Inferring Expected Runtimes Using Sizes


KoAT2 Proof MAYBE

Initial Complexity Problem (after preprocessing)

Start:f
Program_Vars:Arg_0, Arg_1, Arg_2, Arg_3
Temp_Vars:numShare
Locations:f, g, h, i
Transitions:
f(Arg_0,Arg_1,Arg_2,Arg_3) -> g(Arg_0,Arg_1,Arg_2,0) :|: 0<=Arg_0
h(Arg_0,Arg_1,Arg_2,Arg_3) -> i(Arg_0,Arg_1,Uniform (0, 3),Arg_3) :|: 0<=Arg_3 && 0<=Arg_1+Arg_3 && 0<=Arg_0+Arg_3 && 0<=Arg_1 && 0<=Arg_0+Arg_1 && Arg_0<=Arg_1 && 0<=Arg_0 && 0<=Arg_3 && 0<=Arg_1+Arg_3 && 0<=Arg_0+Arg_3 && 0<=Arg_1 && 0<=Arg_0+Arg_1 && Arg_0<=Arg_1 && 0<=Arg_0
i(Arg_0,Arg_1,Arg_2,Arg_3) -> i(Arg_0,Arg_1,Arg_2-1,Arg_3+Arg_1) :|: 0<Arg_2 && 0<=Arg_3 && 0<=Arg_1+Arg_3 && 0<=Arg_0+Arg_3 && 0<=Arg_1 && 0<=Arg_0+Arg_1 && Arg_0<=Arg_1 && 0<=Arg_0 && 0<=Arg_3 && 0<=Arg_1+Arg_3 && 0<=Arg_0+Arg_3 && 0<=Arg_1 && 0<=Arg_0+Arg_1 && Arg_0<=Arg_1 && 0<=Arg_0
i(Arg_0,Arg_1,Arg_2,Arg_3) -> g(Arg_0,Arg_1,Arg_2,Arg_3) :|: numShare<1 && 0<=Arg_3 && 0<=Arg_1+Arg_3 && 0<=Arg_0+Arg_3 && 0<=Arg_1 && 0<=Arg_0+Arg_1 && Arg_0<=Arg_1 && 0<=Arg_0 && 0<=Arg_3 && 0<=Arg_1+Arg_3 && 0<=Arg_0+Arg_3 && 0<=Arg_1 && 0<=Arg_0+Arg_1 && Arg_0<=Arg_1 && 0<=Arg_0
f(Arg_0,Arg_1,Arg_2,Arg_3) -{2}> 1/4:h(Arg_0,Arg_1+1,Arg_2,0) :+: 3/4:h(Arg_0,Arg_1-1,Arg_2,0) :|: 0<=Arg_0 && Arg_0<Arg_1 && 0<=0 && 0<=Arg_0 && 0<=Arg_0
i(Arg_0,Arg_1,Arg_2,Arg_3) -{2}> 1/4:h(Arg_0,Arg_1+1,Arg_2,Arg_3) :+: 3/4:h(Arg_0,Arg_1-1,Arg_2,Arg_3) :|: numShare<1 && 0<=Arg_3 && 0<=Arg_1+Arg_3 && 0<=Arg_0+Arg_3 && 0<=Arg_1 && 0<=Arg_0+Arg_1 && Arg_0<=Arg_1 && 0<=Arg_0 && Arg_0<Arg_1 && 0<=Arg_3 && 0<=Arg_0+Arg_3 && 0<=Arg_0 && 0<=Arg_3 && 0<=Arg_1+Arg_3 && 0<=Arg_0+Arg_3 && 0<=Arg_1 && 0<=Arg_0+Arg_1 && Arg_0<=Arg_1 && 0<=Arg_0

G f f g g f->g t₀ ∈ g₀ η (Arg_3) = 0 τ = 0<=Arg_0 h h f->h t₆ ∈ g₅ p = 1/4 η (Arg_1) = Arg_1+1 η (Arg_3) = 0 τ = 0<=Arg_0 && Arg_0<Arg_1 {2} f->h t₇ ∈ g₅ p = 3/4 η (Arg_1) = Arg_1-1 η (Arg_3) = 0 τ = 0<=Arg_0 && Arg_0<Arg_1 {2} i i h->i t₃ ∈ g₂ η (Arg_2) = Uniform (0, 3) i->g t₅ ∈ g₄ τ = numShare<1 i->h t₈ ∈ g₆ p = 1/4 η (Arg_1) = Arg_1+1 τ = numShare<1 && Arg_0<Arg_1 {2} i->h t₉ ∈ g₆ p = 3/4 η (Arg_1) = Arg_1-1 τ = numShare<1 && Arg_0<Arg_1 {2} i->i t₄ ∈ g₃ η (Arg_2) = Arg_2-1 η (Arg_3) = Arg_3+Arg_1 τ = 0<Arg_2

