* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
f(X,Y,g(X,Y)) -> h(0(),g(X,Y))
g(X,s(Y)) -> g(X,Y)
g(0(),Y) -> 0()
h(X,Z) -> f(X,s(X),Z)
- Signature:
{f/3,g/2,h/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
f(X,Y,g(X,Y)) -> h(0(),g(X,Y))
g(X,s(Y)) -> g(X,Y)
g(0(),Y) -> 0()
h(X,Z) -> f(X,s(X),Z)
- Signature:
{f/3,g/2,h/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
g(x,y){y -> s(y)} =
g(x,s(y)) ->^+ g(x,y)
= C[g(x,y) = g(x,y){}]
** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(X,Y,g(X,Y)) -> h(0(),g(X,Y))
g(X,s(Y)) -> g(X,Y)
g(0(),Y) -> 0()
h(X,Z) -> f(X,s(X),Z)
- Signature:
{f/3,g/2,h/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,s}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:
Strict DPs
f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)))
g#(X,s(Y)) -> c_2(g#(X,Y))
g#(0(),Y) -> c_3()
h#(X,Z) -> c_4(f#(X,s(X),Z))
Weak DPs
and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)))
g#(X,s(Y)) -> c_2(g#(X,Y))
g#(0(),Y) -> c_3()
h#(X,Z) -> c_4(f#(X,s(X),Z))
- Strict TRS:
f(X,Y,g(X,Y)) -> h(0(),g(X,Y))
g(X,s(Y)) -> g(X,Y)
g(0(),Y) -> 0()
h(X,Z) -> f(X,s(X),Z)
- Signature:
{f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
g(X,s(Y)) -> g(X,Y)
g(0(),Y) -> 0()
f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)))
g#(X,s(Y)) -> c_2(g#(X,Y))
g#(0(),Y) -> c_3()
h#(X,Z) -> c_4(f#(X,s(X),Z))
** Step 1.b:3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)))
g#(X,s(Y)) -> c_2(g#(X,Y))
g#(0(),Y) -> c_3()
h#(X,Z) -> c_4(f#(X,s(X),Z))
- Strict TRS:
g(X,s(Y)) -> g(X,Y)
g(0(),Y) -> 0()
- Signature:
{f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,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(c_1) = {1},
uargs(c_2) = {1},
uargs(c_4) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(f) = [0]
p(g) = [4] x2 + [2]
p(h) = [1] x1 + [2]
p(s) = [1] x1 + [4]
p(f#) = [10] x1 + [2] x2 + [4] x3 + [0]
p(g#) = [8] x1 + [1] x2 + [1]
p(h#) = [13] x1 + [4] x2 + [2]
p(c_1) = [1] x1 + [4]
p(c_2) = [1] x1 + [3]
p(c_3) = [2]
p(c_4) = [1] x1 + [8]
Following rules are strictly oriented:
g#(X,s(Y)) = [8] X + [1] Y + [5]
> [8] X + [1] Y + [4]
= c_2(g#(X,Y))
g(X,s(Y)) = [4] Y + [18]
> [4] Y + [2]
= g(X,Y)
g(0(),Y) = [4] Y + [2]
> [0]
= 0()
Following rules are (at-least) weakly oriented:
f#(X,Y,g(X,Y)) = [10] X + [18] Y + [8]
>= [16] Y + [14]
= c_1(h#(0(),g(X,Y)))
g#(0(),Y) = [1] Y + [1]
>= [2]
= c_3()
h#(X,Z) = [13] X + [4] Z + [2]
>= [12] X + [4] Z + [16]
= c_4(f#(X,s(X),Z))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: RemoveInapplicable WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)))
g#(0(),Y) -> c_3()
h#(X,Z) -> c_4(f#(X,s(X),Z))
- Weak DPs:
g#(X,s(Y)) -> c_2(g#(X,Y))
- Weak TRS:
g(X,s(Y)) -> g(X,Y)
g(0(),Y) -> 0()
- Signature:
{f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
+ Applied Processor:
RemoveInapplicable
+ Details:
Only the nodes
{2,3,4}
are reachable from nodes
{2,3,4}
that start derivation from marked basic terms.
