1) = [1] x1 + [13]
c_2(x1) = [1] x1 + [11]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, #($(x1)) -> *($(x1))}
Weak Rules:
{ 0(*(x1)) -> *(1(x1))
, #^#(0(x1)) -> c_2(0^#(#(x1)))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 1(*(x1)) -> 0(#(x1))
, #^#(#(x1)) -> c_5(#^#(x1))
, 0^#(*(x1)) -> c_0(1^#(x1))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, #($(x1)) -> *($(x1))}
Weak Rules:
{ 0(*(x1)) -> *(1(x1))
, #^#(0(x1)) -> c_2(0^#(#(x1)))
, 1^#(*(x1)) -> c_1(0^#(#(x1)))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 1(*(x1)) -> 0(#(x1))
, #^#(#(x1)) -> c_5(#^#(x1))
, 0^#(*(x1)) -> c_0(1^#(x1))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ *_0(2) -> 2
, *_0(2) -> 12
, *_0(5) -> 2
, *_0(5) -> 12
, *_1(2) -> 15
, *_1(5) -> 15
, *_1(13) -> 12
, *_1(13) -> 15
, #_0(2) -> 12
, #_0(5) -> 12
, #_1(2) -> 15
, #_1(5) -> 15
, $_0(2) -> 5
, $_0(5) -> 5
, $_1(2) -> 13
, $_1(5) -> 13
, 0^#_0(2) -> 6
, 0^#_0(5) -> 6
, 0^#_0(12) -> 11
, 0^#_1(15) -> 14
, c_0_0(8) -> 6
, c_0_0(8) -> 11
, c_0_1(16) -> 14
, c_0_1(17) -> 11
, c_0_1(17) -> 14
, 1^#_0(2) -> 8
, 1^#_0(5) -> 8
, 1^#_1(2) -> 16
, 1^#_1(5) -> 16
, 1^#_1(13) -> 17
, c_1_0(11) -> 8
, c_1_1(14) -> 8
, c_1_1(14) -> 16
, #^#_0(2) -> 10
, #^#_0(5) -> 10}
3) { #^#(#(x1)) -> c_5(#^#(x1))
, #^#(1(x1)) -> c_3(1^#(#(x1)))}
The usable rules for this path are the following:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, #($(x1)) -> *($(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 0(*(x1)) -> *(1(x1))
, 1(*(x1)) -> 0(#(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, #($(x1)) -> *($(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 0(*(x1)) -> *(1(x1))
, 1(*(x1)) -> 0(#(x1))
, #^#(#(x1)) -> c_5(#^#(x1))
, #^#(1(x1)) -> c_3(1^#(#(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [0]
*(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
#(x1) = [1] x1 + [1]
$(x1) = [1] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
#^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [1]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{#^#(#(x1)) -> c_5(#^#(x1))}
and weakly orienting the rules
{ #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{#^#(#(x1)) -> c_5(#^#(x1))}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [0]
*(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
#(x1) = [1] x1 + [1]
$(x1) = [1] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
#^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{#^#(1(x1)) -> c_3(1^#(#(x1)))}
and weakly orienting the rules
{ #^#(#(x1)) -> c_5(#^#(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{#^#(1(x1)) -> c_3(1^#(#(x1)))}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [0]
*(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
#(x1) = [1] x1 + [1]
$(x1) = [1] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
#^#(x1) = [1] x1 + [9]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{1(*(x1)) -> 0(#(x1))}
and weakly orienting the rules
{ #^#(1(x1)) -> c_3(1^#(#(x1)))
, #^#(#(x1)) -> c_5(#^#(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1(*(x1)) -> 0(#(x1))}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [0]
*(x1) = [1] x1 + [5]
1(x1) = [1] x1 + [0]
#(x1) = [1] x1 + [1]
$(x1) = [1] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
#^#(x1) = [1] x1 + [4]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [1]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{#($(x1)) -> *($(x1))}
and weakly orienting the rules
{ 1(*(x1)) -> 0(#(x1))
, #^#(1(x1)) -> c_3(1^#(#(x1)))
, #^#(#(x1)) -> c_5(#^#(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{#($(x1)) -> *($(x1))}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [0]
*(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [4]
#(x1) = [1] x1 + [1]
$(x1) = [1] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
#^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, 0(*(x1)) -> *(1(x1))}
Weak Rules:
{ #($(x1)) -> *($(x1))
, 1(*(x1)) -> 0(#(x1))
, #^#(1(x1)) -> c_3(1^#(#(x1)))
, #^#(#(x1)) -> c_5(#^#(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, 0(*(x1)) -> *(1(x1))}
Weak Rules:
{ #($(x1)) -> *($(x1))
, 1(*(x1)) -> 0(#(x1))
, #^#(1(x1)) -> c_3(1^#(#(x1)))
, #^#(#(x1)) -> c_5(#^#(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ *_0(2) -> 2
, *_0(5) -> 2
, $_0(2) -> 5
, $_0(5) -> 5
, 1^#_0(2) -> 8
, 1^#_0(5) -> 8
, #^#_0(2) -> 10
, #^#_0(5) -> 10}
4) { #^#(#(x1)) -> c_5(#^#(x1))
, #^#(0(x1)) -> c_2(0^#(#(x1)))}
The usable rules for this path are the following:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, #($(x1)) -> *($(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 0(*(x1)) -> *(1(x1))
