Inferring Expected Runtimes Using Sizes

KoAT2 Proof MAYBE

Initial Complexity Problem (after preprocessing)

Start:f
Program_Vars:Arg_0, Arg_1
Temp_Vars:
Locations:f, g, h
Transitions:
f(Arg_0,Arg_1) -> g(Arg_0,Arg_1)
g(Arg_0,Arg_1) -> h(Uniform (-1, 0),Arg_0) :|: 0<Arg_0
h(Arg_0,Arg_1) -> 1/4:h(Arg_0,Arg_1) :+: 3/4:h(Arg_0,Arg_1-1) :|: 0<Arg_1 && Arg_1<=1+Arg_0 && 0<=Arg_1 && 0<=Arg_0+Arg_1 && 0<=Arg_0
h(Arg_0,Arg_1) -> g(Arg_0,Arg_1) :|: Arg_1<1 && Arg_1<=1+Arg_0 && 0<=Arg_1 && 0<=Arg_0+Arg_1 && 0<=Arg_0

G f f g g f->g t₀ ∈ g₀ h h g->h t₁ ∈ g₁ η (Arg_0) = Uniform (-1, 0) η (Arg_1) = Arg_0 τ = 0<Arg_0 h->g t₄ ∈ g₃ τ = Arg_1<1 h->h t₂ ∈ g₂ p = 1/4 τ = 0<Arg_1 h->h t₃ ∈ g₂ p = 3/4 η (Arg_1) = Arg_1-1 τ = 0<Arg_1

Timebounds:

Overall timebound:inf {Infinity}
0,0: f->g: 1 {O(1)}
1,1: g->h: inf {Infinity}
2,2: h->h: inf {Infinity}
3,2: h->h: inf {Infinity}
4,3: h->g: inf {Infinity}

Expected Timebounds:

Overall expected timebound: inf {Infinity}
0: f->[1:g]: 1 {O(1)}
1: g->[1:h]: 2*Arg_0 {O(n)}
2: h->[1/4:h; 3/4:h]: inf {Infinity}
3: h->[1:g]: 2*Arg_0 {O(n)}

Costbounds:

Overall costbound: inf {Infinity}
0,0: f->g: inf {Infinity}
1,1: g->h: inf {Infinity}
2,2: h->h: inf {Infinity}
3,2: h->h: inf {Infinity}
4,3: h->g: inf {Infinity}

Expected Costbounds:

Overall expected costbound: inf {Infinity}
0: f->[1:g]: 1 {O(1)}
1: g->[1:h]: 2*Arg_0 {O(n)}
2: h->[1/4:h; 3/4:h]: inf {Infinity}
3: h->[1:g]: 2*Arg_0 {O(n)}

Sizebounds:

0,0: f->g, Arg_0: Arg_0 {O(n)}
0,0: f->g, Arg_1: Arg_1 {O(n)}
1,1: g->h, Arg_0: Arg_0 {O(n)}
1,1: g->h, Arg_1: Arg_0 {O(n)}
2,2: h->h, Arg_0: Arg_0 {O(n)}
2,2: h->h, Arg_1: Arg_0 {O(n)}
3,2: h->h, Arg_0: Arg_0 {O(n)}
3,2: h->h, Arg_1: Arg_0 {O(n)}
4,3: h->g, Arg_0: Arg_0 {O(n)}
4,3: h->g, Arg_1: 0 {O(1)}

ExpSizeBounds:

(0: f->[1:g], g), Arg_0: Arg_0 {O(n)}
(0: f->[1:g], g), Arg_1: Arg_1 {O(n)}
(1: g->[1:h], h), Arg_0: Arg_0 {O(n)}
(1: g->[1:h], h), Arg_1: Arg_0 {O(n)}
(2: h->[1/4:h; 3/4:h], h), Arg_0: Arg_0 {O(n)}
(2: h->[1/4:h; 3/4:h], h), Arg_1: Arg_0 {O(n)}
(3: h->[1:g], g), Arg_0: Arg_0 {O(n)}
(3: h->[1:g], g), Arg_1: 0 {O(1)}