Preprocessing

Probabilistic Analysis

Probabilistic Program after Preprocessing

Start: f
Program_Vars: X₀, X₁
Temp_Vars:
Locations: f, g
Transitions:
g₁:f(X₀,X₁) -{0}> t₂:g(X₀,X₁) :|:
g₃:g(X₀,X₁) → t₄:g(X₀+Binomial (1, 1/2),X₁) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ 1

Run classical analysis on SCC: [f]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:inf {Infinity}
g₁: 1 {O(1)}
g₃: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
g₁: inf {Infinity}
g₃: inf {Infinity}

Sizebounds

(g₁,g), X₀: X₀ {O(n)}
(g₁,g), X₁: X₁ {O(n)}

Run probabilistic analysis on SCC: [f]

Run classical analysis on SCC: [g]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:inf {Infinity}
g₁: 1 {O(1)}
g₃: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
g₁: inf {Infinity}
g₃: inf {Infinity}

Sizebounds

(g₁,g), X₀: X₀ {O(n)}
(g₁,g), X₁: X₁ {O(n)}
(g₃,g), X₁: X₁ {O(n)}

Run probabilistic analysis on SCC: [g]

Plrf for transition g₃:g(X₀,X₁) → t₄:g(X₀+Binomial (1, 1/2),X₁) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ 1:

new bound:

2⋅X₀+2⋅X₁ {O(n)}

PLRF:

• g: 2⋅X₁-2⋅X₀

Use expected size bounds for entry point (g₁:f→[t₂:1:g],g)
Use expected size bounds for entry point (g₁:f→[t₂:1:g],g)
Use classical time bound for entry point (g₁:f→[t₂:1:g],g)

Results of Probabilistic Analysis

All Bounds

Timebounds

Overall timebound:2⋅X₀+2⋅X₁+1 {O(n)}
g₁: 1 {O(1)}
g₃: 2⋅X₀+2⋅X₁ {O(n)}

Costbounds

Overall costbound: 2⋅X₀+2⋅X₁ {O(n)}
g₁: 0 {O(1)}
g₃: 2⋅X₀+2⋅X₁ {O(n)}

Sizebounds

(g₁,g), X₀: X₀ {O(n)}
(g₁,g), X₁: X₁ {O(n)}
(g₃,g), X₀: 2⋅X₀+X₁ {O(n)}
(g₃,g), X₁: X₁ {O(n)}