Formalising Semantics for Expected Running Time of Probabilistic - PowerPoint PPT Presentation

Johannes Hölzl TU München, Germany Formalising Semantics for Expected Running Time of Probabilistic Programs (Rough Diamond)

Kaminski, Katoen, Matheja, and Olmedo [ESOP 2016] ⋆ Denotational: Operational: Correspondence: Denotational 2 clarified semantics – different proofs – fixed proofs • Coupon Collector • Simple Random Walk  • Examples: Operational pgcl stream measure pgcl • Probabilistic programs (pGCL) + expected running time. 0 0 pgcl • Two semantics: This Talk

Denotational: Operational: Correspondence: Denotational 2 stream measure clarified semantics – different proofs – fixed proofs • Coupon Collector • Simple Random Walk  • Examples: Operational pgcl pgcl • Probabilistic programs (pGCL) + expected running time. 0 0 pgcl • Two semantics: ⋆ This Talk Kaminski, Katoen, Matheja, and Olmedo [ESOP 2016]

Denotational: Operational: Correspondence: Denotational 2 stream measure clarified semantics – different proofs – fixed proofs • Coupon Collector • Simple Random Walk  • Examples: Operational pgcl pgcl • Probabilistic programs (pGCL) + expected running time. 0 0 pgcl • Two semantics: This Talk Kaminski, Katoen, Matheja, and Olmedo [ESOP 2016] ⋆

Denotational: Operational: Correspondence: Denotational 2 stream measure clarified semantics – different proofs – fixed proofs • Coupon Collector • Simple Random Walk  • Examples: Operational pgcl pgcl • Probabilistic programs (pGCL) + expected running time. 0 0 pgcl • Two semantics: This Talk Kaminski, Katoen, Matheja, and Olmedo [ESOP 2016] ⋆

Operational: Correspondence: Denotational 2 pgcl clarified semantics – different proofs – fixed proofs • Two semantics: • Coupon Collector • Simple Random Walk  • Examples: Operational stream measure pgcl • Probabilistic programs (pGCL) + expected running time. This Talk Kaminski, Katoen, Matheja, and Olmedo [ESOP 2016] ⋆ ( ) ( ) Denotational: σ pgcl ⇒ σ ⇒ R ≥ 0 ⇒ σ ⇒ R ≥ 0

Correspondence: Denotational 2 • Probabilistic programs (pGCL) + expected running time. clarified semantics – different proofs – fixed proofs • Two semantics: • Coupon Collector • Simple Random Walk  • Examples: Operational stream measure This Talk Kaminski, Katoen, Matheja, and Olmedo [ESOP 2016] ⋆ ( ) ( ) Denotational: σ pgcl ⇒ σ ⇒ R ≥ 0 ⇒ σ ⇒ R ≥ 0 ( ) ( ) Operational: σ pgcl × σ ⇒ σ pgcl × σ

2 • Probabilistic programs (pGCL) + expected running time. clarified semantics – different proofs – fixed proofs • Two semantics: • Coupon Collector • Simple Random Walk  • Examples: stream measure This Talk Kaminski, Katoen, Matheja, and Olmedo [ESOP 2016] ⋆ ( ) ( ) Denotational: σ pgcl ⇒ σ ⇒ R ≥ 0 ⇒ σ ⇒ R ≥ 0 ( ) ( ) Operational: σ pgcl × σ ⇒ σ pgcl × σ Correspondence: Denotational ⇔ Operational

2 • Probabilistic programs (pGCL) + expected running time. clarified semantics – different proofs – fixed proofs • Two semantics: • Coupon Collector • Simple Random Walk  • Examples: stream measure This Talk Kaminski, Katoen, Matheja, and Olmedo [ESOP 2016] ⋆ ( ) ( ) Denotational: σ pgcl ⇒ σ ⇒ R ≥ 0 ⇒ σ ⇒ R ≥ 0 ( ) ( ) Operational: σ pgcl × σ ⇒ σ pgcl × σ Correspondence: Denotational ⇔ Operational

