Inferring Expected Runtimes Using Sizes

KoAT2 Proof WORST_CASE( ?, 2/5*Arg_0+23/5 {O(n)})

Initial Complexity Problem (after preprocessing)

Start:f
Program_Vars:Arg_0, Arg_1, Arg_2, Arg_3, Arg_4, Arg_5
Temp_Vars:valToFill
Locations:f, g, h, i, j
Transitions:
f(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -{0}> g(Arg_0,10,3,1,Arg_4,0)
g(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -{0}> h(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Uniform (0, 1)) :|: Arg_5<=0 && 1+Arg_5<=Arg_3 && Arg_3+Arg_5<=1 && 3+Arg_5<=Arg_2 && Arg_2+Arg_5<=3 && 10+Arg_5<=Arg_1 && Arg_1+Arg_5<=10 && 0<=Arg_5 && 1<=Arg_3+Arg_5 && Arg_3<=1+Arg_5 && 3<=Arg_2+Arg_5 && Arg_2<=3+Arg_5 && 10<=Arg_1+Arg_5 && Arg_1<=10+Arg_5 && Arg_3<=1 && 2+Arg_3<=Arg_2 && Arg_2+Arg_3<=4 && 9+Arg_3<=Arg_1 && Arg_1+Arg_3<=11 && 1<=Arg_3 && 4<=Arg_2+Arg_3 && Arg_2<=2+Arg_3 && 11<=Arg_1+Arg_3 && Arg_1<=9+Arg_3 && Arg_2<=3 && 7+Arg_2<=Arg_1 && Arg_1+Arg_2<=13 && 3<=Arg_2 && 13<=Arg_1+Arg_2 && Arg_1<=7+Arg_2 && Arg_1<=10 && 10<=Arg_1
h(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -{0}> i(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) :|: Arg_5<=Arg_0 && 0<=Arg_5 && 1<=Arg_3+Arg_5 && Arg_3<=1+Arg_5 && 3<=Arg_2+Arg_5 && Arg_2<=3+Arg_5 && 10<=Arg_1+Arg_5 && Arg_1<=10+Arg_5 && Arg_3<=1 && 2+Arg_3<=Arg_2 && Arg_2+Arg_3<=4 && 9+Arg_3<=Arg_1 && Arg_1+Arg_3<=11 && 1<=Arg_3 && 4<=Arg_2+Arg_3 && Arg_2<=2+Arg_3 && 11<=Arg_1+Arg_3 && Arg_1<=9+Arg_3 && Arg_2<=3 && 7+Arg_2<=Arg_1 && Arg_1+Arg_2<=13 && 3<=Arg_2 && 13<=Arg_1+Arg_2 && Arg_1<=7+Arg_2 && Arg_1<=10 && 10<=Arg_1
i(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -{0}> j(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Uniform (9, 10)) :|: valToFill<Arg_5+Arg_1 && Arg_5<=Arg_0 && 0<=Arg_5 && 1<=Arg_3+Arg_5 && Arg_3<=1+Arg_5 && 3<=Arg_2+Arg_5 && Arg_2<=3+Arg_5 && 10<=Arg_1+Arg_5 && Arg_1<=10+Arg_5 && 0<=Arg_0+Arg_5 && Arg_3<=1 && 2+Arg_3<=Arg_2 && Arg_2+Arg_3<=4 && 9+Arg_3<=Arg_1 && Arg_1+Arg_3<=11 && Arg_3<=1+Arg_0 && 1<=Arg_3 && 4<=Arg_2+Arg_3 && Arg_2<=2+Arg_3 && 11<=Arg_1+Arg_3 && Arg_1<=9+Arg_3 && 1<=Arg_0+Arg_3 && Arg_2<=3 && 7+Arg_2<=Arg_1 && Arg_1+Arg_2<=13 && Arg_2<=3+Arg_0 && 3<=Arg_2 && 13<=Arg_1+Arg_2 && Arg_1<=7+Arg_2 && 3<=Arg_0+Arg_2 && Arg_1<=10 && Arg_1<=10+Arg_0 && 10<=Arg_1 && 10<=Arg_0+Arg_1 && 0<=Arg_0