Timebounds:

Overall timebound:inf {Infinity}
0,0: f->g: 1 {O(1)}
6,5: f->h: 1 {O(1)}
7,5: f->h: 1 {O(1)}
3,2: h->i: inf {Infinity}
4,3: i->i: inf {Infinity}
5,4: i->g: 1 {O(1)}
8,6: i->h: inf {Infinity}
9,6: i->h: inf {Infinity}

Expected Timebounds:

Overall expected timebound: inf {Infinity}
0: f->[1:g]: 1 {O(1)}
2: h->[1:i]: 2*(1+4*Arg_1) {O(n)}
3: i->[1:i]: inf {Infinity}
4: i->[1:g]: 1 {O(1)}
5: f->[1/4:h; 3/4:h]: 1 {O(1)}
6: i->[1/4:h; 3/4:h]: 8*Arg_1 {O(n)}

Costbounds:

Overall costbound: inf {Infinity}
0,0: f->g: inf {Infinity}
6,5: f->h: inf {Infinity}
7,5: f->h: inf {Infinity}
3,2: h->i: inf {Infinity}
4,3: i->i: inf {Infinity}
5,4: i->g: inf {Infinity}
8,6: i->h: inf {Infinity}
9,6: i->h: inf {Infinity}

Expected Costbounds:

Overall expected costbound: inf {Infinity}
0: f->[1:g]: 1 {O(1)}
2: h->[1:i]: 2*(1+4*Arg_1) {O(n)}
3: i->[1:i]: inf {Infinity}
4: i->[1:g]: 1 {O(1)}
5: f->[1/4:h; 3/4:h]: 2 {O(1)}
6: i->[1/4:h; 3/4:h]: 16*Arg_1 {O(n)}

Sizebounds:

0,0: f->g, Arg_0: Arg_0 {O(n)}
0,0: f->g, Arg_1: Arg_1 {O(n)}
0,0: f->g, Arg_2: Arg_2 {O(n)}
0,0: f->g, Arg_3: 0 {O(1)}
6,5: f->h, Arg_0: Arg_0 {O(n)}
6,5: f->h, Arg_1: 2*Arg_1 {O(n)}
6,5: f->h, Arg_2: Arg_2 {O(n)}
6,5: f->h, Arg_3: 0 {O(1)}
7,5: f->h, Arg_0: Arg_0 {O(n)}
7,5: f->h, Arg_1: Arg_1 {O(n)}
7,5: f->h, Arg_2: Arg_2 {O(n)}
7,5: f->h, Arg_3: 0 {O(1)}
3,2: h->i, Arg_0: Arg_0 {O(n)}
4,3: i->i, Arg_0: Arg_0 {O(n)}
5,4: i->g, Arg_0: Arg_0 {O(n)}
8,6: i->h, Arg_0: Arg_0 {O(n)}
9,6: i->h, Arg_0: Arg_0 {O(n)}

ExpSizeBounds:

(0: f->[1:g], g), numShare: numShare {O(n)}
(0: f->[1:g], g), Arg_0: Arg_0 {O(n)}
(0: f->[1:g], g), Arg_1: Arg_1 {O(n)}
(0: f->[1:g], g), Arg_2: Arg_2 {O(n)}
(0: f->[1:g], g), Arg_3: 0 {O(1)}
(2: h->[1:i], i), numShare: numShare {O(n)}
(2: h->[1:i], i), Arg_0: Arg_0 {O(n)}
(2: h->[1:i], i), Arg_1: 10*Arg_1 {O(n)}
(3: i->[1:i], i), numShare: numShare {O(n)}
(3: i->[1:i], i), Arg_0: Arg_0 {O(n)}
(3: i->[1:i], i), Arg_1: 10*Arg_1 {O(n)}
(4: i->[1:g], g), numShare: 2*numShare {O(n)}
(4: i->[1:g], g), Arg_0: Arg_0 {O(n)}
(4: i->[1:g], g), Arg_1: 20*Arg_1 {O(n)}
(5: f->[1/4:h; 3/4:h], h), numShare: numShare {O(n)}
(5: f->[1/4:h; 3/4:h], h), Arg_0: Arg_0 {O(n)}
(5: f->[1/4:h; 3/4:h], h), Arg_1: 2*Arg_1 {O(n)}
(5: f->[1/4:h; 3/4:h], h), Arg_2: Arg_2 {O(n)}
(5: f->[1/4:h; 3/4:h], h), Arg_3: 0 {O(1)}
(6: i->[1/4:h; 3/4:h], h), numShare: numShare {O(n)}
(6: i->[1/4:h; 3/4:h], h), Arg_0: Arg_0 {O(n)}
(6: i->[1/4:h; 3/4:h], h), Arg_1: 10*Arg_1 {O(n)}