2 • Probabilistic programs (pGCL) + expected running time. clarified semantics – different proofs – fixed proofs • Two semantics: • Coupon Collector • Examples: stream measure This Talk Kaminski, Katoen, Matheja, and Olmedo [ESOP 2016] ⋆ ( ) ( ) Denotational: σ pgcl ⇒ σ ⇒ R ≥ 0 ⇒ σ ⇒ R ≥ 0 ( ) ( ) Operational: σ pgcl × σ ⇒ σ pgcl × σ Correspondence: Denotational ⇔ Operational • Simple Random Walk 

2 • Probabilistic programs (pGCL) + expected running time. – different proofs – fixed proofs • Two semantics: clarified semantics • Coupon Collector • Examples: stream measure This Talk Kaminski, Katoen, Matheja, and Olmedo [ESOP 2016] ⋆ ( ) ( ) Denotational: σ pgcl ⇒ σ ⇒ R ≥ 0 ⇒ σ ⇒ R ≥ 0 ( ) ( ) Operational: σ pgcl × σ ⇒ σ pgcl × σ Correspondence: Denotational ⇔ Operational • Simple Random Walk 

2 • Probabilistic programs (pGCL) + expected running time. – fixed proofs • Two semantics: clarified semantics – different proofs • Coupon Collector • Examples: stream measure This Talk Kaminski, Katoen, Matheja, and Olmedo [ESOP 2016] ⋆ ( ) ( ) Denotational: σ pgcl ⇒ σ ⇒ R ≥ 0 ⇒ σ ⇒ R ≥ 0 ( ) ( ) Operational: σ pgcl × σ ⇒ σ pgcl × σ Correspondence: Denotational ⇔ Operational • Simple Random Walk 

2 • Probabilistic programs (pGCL) + expected running time. clarified semantics – different proofs – fixed proofs • Two semantics: • Coupon Collector • Examples: stream measure This Talk Kaminski, Katoen, Matheja, and Olmedo [ESOP 2016] ⋆ ( ) ( ) Denotational: σ pgcl ⇒ σ ⇒ R ≥ 0 ⇒ σ ⇒ R ≥ 0 ( ) ( ) Operational: σ pgcl × σ ⇒ σ pgcl × σ Correspondence: Denotational ⇔ Operational • Simple Random Walk 

x x p 1 p 2 p 1 p 2 ITE g p 1 p 2 g WHILE g DO p 3 bool Assign pmf expr or   Probabilistic Guarded Command Language (pGCL) σ pgcl = ⊥

x x p 1 p 2 p 1 p 2 ITE g p 1 p 2 g WHILE g DO p 3 bool Assign pmf expr or  Probabilistic Guarded Command Language (pGCL) σ pgcl = ⊥ | 

x x p 1 p 2 p 1 p 2 ITE g p 1 p 2 g WHILE g DO p 3 bool pmf Assign expr or Probabilistic Guarded Command Language (pGCL) σ pgcl = ⊥ |  | 

p 1 p 2 p 1 p 2 ITE g p 1 p 2 g WHILE g DO p 3 bool or Probabilistic Guarded Command Language (pGCL) σ pgcl = ⊥ |  |  x : ∼ D x := expr | ” Assign ( σ ⇒ σ pmf )”

p 1 p 2 ITE g p 1 p 2 g WHILE g DO p 3 bool or Probabilistic Guarded Command Language (pGCL) σ pgcl = ⊥ |  |  x : ∼ D x := expr | ” Assign ( σ ⇒ σ pmf )” p 1 ; p 2 |

ITE g p 1 p 2 g WHILE g DO p 3 or bool Probabilistic Guarded Command Language (pGCL) σ pgcl = ⊥ |  |  x : ∼ D x := expr | ” Assign ( σ ⇒ σ pmf )” p 1 ; p 2 | p 1 | p 2 |

