Inferring Expected Runtimes Using Sizes


KoAT2 Proof MAYBE

Initial Complexity Problem (after preprocessing)

Start:f
Program_Vars:Arg_0, Arg_1, Arg_2
Temp_Vars:
Locations:f, g, h
Transitions:
f(Arg_0,Arg_1,Arg_2) -> g(1,Arg_1,Arg_2)
g(Arg_0,Arg_1,Arg_2) -> h(Arg_0,Arg_2,Arg_2) :|: 0<Arg_0 && Arg_0<=1 && 0<=Arg_0
h(Arg_0,Arg_1,Arg_2) -> h(Arg_0,Arg_1-Arg_0,Arg_2) :|: 0<Arg_1 && Arg_1<=Arg_2 && Arg_0<=1 && 1<=Arg_0
h(Arg_0,Arg_1,Arg_2) -> 1/2:g(1,Arg_1,Arg_2) :+: 1/2:g(0,Arg_1,Arg_2) :|: Arg_1<1 && Arg_1<=Arg_2 && Arg_0<=1 && 1<=Arg_0

G f f g g f->g t₀ ∈ g₀ η (Arg_0) = 1 h h g->h t₁ ∈ g₁ η (Arg_1) = Arg_2 τ = 0<Arg_0 h->g t₃ ∈ g₃ p = 1/2 η (Arg_0) = 1 τ = Arg_1<1 h->g t₄ ∈ g₃ p = 1/2 η (Arg_0) = 0 τ = Arg_1<1 h->h t₂ ∈ g₂ η (Arg_1) = Arg_1-Arg_0 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}
3,3: h->g: inf {Infinity}
4,3: h->g: 1 {O(1)}

Expected Timebounds:

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

Costbounds:

Overall costbound: inf {Infinity}
0,0: f->g: inf {Infinity}
1,1: g->h: inf {Infinity}
2,2: h->h: inf {Infinity}
3,3: h->g: 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 {O(1)}
2: h->[1:h]: inf {Infinity}
3: h->[1/2:g; 1/2:g]: 2 {O(1)}

Sizebounds:

0,0: f->g, Arg_0: 1 {O(1)}
0,0: f->g, Arg_1: Arg_1 {O(n)}
0,0: f->g, Arg_2: Arg_2 {O(n)}
1,1: g->h, Arg_0: 1 {O(1)}
1,1: g->h, Arg_1: Arg_2 {O(n)}
1,1: g->h, Arg_2: Arg_2 {O(n)}
2,2: h->h, Arg_0: 1 {O(1)}
2,2: h->h, Arg_1: Arg_2 {O(n)}
2,2: h->h, Arg_2: Arg_2 {O(n)}
3,3: h->g, Arg_0: 1 {O(1)}
3,3: h->g, Arg_1: 0 {O(1)}
3,3: h->g, Arg_2: Arg_2 {O(n)}
4,3: h->g, Arg_0: 0 {O(1)}
4,3: h->g, Arg_1: 0 {O(1)}
4,3: h->g, Arg_2: Arg_2 {O(n)}

ExpSizeBounds:

(0: f->[1:g], g), Arg_0: 1 {O(1)}
(0: f->[1:g], g), Arg_1: Arg_1 {O(n)}
(0: f->[1:g], g), Arg_2: Arg_2 {O(n)}
(1: g->[1:h], h), Arg_0: 1 {O(1)}
(1: g->[1:h], h), Arg_1: Arg_2 {O(n)}
(1: g->[1:h], h), Arg_2: Arg_2 {O(n)}
(2: h->[1:h], h), Arg_0: 1 {O(1)}
(2: h->[1:h], h), Arg_1: Arg_2 {O(n)}
(2: h->[1:h], h), Arg_2: Arg_2 {O(n)}
(3: h->[1/2:g; 1/2:g], g), Arg_0: 1 {O(1)}
(3: h->[1/2:g; 1/2:g], g), Arg_1: 0 {O(1)}
(3: h->[1/2:g; 1/2:g], g), Arg_2: Arg_2 {O(n)}