, 1(*(x1)) -> 0(#(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, #($(x1)) -> *($(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 0(*(x1)) -> *(1(x1))
, 1(*(x1)) -> 0(#(x1))
, #^#(#(x1)) -> c_5(#^#(x1))
, #^#(0(x1)) -> c_2(0^#(#(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 1(*(x1)) -> 0(#(x1))
, #^#(#(x1)) -> c_5(#^#(x1))
, #^#(0(x1)) -> c_2(0^#(#(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 1(*(x1)) -> 0(#(x1))
, #^#(#(x1)) -> c_5(#^#(x1))
, #^#(0(x1)) -> c_2(0^#(#(x1)))}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [0]
*(x1) = [1] x1 + [15]
1(x1) = [1] x1 + [0]
#(x1) = [1] x1 + [1]
$(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
#^#(x1) = [1] x1 + [3]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{#($(x1)) -> *($(x1))}
and weakly orienting the rules
{ #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 1(*(x1)) -> 0(#(x1))
, #^#(#(x1)) -> c_5(#^#(x1))
, #^#(0(x1)) -> c_2(0^#(#(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{#($(x1)) -> *($(x1))}
Details:
Interpretation Functions:
0(x1) = [1] x1 + [0]
*(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [8]
#(x1) = [1] x1 + [1]
$(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
#^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [1]
c_6() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, 0(*(x1)) -> *(1(x1))}
Weak Rules:
{ #($(x1)) -> *($(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 1(*(x1)) -> 0(#(x1))
, #^#(#(x1)) -> c_5(#^#(x1))
, #^#(0(x1)) -> c_2(0^#(#(x1)))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ #(0(x1)) -> 0(#(x1))
, #(1(x1)) -> 1(#(x1))
, 0(*(x1)) -> *(1(x1))}
Weak Rules:
{ #($(x1)) -> *($(x1))
, #(#(x1)) -> #(x1)
, #(*(x1)) -> *(x1)
, 1(*(x1)) -> 0(#(x1))
, #^#(#(x1)) -> c_5(#^#(x1))
, #^#(0(x1)) -> c_2(0^#(#(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ *_0(2) -> 2
, *_0(5) -> 2
, $_0(2) -> 5
, $_0(5) -> 5
, 0^#_0(2) -> 6
, 0^#_0(5) -> 6
, #^#_0(2) -> 10
, #^#_0(5) -> 10}
5) { #^#(#(x1)) -> c_5(#^#(x1))
, #^#($(x1)) -> c_4()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
0(x1) = [0] x1 + [0]
*(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
#(x1) = [0] x1 + [0]
$(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
#^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {#^#($(x1)) -> c_4()}
Weak Rules: {#^#(#(x1)) -> c_5(#^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{#^#($(x1)) -> c_4()}
and weakly orienting the rules
{#^#(#(x1)) -> c_5(#^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{#^#($(x1)) -> c_4()}
Details:
Interpretation Functions:
0(x1) = [0] x1 + [0]
*(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
#(x1) = [1] x1 + [0]
$(x1) = [1] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
#^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules:
{ #^#($(x1)) -> c_4()
, #^#(#(x1)) -> c_5(#^#(x1))}
Details:
The given problem does not contain any strict rules
6) { #^#(#(x1)) -> c_5(#^#(x1))
, #^#(*(x1)) -> c_6()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
0(x1) = [0] x1 + [0]
*(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
#(x1) = [0] x1 + [0]
$(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
#^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {#^#(*(x1)) -> c_6()}
Weak Rules: {#^#(#(x1)) -> c_5(#^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{#^#(*(x1)) -> c_6()}
and weakly orienting the rules
{#^#(#(x1)) -> c_5(#^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{#^#(*(x1)) -> c_6()}
Details:
Interpretation Functions:
0(x1) = [0] x1 + [0]
*(x1) = [1] x1 + [0]
1(x1) = [0] x1 + [0]
#(x1) = [1] x1 + [0]
$(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
#^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules:
{ #^#(*(x1)) -> c_6()
, #^#(#(x1)) -> c_5(#^#(x1))}
Details:
The given problem does not contain any strict rules
7) {#^#(#(x1)) -> c_5(#^#(x1))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
0(x1) = [0] x1 + [0]
*(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
#(x1) = [0] x1 + [0]
$(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
#^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {#^#(#(x1)) -> c_5(#^#(x1))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{#^#(#(x1)) -> c_5(#^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{#^#(#(x1)) -> c_5(#^#(x1))}
Details:
Interpretation Functions:
0(x1) = [0] x1 + [0]
*(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
#(x1) = [1] x1 + [8]
$(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
#^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [3]
c_6() = [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {#^#(#(x1)) -> c_5(#^#(x1))}
Details:
The given problem does not contain any strict rules