i(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -{0}> j(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Uniform (2, 4)) :|: valToFill<Arg_5+Arg_2 && Arg_5+Arg_1<=Arg_0 && Arg_5<=Arg_0 && 0<=Arg_5 && 1<=Arg_3+Arg_5 && Arg_3<=1+Arg_5 && 3<=Arg_2+Arg_5 && Arg_2<=3+Arg_5 && 10<=Arg_1+Arg_5 && Arg_1<=10+Arg_5 && 0<=Arg_0+Arg_5 && Arg_3<=1 && 2+Arg_3<=Arg_2 && Arg_2+Arg_3<=4 && 9+Arg_3<=Arg_1 && Arg_1+Arg_3<=11 && Arg_3<=1+Arg_0 && 1<=Arg_3 && 4<=Arg_2+Arg_3 && Arg_2<=2+Arg_3 && 11<=Arg_1+Arg_3 && Arg_1<=9+Arg_3 && 1<=Arg_0+Arg_3 && Arg_2<=3 && 7+Arg_2<=Arg_1 && Arg_1+Arg_2<=13 && Arg_2<=3+Arg_0 && 3<=Arg_2 && 13<=Arg_1+Arg_2 && Arg_1<=7+Arg_2 && 3<=Arg_0+Arg_2 && Arg_1<=10 && Arg_1<=10+Arg_0 && 10<=Arg_1 && 10<=Arg_0+Arg_1 && 0<=Arg_0
i(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -{0}> j(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Uniform (0, 4)) :|: Arg_5+Arg_2<=valToFill && Arg_5+Arg_1<=Arg_0 && Arg_5<=Arg_0 && 0<=Arg_5 && 1<=Arg_3+Arg_5 && Arg_3<=1+Arg_5 && 3<=Arg_2+Arg_5 && Arg_2<=3+Arg_5 && 10<=Arg_1+Arg_5 && Arg_1<=10+Arg_5 && 0<=Arg_0+Arg_5 && Arg_3<=1 && 2+Arg_3<=Arg_2 && Arg_2+Arg_3<=4 && 9+Arg_3<=Arg_1 && Arg_1+Arg_3<=11 && Arg_3<=1+Arg_0 && 1<=Arg_3 && 4<=Arg_2+Arg_3 && Arg_2<=2+Arg_3 && 11<=Arg_1+Arg_3 && Arg_1<=9+Arg_3 && 1<=Arg_0+Arg_3 && Arg_2<=3 && 7+Arg_2<=Arg_1 && Arg_1+Arg_2<=13 && Arg_2<=3+Arg_0 && 3<=Arg_2 && 13<=Arg_1+Arg_2 && Arg_1<=7+Arg_2 && 3<=Arg_0+Arg_2 && Arg_1<=10 && Arg_1<=10+Arg_0 && 10<=Arg_1 && 10<=Arg_0+Arg_1 && 0<=Arg_0
j(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -> h(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Uniform (0, 1)) :|: Arg_5<=10+Arg_0 && 0<=Arg_5 && 1<=Arg_3+Arg_5 && Arg_3<=1+Arg_5 && 3<=Arg_2+Arg_5 && Arg_2<=3+Arg_5 && 10<=Arg_1+Arg_5 && Arg_1<=10+Arg_5 && 9<=Arg_0+Arg_5 && Arg_3<=1 && 2+Arg_3<=Arg_2 && Arg_2+Arg_3<=4 && 9+Arg_3<=Arg_1 && Arg_1+Arg_3<=11 && Arg_3<=1+Arg_0 && 1<=Arg_3 && 4<=Arg_2+Arg_3 && Arg_2<=2+Arg_3 && 11<=Arg_1+Arg_3 && Arg_1<=9+Arg_3 && 1<=Arg_0+Arg_3 && Arg_2<=3 && 7+Arg_2<=Arg_1 && Arg_1+Arg_2<=13 && Arg_2<=3+Arg_0 && 3<=Arg_2 && 13<=Arg_1+Arg_2 && Arg_1<=7+Arg_2 && 3<=Arg_0+Arg_2 && Arg_1<=10 && Arg_1<=10+Arg_0 && 10<=Arg_1 && 10<=Arg_0+Arg_1 && 0<=Arg_0

