A Formalized Hierarchy of Probabilistic System Types Proof Pearl - PowerPoint PPT Presentation

A Formalized Hierarchy of Probabilistic System Types Proof Pearl olzl 1 , Andreas Lochbihler 2 , and Dmitriy Traytel 1 , 2 Johannes H¨ 1 Institut f¨ ur Informatik 2 Institute of Information Security TU M¨ unchen, Germany ETH Zurich, Switzerland ITP 2015

Zoo of Probabilistic System Types Det. automaton H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

Zoo of Probabilistic System Types Non-det. automaton Det. automaton H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

Zoo of Probabilistic System Types Labeled Markov chain Non-det. automaton Det. automaton H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

Zoo of Probabilistic System Types Labeled Markov decision process Labeled Markov chain Non-det. automaton Det. automaton H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

Zoo of Probabilistic System Types Labeled Markov decision process Reactive system Labeled Markov chain Non-det. automaton Det. automaton H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

Zoo of Probabilistic System Types Labeled Markov decision process Reactive system Labeled Markov chain Non-det. automaton Generative system Det. automaton H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

Zoo of Probabilistic System Types Labeled Markov decision process Reactive system Stratified system Labeled Markov chain Non-det. automaton Generative system Det. automaton H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

Zoo of Probabilistic System Types Labeled Markov decision process Reactive system Stratified system Alternating system Labeled Markov chain Non-det. automaton Generative system Det. automaton H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

Zoo of Probabilistic System Types Labeled Markov decision process Reactive system Stratified system Alternating system Segala system Labeled Markov chain Non-det. automaton Generative system Det. automaton H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

Zoo of Probabilistic System Types Labeled Markov decision process Reactive system Stratified system Alternating system Simple Segala system Segala system Labeled Markov chain Non-det. automaton Generative system Det. automaton H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

Zoo of Probabilistic System Types Labeled Markov decision process Reactive system Stratified system Alternating system Bundle system Simple Segala system Segala system Labeled Markov chain Non-det. automaton Generative system Det. automaton H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

Zoo of Probabilistic System Types Labeled Markov decision process Reactive system Stratified system Alternating system Pnueli-Zuck system Bundle system Simple Segala system Segala system Labeled Markov chain Non-det. automaton Generative system Det. automaton H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

Zoo of Probabilistic System Types Labeled Markov Most general system decision process Reactive system Stratified system Alternating system Pnueli-Zuck system Bundle system Simple Segala system Segala system Labeled Markov chain Non-det. automaton Generative system Det. automaton H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

Hierarchy of Probabilistic System Types Ana Sokolva – Coalgebraic Analysis of Probabilistic Systems (2005): H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 3 / 19

Hierarchy of Probabilistic Systems Types How to . . . H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

Hierarchy of Probabilistic Systems Types How to . . . . . . model system types? H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

Hierarchy of Probabilistic Systems Types How to . . . . . . model system types? . . . compare systems of same type? H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

Hierarchy of Probabilistic Systems Types How to . . . . . . model system types? . . . compare systems of same type? . . . compare different system types? H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

Hierarchy of Probabilistic Systems Types How to . . . . . . model system types? Coalgebras . . . compare systems of same type? . . . compare different system types? H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

Hierarchy of Probabilistic Systems Types How to . . . . . . model system types? Coalgebras . . . compare systems of same type? Bisimulation . . . compare different system types? H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

Hierarchy of Probabilistic Systems Types How to . . . . . . model system types? Coalgebras . . . compare systems of same type? Bisimulation . . . compare different system types? Embedding respecting bisimulation H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

Hierarchy of Probabilistic Systems Types How to . . . . . . model system types? Coalgebras . . . compare systems of same type? Bisimulation . . . compare different system types? Embedding respecting bisimulation H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

Hierarchy of Probabilistic Systems Types How to . . . . . . model system types? Coalgebras . . . compare systems of same type? Bisimulation . . . compare different system types? Embedding respecting bisimulation . . . formalize it in Isabelle/HOL? H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

Hierarchy of Probabilistic Systems Types How to . . . . . . model system types? Coalgebras . . . compare systems of same type? Bisimulation . . . compare different system types? Embedding respecting bisimulation . . . formalize it in Isabelle/HOL? codatatype + Probability Mass Func. + Eisbach H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

Coalgebras ◮ Functor F describes the system type H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 5 / 19

Coalgebras ◮ Functor F describes the system type ◮ Examples: H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 5 / 19

Coalgebras ◮ Functor F describes the system type ◮ Examples: Deterministic System α × ( β ⇒ � ) H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 5 / 19

Coalgebras ◮ Functor F describes the system type ◮ Examples: Deterministic System α × ( β ⇒ � ) Non-Deterministic System α × ( β ⇒ � set ) H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 5 / 19

Coalgebras ◮ Functor F describes the system type ◮ Examples: Deterministic System α × ( β ⇒ � ) Non-Deterministic System α × ( β ⇒ � set ) ◮ System ( σ, s ) of type F : σ type of states, s :: σ ⇒ σ F transition system H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 5 / 19

Coalgebras ◮ Functor F describes the system type ◮ Examples: Deterministic System α × ( β ⇒ � ) Non-Deterministic System α × ( β ⇒ � set ) ◮ System ( σ, s ) of type F : σ type of states, s :: σ ⇒ σ F transition system ◮ ( σ, s ) is a F -coalgebra H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 5 / 19

Types of Transition System � Functor Property H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 6 / 19

Types of Transition System � q � p r � Functor Property ◮ Probability p � pmf H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 6 / 19

Types of Transition System � q α � p r � Functor Property ◮ Probability p � pmf ◮ Label α α × ( � pmf ) H¨ olzl, Lochbihler & Traytel A Formalized Hierarchy of Probabilistic System Types ITP 2015 6 / 19