Preprocessing

Found invariant 0 ≤ X₀ for location l1

Found invariant 1 ≤ X₀ for location l2

Probabilistic Analysis

Probabilistic Program after Preprocessing

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
g₀:l0(X₀,X₁) → t₁:l1(1,X₁) :|:
g₂:l1(X₀,X₁) → t₃:l2(X₀,X₁) :|: 1 ≤ X₀ ∧ 0 ≤ X₀
g₄:l2(X₀,X₁) → t₅:l1(X₀-1,X₁-10) :|: 101 ≤ X₁ ∧ 1 ≤ X₀
g₆:l2(X₀,X₁) → t₇:l1(1+X₀,11+X₁) :|: X₁ ≤ 100 ∧ 1 ≤ X₀

Run classical analysis on SCC: [l0]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₆: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₆: inf {Infinity}

Sizebounds

(g₀,l1), X₀: 1 {O(1)}
(g₀,l1), X₁: X₁ {O(n)}

Run probabilistic analysis on SCC: [l0]

Run classical analysis on SCC: [l1; l2]

MPRF for transition t₇: l2(X₀,X₁) → l1(1+X₀,11+X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ 100 of depth 1:

new bound:

X₁+101 {O(n)}

MPRF:

• l1: [91+10⋅X₀-X₁]
• l2: [91+10⋅X₀-X₁]

MPRF for transition t₃: l1(X₀,X₁) → l2(X₀,X₁) :|: 0 ≤ X₀ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₁⋅X₁+204⋅X₁+10405 {O(n^2)}

MPRF:

• l1: [1+X₀]
• l2: [X₀]

MPRF for transition t₅: l2(X₀,X₁) → l1(X₀-1,X₁-10) :|: 1 ≤ X₀ ∧ 101 ≤ X₁ of depth 1:

new bound:

X₁⋅X₁+203⋅X₁+10303 {O(n^2)}

MPRF:

• l1: [X₀]
• l2: [X₀]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:2⋅X₁⋅X₁+408⋅X₁+20810 {O(n^2)}
g₀: 1 {O(1)}
g₂: X₁⋅X₁+204⋅X₁+10405 {O(n^2)}
g₄: X₁⋅X₁+203⋅X₁+10303 {O(n^2)}
g₆: X₁+101 {O(n)}

Costbounds

Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₆: inf {Infinity}

Sizebounds

(g₀,l1), X₀: 1 {O(1)}
(g₀,l1), X₁: X₁ {O(n)}
(g₂,l2), X₀: X₁+102 {O(n)}
(g₂,l2), X₁: 12⋅X₁+1111 {O(n)}
(g₄,l1), X₀: X₁+102 {O(n)}
(g₄,l1), X₁: 12⋅X₁+1111 {O(n)}
(g₆,l1), X₀: X₁+102 {O(n)}
(g₆,l1), X₁: 12⋅X₁+1111 {O(n)}

Run probabilistic analysis on SCC: [l1; l2]

Results of Probabilistic Analysis

All Bounds

Timebounds

Overall timebound:2⋅X₁⋅X₁+408⋅X₁+20810 {O(n^2)}
g₀: 1 {O(1)}
g₂: X₁⋅X₁+204⋅X₁+10405 {O(n^2)}
g₄: X₁⋅X₁+203⋅X₁+10303 {O(n^2)}
g₆: X₁+101 {O(n)}

Costbounds

Overall costbound: 2⋅X₁⋅X₁+408⋅X₁+20810 {O(n^2)}
g₀: 1 {O(1)}
g₂: X₁⋅X₁+204⋅X₁+10405 {O(n^2)}
g₄: X₁⋅X₁+203⋅X₁+10303 {O(n^2)}
g₆: X₁+101 {O(n)}

Sizebounds

(g₀,l1), X₀: 1 {O(1)}
(g₀,l1), X₁: X₁ {O(n)}
(g₂,l2), X₀: X₁+102 {O(n)}
(g₂,l2), X₁: 12⋅X₁+1111 {O(n)}
(g₄,l1), X₀: X₁+102 {O(n)}
(g₄,l1), X₁: 12⋅X₁+1111 {O(n)}
(g₆,l1), X₀: X₁+102 {O(n)}
(g₆,l1), X₁: 12⋅X₁+1111 {O(n)}