G f f g g f->g t₀ ∈ g₀ η (Arg_1) = 10 η (Arg_2) = 3 η (Arg_3) = 1 η (Arg_5) = 0 {0} h h g->h t₁ ∈ g₁ η (Arg_5) = Uniform (0, 1) {0} i i h->i t₂ ∈ g₂ τ = Arg_5<=Arg_0 {0} j j i->j t₃ ∈ g₃ η (Arg_5) = Uniform (9, 10) τ = valToFill<Arg_5+Arg_1 {0} i->j t₄ ∈ g₄ η (Arg_5) = Uniform (2, 4) τ = valToFill<Arg_5+Arg_2 && Arg_5+Arg_1<=Arg_0 {0} i->j t₅ ∈ g₅ η (Arg_5) = Uniform (0, 4) τ = Arg_5+Arg_2<=valToFill && Arg_5+Arg_1<=Arg_0 {0} j->h t₆ ∈ g₆ η (Arg_5) = Uniform (0, 1)

Timebounds:

Overall timebound:inf {Infinity}
0,0: f->g: 1 {O(1)}
1,1: g->h: 1 {O(1)}
2,2: h->i: inf {Infinity}
3,3: i->j: max([0, 10+Arg_0]) {O(n)}
4,4: i->j: max([0, Arg_0]) {O(n)}
5,5: i->j: inf {Infinity}
6,6: j->h: inf {Infinity}

Expected Timebounds:

Overall expected timebound: 129/5+16/5*Arg_0 {O(n)}
0: f->[1:g]: 1 {O(1)}
1: g->[1:h]: 1 {O(1)}
2: h->[1:i]: 2/5*Arg_0+23/5 {O(n)}
3: i->[1:j]: 10+Arg_0 {O(n)}
4: i->[1:j]: Arg_0 {O(n)}
5: i->[1:j]: 2/5*Arg_0+23/5 {O(n)}
6: j->[1:h]: 2/5*Arg_0+23/5 {O(n)}

Costbounds:

Overall costbound: inf {Infinity}
0,0: f->g: inf {Infinity}
1,1: g->h: inf {Infinity}
2,2: h->i: inf {Infinity}
3,3: i->j: inf {Infinity}
4,4: i->j: inf {Infinity}
5,5: i->j: inf {Infinity}
6,6: j->h: inf {Infinity}

Expected Costbounds:

Overall expected costbound: 2/5*Arg_0+23/5 {O(n)}
0: f->[1:g]: 0 {O(1)}
1: g->[1:h]: 0 {O(1)}
2: h->[1:i]: 0 {O(1)}
3: i->[1:j]: 0 {O(1)}
4: i->[1:j]: 0 {O(1)}
5: i->[1:j]: 0 {O(1)}
6: j->[1:h]: 2/5*Arg_0+23/5 {O(n)}

Sizebounds:

0,0: f->g, Arg_0: Arg_0 {O(n)}
0,0: f->g, Arg_1: 10 {O(1)}
0,0: f->g, Arg_2: 3 {O(1)}
0,0: f->g, Arg_3: 1 {O(1)}
0,0: f->g, Arg_4: Arg_4 {O(n)}
0,0: f->g, Arg_5: 0 {O(1)}
1,1: g->h, Arg_0: Arg_0 {O(n)}
1,1: g->h, Arg_1: 10 {O(1)}
1,1: g->h, Arg_2: 3 {O(1)}
1,1: g->h, Arg_3: 1 {O(1)}
1,1: g->h, Arg_4: Arg_4 {O(n)}
1,1: g->h, Arg_5: 1 {O(1)}
2,2: h->i, Arg_0: Arg_0 {O(n)}
2,2: h->i, Arg_1: 10 {O(1)}
2,2: h->i, Arg_2: 3 {O(1)}
2,2: h->i, Arg_3: 1 {O(1)}
2,2: h->i, Arg_4: Arg_4 {O(n)}
3,3: i->j, Arg_0: Arg_0 {O(n)}
3,3: i->j, Arg_1: 10 {O(1)}
3,3: i->j, Arg_2: 3 {O(1)}
3,3: i->j, Arg_3: 1 {O(1)}
3,3: i->j, Arg_4: Arg_4 {O(n)}
4,4: i->j, Arg_0: Arg_0 {O(n)}
4,4: i->j, Arg_1: 10 {O(1)}
4,4: i->j, Arg_2: 3 {O(1)}
4,4: i->j, Arg_3: 1 {O(1)}
4,4: i->j, Arg_4: Arg_4 {O(n)}
5,5: i->j, Arg_0: Arg_0 {O(n)}
5,5: i->j, Arg_1: 10 {O(1)}
5,5: i->j, Arg_2: 3 {O(1)}
5,5: i->j, Arg_3: 1 {O(1)}
5,5: i->j, Arg_4: Arg_4 {O(n)}
6,6: j->h, Arg_0: Arg_0 {O(n)}
6,6: j->h, Arg_1: 10 {O(1)}
6,6: j->h, Arg_2: 3 {O(1)}
6,6: j->h, Arg_3: 1 {O(1)}
6,6: j->h, Arg_4: Arg_4 {O(n)}