The nodes not reachable are removed from the problem.
** Step 1.b:5: PredecessorEstimation WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
g#(0(),Y) -> c_3()
h#(X,Z) -> c_4(f#(X,s(X),Z))
- Weak DPs:
g#(X,s(Y)) -> c_2(g#(X,Y))
- Weak TRS:
g(X,s(Y)) -> g(X,Y)
g(0(),Y) -> 0()
- Signature:
{f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2}
by application of
Pre({2}) = {}.
Here rules are labelled as follows:
1: g#(0(),Y) -> c_3()
2: h#(X,Z) -> c_4(f#(X,s(X),Z))
3: g#(X,s(Y)) -> c_2(g#(X,Y))
** Step 1.b:6: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
g#(0(),Y) -> c_3()
- Weak DPs:
g#(X,s(Y)) -> c_2(g#(X,Y))
h#(X,Z) -> c_4(f#(X,s(X),Z))
- Weak TRS:
g(X,s(Y)) -> g(X,Y)
g(0(),Y) -> 0()
- Signature:
{f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:g#(0(),Y) -> c_3()
2:W:g#(X,s(Y)) -> c_2(g#(X,Y))
-->_1 g#(X,s(Y)) -> c_2(g#(X,Y)):2
-->_1 g#(0(),Y) -> c_3():1
3:W:h#(X,Z) -> c_4(f#(X,s(X),Z))
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: h#(X,Z) -> c_4(f#(X,s(X),Z))
** Step 1.b:7: UsableRules WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
g#(0(),Y) -> c_3()
- Weak DPs:
g#(X,s(Y)) -> c_2(g#(X,Y))
- Weak TRS:
g(X,s(Y)) -> g(X,Y)
g(0(),Y) -> 0()
- Signature:
{f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
g#(X,s(Y)) -> c_2(g#(X,Y))
g#(0(),Y) -> c_3()
** Step 1.b:8: PredecessorEstimationCP WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
g#(0(),Y) -> c_3()
- Weak DPs:
g#(X,s(Y)) -> c_2(g#(X,Y))
- Signature:
{f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: g#(0(),Y) -> c_3()
The strictly oriented rules are moved into the weak component.
*** Step 1.b:8.a:1: NaturalMI WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
g#(0(),Y) -> c_3()
- Weak DPs:
g#(X,s(Y)) -> c_2(g#(X,Y))
- Signature:
{f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 0, 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 0 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:
{f#,g#,h#}
TcT has computed the following interpretation:
p(0) = [0]
p(f) = [0]
p(g) = [0]
p(h) = [0]
p(s) = [0]
p(f#) = [0]
p(g#) = [5]
p(h#) = [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [1]
p(c_4) = [0]
Following rules are strictly oriented:
g#(0(),Y) = [5]
> [1]
= c_3()
Following rules are (at-least) weakly oriented:
g#(X,s(Y)) = [5]
>= [5]
= c_2(g#(X,Y))
*** Step 1.b:8.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
g#(X,s(Y)) -> c_2(g#(X,Y))
g#(0(),Y) -> c_3()
- Signature:
{f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 1.b:8.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
g#(X,s(Y)) -> c_2(g#(X,Y))
g#(0(),Y) -> c_3()
- Signature:
{f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:g#(X,s(Y)) -> c_2(g#(X,Y))
-->_1 g#(0(),Y) -> c_3():2
-->_1 g#(X,s(Y)) -> c_2(g#(X,Y)):1
2:W:g#(0(),Y) -> c_3()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: g#(X,s(Y)) -> c_2(g#(X,Y))
2: g#(0(),Y) -> c_3()
*** Step 1.b:8.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))