Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, 10*Arg_0+8 {O(n)})

Initial Complexity Problem (after preprocessing)

Start:f
Program_Vars:Arg_0
Temp_Vars:
Locations:f, g, h
Transitions:
f(Arg_0) -{0}> g(Arg_0)
g(Arg_0) -{10}> g(Uniform (-8, -6)) :|: 8<=Arg_0
g(Arg_0) -> h(Arg_0) :|: Arg_0<8
h(Arg_0) -> h(Arg_0-1) :|: 0<Arg_0 && Arg_0<=7

G f f g g f->g t₀ ∈ g₀ {0} g->g t₁ ∈ g₁ η (Arg_0) = Uniform (-8, -6) τ = 8<=Arg_0 {10} h h g->h t₂ ∈ g₂ τ = Arg_0<8 h->h t₃ ∈ g₃ η (Arg_0) = Arg_0-1 τ = 0<Arg_0

Timebounds:

Overall timebound:max([9, 7+2+Arg_0]) {O(n)}
0,0: f->g: 1 {O(1)}
1,1: g->g: max([0, Arg_0]) {O(n)}
2,2: g->h: 1 {O(1)}
3,3: h->h: 7 {O(1)}

Expected Timebounds:

Overall expected timebound: 9+Arg_0 {O(n)}
0: f->[1:g]: 1 {O(1)}
1: g->[1:g]: Arg_0 {O(n)}
2: g->[1:h]: 1 {O(1)}
3: h->[1:h]: 7 {O(1)}

Costbounds:

Overall costbound: inf {Infinity}
0,0: f->g: inf {Infinity}
1,1: g->g: inf {Infinity}
2,2: g->h: inf {Infinity}
3,3: h->h: inf {Infinity}

Expected Costbounds:

Overall expected costbound: 10*Arg_0+8 {O(n)}
0: f->[1:g]: 0 {O(1)}
1: g->[1:g]: 10*Arg_0 {O(n)}
2: g->[1:h]: 1 {O(1)}
3: h->[1:h]: 7 {O(1)}

Sizebounds:

0,0: f->g, Arg_0: Arg_0 {O(n)}
1,1: g->g, Arg_0: Arg_0 {O(n)}
2,2: g->h, Arg_0: 7 {O(1)}
3,3: h->h, Arg_0: 6 {O(1)}

ExpSizeBounds:

(0: f->[1:g], g), Arg_0: Arg_0 {O(n)}
(1: g->[1:g], g), Arg_0: Arg_0 {O(n)}
(2: g->[1:h], h), Arg_0: 7 {O(1)}
(3: h->[1:h], h), Arg_0: 6 {O(1)}