WORST_CASE(Omega(n^1),O(n^2))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
conv(0()) -> cons(nil(),0())
conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lastbit(0()) -> 0()
lastbit(s(0())) -> s(0())
lastbit(s(s(x))) -> lastbit(x)
- Signature:
{conv/1,half/1,lastbit/1} / {0/0,cons/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv,half,lastbit} and constructors {0,cons,nil,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
conv(0()) -> cons(nil(),0())
conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lastbit(0()) -> 0()
lastbit(s(0())) -> s(0())
lastbit(s(s(x))) -> lastbit(x)
- Signature:
{conv/1,half/1,lastbit/1} / {0/0,cons/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv,half,lastbit} and constructors {0,cons,nil,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
half(x){x -> s(s(x))} =
half(s(s(x))) ->^+ s(half(x))
= C[half(x) = half(x){}]
** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
conv(0()) -> cons(nil(),0())
conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lastbit(0()) -> 0()
lastbit(s(0())) -> s(0())
lastbit(s(s(x))) -> lastbit(x)
- Signature:
{conv/1,half/1,lastbit/1} / {0/0,cons/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv,half,lastbit} and constructors {0,cons,nil,s}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:
Strict DPs
conv#(0()) -> c_1()
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
half#(0()) -> c_3()
half#(s(0())) -> c_4()
half#(s(s(x))) -> c_5(half#(x))
lastbit#(0()) -> c_6()
lastbit#(s(0())) -> c_7()
lastbit#(s(s(x))) -> c_8(lastbit#(x))
Weak DPs
and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
conv#(0()) -> c_1()
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
half#(0()) -> c_3()
half#(s(0())) -> c_4()
half#(s(s(x))) -> c_5(half#(x))
lastbit#(0()) -> c_6()
lastbit#(s(0())) -> c_7()
lastbit#(s(s(x))) -> c_8(lastbit#(x))
- Strict TRS:
conv(0()) -> cons(nil(),0())
conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
lastbit(0()) -> 0()
lastbit(s(0())) -> s(0())
lastbit(s(s(x))) -> lastbit(x)
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
conv#(0()) -> c_1()
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
half#(0()) -> c_3()
half#(s(0())) -> c_4()
half#(s(s(x))) -> c_5(half#(x))
lastbit#(0()) -> c_6()
lastbit#(s(0())) -> c_7()
lastbit#(s(s(x))) -> c_8(lastbit#(x))
** Step 1.b:3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
conv#(0()) -> c_1()
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
half#(0()) -> c_3()
half#(s(0())) -> c_4()
half#(s(s(x))) -> c_5(half#(x))
lastbit#(0()) -> c_6()
lastbit#(s(0())) -> c_7()
lastbit#(s(s(x))) -> c_8(lastbit#(x))
- Strict TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(s) = {1},
uargs(conv#) = {1},
uargs(c_2) = {1,2},
uargs(c_5) = {1},
uargs(c_8) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(cons) = [1] x1 + [1] x2 + [0]
p(conv) = [8] x1 + [2]
p(half) = [1] x1 + [7]
p(lastbit) = [0]
p(nil) = [0]
p(s) = [1] x1 + [8]
p(conv#) = [1] x1 + [0]
p(half#) = [0]
p(lastbit#) = [1]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [1] x1 + [0]
Following rules are strictly oriented:
lastbit#(0()) = [1]
> [0]
= c_6()
lastbit#(s(0())) = [1]
> [0]
= c_7()
half(0()) = [7]
> [0]
= 0()
half(s(0())) = [15]
> [0]
= 0()
half(s(s(x))) = [1] x + [23]
> [1] x + [15]
= s(half(x))
Following rules are (at-least) weakly oriented:
conv#(0()) = [0]
>= [0]
= c_1()
conv#(s(x)) = [1] x + [8]
>= [1] x + [16]
= c_2(conv#(half(s(x))),lastbit#(s(x)))
half#(0()) = [0]
>= [0]
= c_3()
half#(s(0())) = [0]
>= [0]
= c_4()
half#(s(s(x))) = [0]
>= [0]
= c_5(half#(x))
lastbit#(s(s(x))) = [1]
>= [1]
= c_8(lastbit#(x))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
conv#(0()) -> c_1()
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
half#(0()) -> c_3()
half#(s(0())) -> c_4()
half#(s(s(x))) -> c_5(half#(x))
lastbit#(s(s(x))) -> c_8(lastbit#(x))
- Weak DPs:
lastbit#(0()) -> c_6()
lastbit#(s(0())) -> c_7()
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,4}
by application of
Pre({1,3,4}) = {2,5}.
