Preprocessing
Found invariant 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location h
Found invariant 1 ≤ X₀ for location g
Probabilistic Analysis
Probabilistic Program after Preprocessing
Start: f
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: f, g, h
Transitions:
g₀:f(X₀,X₁,X₂) → t₁:g(1,X₁,X₂) :|:
g₂:g(X₀,X₁,X₂) → t₃:h(1+X₀,X₀-1,X₂) :|: 1+X₀ ≤ X₂ ∧ 1 ≤ X₀
g₄:h(X₀,X₁,X₂) → [1/2]:t₅:h(X₀,X₁-1,X₂) :+: [1/2]:t₆:h(X₀,X₁,X₂) :|: 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁
g₇:h(X₀,X₁,X₂) → t₈:g(X₀,X₁,X₂) :|: X₁ ≤ 0 ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁
Show Graph
G
f
f
g
g
f->g
p = 1
t₁ ∈ g₀
η (X₀) = 1
h
h
g->h
p = 1
t₃ ∈ g₂
η (X₀) = 1+X₀
η (X₁) = X₀-1
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂
h->g
p = 1
t₈ ∈ g₇
τ = 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
h->h
p = 1/2
t₅ ∈ g₄
η (X₁) = X₁-1
τ = 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁
h->h
p = 1/2
t₆ ∈ g₄
τ = 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁
Run classical analysis on SCC: [f]
Classical Approximation after Lifting Classical Results
All Bounds
Timebounds
Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₇: inf {Infinity}
Costbounds
Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₇: inf {Infinity}
Sizebounds
(g₀,g), X₀: 1 {O(1)}
(g₀,g), X₁: X₁ {O(n)}
(g₀,g), X₂: X₂ {O(n)}
Run probabilistic analysis on SCC: [f]
Run classical analysis on SCC: [g; h]
MPRF for transition t₃: g(X₀,X₁,X₂) → h(1+X₀,X₀-1,X₂) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ of depth 1:
new bound:
X₂+1 {O(n)}
MPRF:
• g: [X₂-X₀]
• h: [X₂-X₀]
Show Graph
G
f
f
g
g
f->g
t₁
η (X₀) = 1
h
h
g->h
t₃
η (X₀) = 1+X₀
η (X₁) = X₀-1
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂
h->g
t₈
τ = 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
h->h
t₅
η (X₁) = X₁-1
τ = 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁
h->h
t₆
τ = 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁
MPRF for transition t₈: h(X₀,X₁,X₂) → g(X₀,X₁,X₂) :|: 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 of depth 1:
new bound:
X₂+1 {O(n)}
MPRF:
• g: [X₂-X₀]
• h: [1+X₂-X₀]
Show Graph
G
f
f
g
g
f->g
t₁
η (X₀) = 1
h
h
g->h
t₃
η (X₀) = 1+X₀
η (X₁) = X₀-1
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂
h->g
t₈
τ = 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
h->h
t₅
η (X₁) = X₁-1
τ = 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁
h->h
t₆
τ = 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁
MPRF for transition t₅: h(X₀,X₁,X₂) → h(X₀,X₁-1,X₂) :|: 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁ of depth 1:
new bound:
X₂⋅X₂+3⋅X₂+3 {O(n^2)}
MPRF:
• g: [X₀]
• h: [1+X₁]
Show Graph
G
f
f
g
g
f->g
t₁
η (X₀) = 1
h
h
g->h
t₃
η (X₀) = 1+X₀
η (X₁) = X₀-1
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂
h->g
t₈
τ = 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
h->h
t₅
η (X₁) = X₁-1
τ = 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁
h->h
t₆
τ = 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁
Classical Approximation after Lifting Classical Results
All Bounds
Timebounds
Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₂: X₂+1 {O(n)}
g₄: inf {Infinity}
g₇: X₂+1 {O(n)}
Costbounds
Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₇: inf {Infinity}
Sizebounds
(g₀,g), X₀: 1 {O(1)}
(g₀,g), X₁: X₁ {O(n)}
(g₀,g), X₂: X₂ {O(n)}
(g₂,h), X₀: X₂+2 {O(n)}
(g₂,h), X₁: X₂+3 {O(n)}
(g₂,h), X₂: X₂ {O(n)}
(g₄,h), X₀: 2⋅X₂+4 {O(n)}
(g₄,h), X₁: 2⋅X₂+6 {O(n)}
(g₄,h), X₂: 2⋅X₂ {O(n)}
(g₇,g), X₀: X₂+2 {O(n)}
(g₇,g), X₁: 0 {O(1)}
(g₇,g), X₂: X₂ {O(n)}
Run probabilistic analysis on SCC: [g; h]
Plrf for transition g₄:h(X₀,X₁,X₂) → [1/2]:t₅:h(X₀,X₁-1,X₂) :+: [1/2]:t₆:h(X₀,X₁,X₂) :|: 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁:
new bound:
2⋅X₂⋅X₂+6⋅X₂+6 {O(n^2)}
PLRF:
• g: 2⋅X₀
• h: 2⋅X₁
Show Graph
G
f
f
g
g
f->g
p = 1
t₁ ∈ g₀
η (X₀) = 1
h
h
g->h
p = 1
t₃ ∈ g₂
η (X₀) = 1+X₀
η (X₁) = X₀-1
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂
h->g
p = 1
t₈ ∈ g₇
τ = 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
h->h
p = 1/2
t₅ ∈ g₄
η (X₁) = X₁-1
τ = 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁
h->h
p = 1/2
t₆ ∈ g₄
τ = 2 ≤ X₀ ∧ 2 ≤ X₀+X₁ ∧ 2+X₁ ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁
Use expected size bounds for entry point (g₀:f→[t₁:1:g],g)
Use classical time bound for entry point (g₀:f→[t₁:1:g],g)
Use expected size bounds for entry point (g₇:h→[t₈:1:g],g)
Use classical time bound for entry point (g₇:h→[t₈:1:g],g)
Results of Probabilistic Analysis
All Bounds
Timebounds
Overall timebound:2⋅X₂⋅X₂+8⋅X₂+9 {O(n^2)}
g₀: 1 {O(1)}
g₂: X₂+1 {O(n)}
g₄: 2⋅X₂⋅X₂+6⋅X₂+6 {O(n^2)}
g₇: X₂+1 {O(n)}
Costbounds
Overall costbound: 4⋅X₂⋅X₂+14⋅X₂+15 {O(n^2)}
g₀: 1 {O(1)}
g₂: X₂+1 {O(n)}
g₄: 4⋅X₂⋅X₂+12⋅X₂+12 {O(n^2)}
g₇: X₂+1 {O(n)}
Sizebounds
(g₀,g), X₀: 1 {O(1)}
(g₀,g), X₁: X₁ {O(n)}
(g₀,g), X₂: X₂ {O(n)}
(g₂,h), X₀: X₂+2 {O(n)}
(g₂,h), X₁: X₂+3 {O(n)}
(g₂,h), X₂: X₂ {O(n)}
(g₄,h), X₀: X₂+2 {O(n)}
(g₄,h), X₁: 2⋅X₂+6 {O(n)}
(g₄,h), X₂: X₂ {O(n)}
(g₇,g), X₀: X₂+2 {O(n)}
(g₇,g), X₁: 0 {O(1)}
(g₇,g), X₂: X₂ {O(n)}