WHILE g DO p 3 or Probabilistic Guarded Command Language (pGCL) σ pgcl = ⊥ |  |  x : ∼ D x := expr | ” Assign ( σ ⇒ σ pmf )” p 1 ; p 2 | p 1 | p 2 | ITE g p 1 p 2 g :: σ ⇒ bool |

3 or Probabilistic Guarded Command Language (pGCL) σ pgcl = ⊥ |  |  x : ∼ D x := expr | ” Assign ( σ ⇒ σ pmf )” p 1 ; p 2 | p 1 | p 2 | ITE g p 1 p 2 g :: σ ⇒ bool | WHILE g DO p |

Values computed for the a starting state Values we want assigned to a terminal state c c c c c ert Assign u c x c y d u x y ert p 1 p 2 c ert p 1 ert p 2 c ert p 1 p 2 c ert p 1 c ert p 2 c ert ITE g p 1 p 2 c x if g x then ert p 1 c x else ert p 2 c x ert WHILE g DO p c W x 1 if g x then ert p W x else c x lfp 1 4 1 0 ert  1 ert  ert Denotational Semantics (Expected Running Time) ert :: σ pgcl ⇒ ( σ ⇒ R ≥ 0 ) ⇒ ( σ ⇒ R ≥ 0 )

Values computed for the a starting state c c c c c ert Assign u c x c y d u x y ert p 1 p 2 c ert p 1 ert p 2 c ert p 1 p 2 c ert p 1 c ert p 2 c ert ITE g p 1 p 2 c x if g x then ert p 1 c x else ert p 2 c x ert WHILE g DO p c W x 1 if g x then ert p W x else c x 1 lfp 4 1 0 ert  1 ert  ert Denotational Semantics (Expected Running Time) ert :: σ pgcl ⇒ ( σ ⇒ R ≥ 0 ) ⇒ ( σ ⇒ R ≥ 0 ) Values we want assigned to a terminal state

c c c c c ert Assign u c x c y d u x y ert p 1 p 2 c ert p 1 ert p 2 c ert p 1 p 2 c ert p 1 c ert p 2 c ert ITE g p 1 p 2 c x if g x then ert p 1 c x else ert p 2 c x ert WHILE g DO p c W x 1 if g x then ert p W x else c x lfp 1 4 1 0 ert  1 ert  ert Denotational Semantics (Expected Running Time) Values computed for the a starting state ert :: σ pgcl ⇒ ( σ ⇒ R ≥ 0 ) ⇒ ( σ ⇒ R ≥ 0 ) Values we want assigned to a terminal state

Values computed for the a starting state Values we want assigned to a terminal state c c c ert Assign u c x c y d u x y ert p 1 p 2 c ert p 1 ert p 2 c ert p 1 p 2 c ert p 1 c ert p 2 c ert ITE g p 1 p 2 c x if g x then ert p 1 c x else ert p 2 c x ert WHILE g DO p c W x 1 if g x then ert p W x else c x lfp 1 4 1 0 ert  1 ert  Denotational Semantics (Expected Running Time) ert :: σ pgcl ⇒ ( σ ⇒ R ≥ 0 ) ⇒ ( σ ⇒ R ≥ 0 ) c c ert ⊥ =

Values computed for the a starting state Values we want assigned to a terminal state c ert Assign u c x c y d u x y ert p 1 p 2 c ert p 1 ert p 2 c ert p 1 p 2 c ert p 1 c ert p 2 c ert ITE g p 1 p 2 c x if g x then ert p 1 c x else ert p 2 c x ert WHILE g DO p c W x 1 if g x then ert p W x else c x 1 lfp 4 1 0 ert  ert  Denotational Semantics (Expected Running Time) ert :: σ pgcl ⇒ ( σ ⇒ R ≥ 0 ) ⇒ ( σ ⇒ R ≥ 0 ) c c ert ⊥ = c 1 + c =