Here rules are labelled as follows:
1: conv#(0()) -> c_1()
2: conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
3: half#(0()) -> c_3()
4: half#(s(0())) -> c_4()
5: half#(s(s(x))) -> c_5(half#(x))
6: lastbit#(s(s(x))) -> c_8(lastbit#(x))
7: lastbit#(0()) -> c_6()
8: lastbit#(s(0())) -> c_7()
** Step 1.b:5: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
half#(s(s(x))) -> c_5(half#(x))
lastbit#(s(s(x))) -> c_8(lastbit#(x))
- Weak DPs:
conv#(0()) -> c_1()
half#(0()) -> c_3()
half#(s(0())) -> c_4()
lastbit#(0()) -> c_6()
lastbit#(s(0())) -> c_7()
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
-->_2 lastbit#(s(s(x))) -> c_8(lastbit#(x)):3
-->_2 lastbit#(s(0())) -> c_7():8
-->_1 conv#(0()) -> c_1():4
-->_1 conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x))):1
2:S:half#(s(s(x))) -> c_5(half#(x))
-->_1 half#(s(0())) -> c_4():6
-->_1 half#(0()) -> c_3():5
-->_1 half#(s(s(x))) -> c_5(half#(x)):2
3:S:lastbit#(s(s(x))) -> c_8(lastbit#(x))
-->_1 lastbit#(s(0())) -> c_7():8
-->_1 lastbit#(0()) -> c_6():7
-->_1 lastbit#(s(s(x))) -> c_8(lastbit#(x)):3
4:W:conv#(0()) -> c_1()
5:W:half#(0()) -> c_3()
6:W:half#(s(0())) -> c_4()
7:W:lastbit#(0()) -> c_6()
8:W:lastbit#(s(0())) -> c_7()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: half#(0()) -> c_3()
6: half#(s(0())) -> c_4()
4: conv#(0()) -> c_1()
7: lastbit#(0()) -> c_6()
8: lastbit#(s(0())) -> c_7()
** Step 1.b:6: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
half#(s(s(x))) -> c_5(half#(x))
lastbit#(s(s(x))) -> c_8(lastbit#(x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
lastbit#(s(s(x))) -> c_8(lastbit#(x))
- Weak DPs:
half#(s(s(x))) -> c_5(half#(x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
Problem (S)
- Strict DPs:
half#(s(s(x))) -> c_5(half#(x))
- Weak DPs:
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
lastbit#(s(s(x))) -> c_8(lastbit#(x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
*** Step 1.b:6.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
lastbit#(s(s(x))) -> c_8(lastbit#(x))
- Weak DPs:
half#(s(s(x))) -> c_5(half#(x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
-->_2 lastbit#(s(s(x))) -> c_8(lastbit#(x)):3
-->_1 conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x))):1
2:W:half#(s(s(x))) -> c_5(half#(x))
-->_1 half#(s(s(x))) -> c_5(half#(x)):2
3:S:lastbit#(s(s(x))) -> c_8(lastbit#(x))
-->_1 lastbit#(s(s(x))) -> c_8(lastbit#(x)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: half#(s(s(x))) -> c_5(half#(x))
*** Step 1.b:6.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
lastbit#(s(s(x))) -> c_8(lastbit#(x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
3: lastbit#(s(s(x))) -> c_8(lastbit#(x))
The strictly oriented rules are moved into the weak component.