ExpSizeBounds:

(0: f->[1:g], g), valToFill: valToFill {O(n)}
(0: f->[1:g], g), Arg_0: Arg_0 {O(n)}
(0: f->[1:g], g), Arg_1: 10 {O(1)}
(0: f->[1:g], g), Arg_2: 3 {O(1)}
(0: f->[1:g], g), Arg_3: 1 {O(1)}
(0: f->[1:g], g), Arg_4: Arg_4 {O(n)}
(0: f->[1:g], g), Arg_5: 0 {O(1)}
(1: g->[1:h], h), valToFill: valToFill {O(n)}
(1: g->[1:h], h), Arg_0: Arg_0 {O(n)}
(1: g->[1:h], h), Arg_1: 10 {O(1)}
(1: g->[1:h], h), Arg_2: 3 {O(1)}
(1: g->[1:h], h), Arg_3: 1 {O(1)}
(1: g->[1:h], h), Arg_4: Arg_4 {O(n)}
(1: g->[1:h], h), Arg_5: 1/2 {O(1)}
(2: h->[1:i], i), valToFill: valToFill {O(n)}
(2: h->[1:i], i), Arg_0: Arg_0 {O(n)}
(2: h->[1:i], i), Arg_1: 10 {O(1)}
(2: h->[1:i], i), Arg_2: 3 {O(1)}
(2: h->[1:i], i), Arg_3: 1 {O(1)}
(2: h->[1:i], i), Arg_4: Arg_4 {O(n)}
(2: h->[1:i], i), Arg_5: 1/2+19/2*(10+Arg_0)+3*Arg_0+5/2*(2/5*Arg_0+23/5) {O(n)}
(3: i->[1:j], j), valToFill: valToFill {O(n)}
(3: i->[1:j], j), Arg_0: Arg_0 {O(n)}
(3: i->[1:j], j), Arg_1: 10 {O(1)}
(3: i->[1:j], j), Arg_2: 3 {O(1)}
(3: i->[1:j], j), Arg_3: 1 {O(1)}
(3: i->[1:j], j), Arg_4: Arg_4 {O(n)}
(3: i->[1:j], j), Arg_5: 1/2+19/2*(10+Arg_0)+3*Arg_0+5/2*(2/5*Arg_0+23/5) {O(n)}
(4: i->[1:j], j), valToFill: valToFill {O(n)}
(4: i->[1:j], j), Arg_0: Arg_0 {O(n)}
(4: i->[1:j], j), Arg_1: 10 {O(1)}
(4: i->[1:j], j), Arg_2: 3 {O(1)}
(4: i->[1:j], j), Arg_3: 1 {O(1)}
(4: i->[1:j], j), Arg_4: Arg_4 {O(n)}
(4: i->[1:j], j), Arg_5: 1/2+19/2*(10+Arg_0)+3*Arg_0+5/2*(2/5*Arg_0+23/5) {O(n)}
(5: i->[1:j], j), valToFill: valToFill {O(n)}
(5: i->[1:j], j), Arg_0: Arg_0 {O(n)}
(5: i->[1:j], j), Arg_1: 10 {O(1)}
(5: i->[1:j], j), Arg_2: 3 {O(1)}
(5: i->[1:j], j), Arg_3: 1 {O(1)}
(5: i->[1:j], j), Arg_4: Arg_4 {O(n)}
(5: i->[1:j], j), Arg_5: 1/2+19/2*(10+Arg_0)+3*Arg_0+5/2*(2/5*Arg_0+23/5) {O(n)}
(6: j->[1:h], h), valToFill: valToFill {O(n)}
(6: j->[1:h], h), Arg_0: Arg_0 {O(n)}
(6: j->[1:h], h), Arg_1: 10 {O(1)}
(6: j->[1:h], h), Arg_2: 3 {O(1)}
(6: j->[1:h], h), Arg_3: 1 {O(1)}
(6: j->[1:h], h), Arg_4: Arg_4 {O(n)}
(6: j->[1:h], h), Arg_5: 1/2+19/2*(10+Arg_0)+3*Arg_0+5/2*(2/5*Arg_0+23/5) {O(n)}