**** Step 1.b:6.a:2.a:1: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
lastbit#(s(s(x))) -> c_8(lastbit#(x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_2) = {1,2},
uargs(c_8) = {1}
Following symbols are considered usable:
{half,conv#,half#,lastbit#}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(cons) = [0]
[0]
[2]
p(conv) = [0]
[0]
[1]
p(half) = [0 1 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(lastbit) = [0 0 0] [2]
[0 0 0] x1 + [0]
[0 0 2] [2]
p(nil) = [1]
[2]
[2]
p(s) = [0 1 3] [2]
[0 1 2] x1 + [1]
[0 0 1] [1]
p(conv#) = [1 0 0] [0]
[0 2 1] x1 + [1]
[0 1 0] [1]
p(half#) = [0 2 0] [0]
[2 0 0] x1 + [2]
[0 0 0] [0]
p(lastbit#) = [0 0 1] [0]
[1 0 0] x1 + [0]
[1 0 0] [1]
p(c_1) = [1]
[0]
[0]
p(c_2) = [1 0 0] [1 0 0] [0]
[1 0 0] x1 + [0 0 1] x2 + [0]
[0 0 0] [2 0 0] [0]
p(c_3) = [0]
[0]
[2]
p(c_4) = [2]
[1]
[0]
p(c_5) = [1]
[1]
[1]
p(c_6) = [1]
[2]
[0]
p(c_7) = [0]
[0]
[0]
p(c_8) = [1 0 0] [0]
[2 0 0] x1 + [1]
[0 0 0] [0]
Following rules are strictly oriented:
lastbit#(s(s(x))) = [0 0 1] [2]
[0 1 5] x + [6]
[0 1 5] [7]
> [0 0 1] [0]
[0 0 2] x + [1]
[0 0 0] [0]
= c_8(lastbit#(x))
Following rules are (at-least) weakly oriented:
conv#(s(x)) = [0 1 3] [2]
[0 2 5] x + [4]
[0 1 2] [2]
>= [0 1 3] [2]
[0 2 5] x + [4]
[0 0 2] [2]
= c_2(conv#(half(s(x))),lastbit#(s(x)))
half(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
half(s(0())) = [1]
[2]
[1]
>= [0]
[0]
[0]
= 0()
half(s(s(x))) = [0 1 4] [4]
[0 1 5] x + [6]
[0 0 1] [2]
>= [0 1 4] [2]
[0 1 3] x + [1]
[0 0 1] [1]
= s(half(x))
**** Step 1.b:6.a:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
- Weak DPs:
lastbit#(s(s(x))) -> c_8(lastbit#(x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 1.b:6.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
- Weak DPs:
lastbit#(s(s(x))) -> c_8(lastbit#(x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
-->_2 lastbit#(s(s(x))) -> c_8(lastbit#(x)):2
-->_1 conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x))):1
2:W:lastbit#(s(s(x))) -> c_8(lastbit#(x))
-->_1 lastbit#(s(s(x))) -> c_8(lastbit#(x)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: lastbit#(s(s(x))) -> c_8(lastbit#(x))
**** Step 1.b:6.a:2.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
-->_1 conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x))):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
conv#(s(x)) -> c_2(conv#(half(s(x))))
**** Step 1.b:6.a:2.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
conv#(s(x)) -> c_2(conv#(half(s(x))))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: conv#(s(x)) -> c_2(conv#(half(s(x))))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.a:2.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
conv#(s(x)) -> c_2(conv#(half(s(x))))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_2) = {1}
Following symbols are considered usable:
{half,conv#,half#,lastbit#}
TcT has computed the following interpretation:
p(0) = [0]
[1]
p(cons) = [0 2] x1 + [0 0] x2 + [0]
[0 0] [0 1] [0]
p(conv) = [0 0] x1 + [1]
[0 1] [0]
p(half) = [0 2] x1 + [0]
[0 1] [0]
p(lastbit) = [2 1] x1 + [1]
[2 0] [1]
p(nil) = [4]
[0]
p(s) = [0 2] x1 + [4]
[0 1] [1]
p(conv#) = [2 0] x1 + [5]
[0 0] [1]
p(half#) = [0 0] x1 + [1]
[0 1] [1]
p(lastbit#) = [1]
[1]
p(c_1) = [0]
[0]
p(c_2) = [1 1] x1 + [1]
[0 0] [1]
p(c_3) = [1]
[0]
p(c_4) = [4]
[2]
p(c_5) = [4 0] x1 + [0]
[0 0] [1]
p(c_6) = [1]
[1]
p(c_7) = [0]
[0]
p(c_8) = [0]
[1]
Following rules are strictly oriented:
conv#(s(x)) = [0 4] x + [13]
[0 0] [1]
> [0 4] x + [11]
[0 0] [1]
= c_2(conv#(half(s(x))))
Following rules are (at-least) weakly oriented:
half(0()) = [2]
[1]
>= [0]
[1]
= 0()
half(s(0())) = [4]
[2]
>= [0]
[1]
= 0()
half(s(s(x))) = [0 2] x + [4]
[0 1] [2]
>= [0 2] x + [4]
[0 1] [1]
= s(half(x))
***** Step 1.b:6.a:2.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
conv#(s(x)) -> c_2(conv#(half(s(x))))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:6.a:2.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
conv#(s(x)) -> c_2(conv#(half(s(x))))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:conv#(s(x)) -> c_2(conv#(half(s(x))))
-->_1 conv#(s(x)) -> c_2(conv#(half(s(x)))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: conv#(s(x)) -> c_2(conv#(half(s(x))))
***** Step 1.b:6.a:2.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 1.b:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(x))) -> c_5(half#(x))
- Weak DPs:
conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
lastbit#(s(s(x))) -> c_8(lastbit#(x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:half#(s(s(x))) -> c_5(half#(x))
-->_1 half#(s(s(x))) -> c_5(half#(x)):1
2:W:conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
-->_2 lastbit#(s(s(x))) -> c_8(lastbit#(x)):3
-->_1 conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x))):2
3:W:lastbit#(s(s(x))) -> c_8(lastbit#(x))
-->_1 lastbit#(s(s(x))) -> c_8(lastbit#(x)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: conv#(s(x)) -> c_2(conv#(half(s(x))),lastbit#(s(x)))
3: lastbit#(s(s(x))) -> c_8(lastbit#(x))
*** Step 1.b:6.b:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(x))) -> c_5(half#(x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
half#(s(s(x))) -> c_5(half#(x))
*** Step 1.b:6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(x))) -> c_5(half#(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: half#(s(s(x))) -> c_5(half#(x))
The strictly oriented rules are moved into the weak component.
**** Step 1.b:6.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(x))) -> c_5(half#(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_5) = {1}
Following symbols are considered usable:
{conv#,half#,lastbit#}
TcT has computed the following interpretation:
p(0) = [1]
p(cons) = [1] x1 + [0]
p(conv) = [8]
p(half) = [1] x1 + [2]
p(lastbit) = [1]
p(nil) = [0]
p(s) = [1] x1 + [1]
p(conv#) = [1] x1 + [1]
p(half#) = [4] x1 + [0]
p(lastbit#) = [1] x1 + [0]
p(c_1) = [1]
p(c_2) = [2]
p(c_3) = [4]
p(c_4) = [0]
p(c_5) = [1] x1 + [4]
p(c_6) = [0]
p(c_7) = [8]
p(c_8) = [1] x1 + [1]
Following rules are strictly oriented:
half#(s(s(x))) = [4] x + [8]
> [4] x + [4]
= c_5(half#(x))
Following rules are (at-least) weakly oriented:
**** Step 1.b:6.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
half#(s(s(x))) -> c_5(half#(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 1.b:6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
half#(s(s(x))) -> c_5(half#(x))
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:half#(s(s(x))) -> c_5(half#(x))
-->_1 half#(s(s(x))) -> c_5(half#(x)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: half#(s(s(x))) -> c_5(half#(x))
**** Step 1.b:6.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{conv/1,half/1,lastbit/1,conv#/1,half#/1,lastbit#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/1
,c_6/0,c_7/0,c_8/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {conv#,half#,lastbit#} and constructors {0,cons,nil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))