How to make a logic probabilistic? Pedro Baltazar SQIG - IT, Lisbon - PowerPoint PPT Presentation

How to make a logic probabilistic? Pedro Baltazar SQIG - IT, Lisbon - Portugal pedro.baltazar@ist.utl.pt CMU, CMACS Seminar - January 14th, 2010

Sources: D. Henriques, M. Biscaia, P. Baltazar, and P. Mateus, Probabilistic quantified linear temporal logic: Model checking, SAT and complete Hilbert calculus . submitted for publication. P. Baltazar and P. Mateus. Temporalization of probabilistic propositional logic . LFCS 2009, LNCS, 2009. P. Baltazar, P. Mateus, R. Nagarajan, and N. Papanikolaou. Exogenous probabilistic computation tree logic . Electronic Notes in Theoretical Computer Science, 190(3) : 95–110, 2007.

CPS : Cyber-Physical Systems � ❅ � ❅ � ❅ � ❅ � ❅

CPS : Cyber-Physical Systems ✬ ✩ System_pc{ // syntax • language; - specification; // theory or/and ✫ ✪ - model(s) } // semantics � ❅ � ❅ � ❅ � ❅ � ❅

CPS : Cyber-Physical Systems ✬ ✩ System_pc{ // syntax • language; - specification; // theory or/and ✫ ✪ - model(s) } // semantics � ❅ � ❅ � ❅ ✛ ✘ � ❅ � ❅ ✚ ✙ System_car{ · · · }

CPS : Cyber-Physical Systems ✬ ✩ ✛ ✘ System_pc{ // syntax • language; ✚ ✙ - specification; // theory System_servers{ · · · } or/and ✫ ✪ - model(s) } // semantics � ❅ � ❅ � ❅ ✛ ✘ � ❅ � ❅ ✚ ✙ System_car{ · · · }

CPS : Cyber-Physical Systems ✬ ✩ ✛ ✘ System_pc{ // syntax • language; ✚ ✙ - specification; // theory System_servers{ · · · } or/and ✫ ✪ - model(s) } // semantics � ❅ � ❅ � ❅ ✛ ✘ ✛ ✘ � ❅ � ❅ ✚ ✙ System_train{ · · · } ✚ ✙ System_car{ · · · }

CPS : Cyber-Physical Systems ✬ ✩ ✛ ✘ System_pc{ // syntax • language; ✚ ✙ - specification; // theory System_servers{ · · · } or/and ✫ ✪ - model(s) } // semantics � ❅ � ❅ � ❅ ✛ ✘ ✛ ✘ � ❅ � ❅ ✚ ✙ System_train{ · · · } ✚ ✙ System_car{ · · · } property: ϕ = “Always ( NOT car_train_crash )”

CPS : Cyber-Physical Systems ✬ ✩ ✛ ✘ System_pc{ // syntax • language; ✚ ✙ - specification; // theory System_servers{ · · · } or/and ✫ ✪ - model(s) } // semantics ϕ 4 ϕ 2 � ❅ � ❅ � ❅ ✛ ✘ ϕ 3 ✛ ✘ � ❅ ϕ 1 � ❅ ✚ ✙ System_train{ · · · } ✚ ✙ System_car{ · · · } property: ϕ = “Always ( NOT car_train_crash )”

CPS : Cyber-Physical Systems ✬ ✩ ✛ ✘ System_pc{ // syntax • language; ✚ ✙ - specification; // theory System_servers{ · · · } ϕ 2 ϕ 4 or/and ✫ ✪ - model(s) } // semantics � ❅ � ❅ � ❅ ϕ 1 ϕ 3 ✛ ✘ ✛ ✘ � ❅ � ❅ ✚ ✙ System_train{ · · · } ✚ ✙ System_car{ · · · } property: ϕ = “ALWAYS ( NOT car_train_crash )”

CPS : Cyber-Physical Systems ✬ ✩ ✛ ✘ System_pc{ // syntax • language; ✚ ✙ - specification; // theory System_servers{ · · · } or/and ✫ ✪ - model(s) } // semantics � ❅ � ❅ � ❅ ✛ ✘ ✛ ✘ � ❅ � ❅ ✚ ✙ ❄ System_train{ · · · } ✚ ✙ System_car{ · · · } YES or NO ϕ

(some) Logics in Verification non-probabilistic probabilistic Propositional logic PCTL and PCTL* Modal logic, CTL, LTL Continuous stochastic logic First-order theories: . . . Presburger arithmetic Pointer logic . . . Separation logic Duration calculus Metric temporal logic Differential dynamic logic . . .

Outline 1 Exogenous Combination of Logics 2 Probabilization of Logics: (generic) SAT completeness 3 Examples: EPPL - Probabilistic propositional logic PTL - Probabilistic temporal logic CTPL - Temporal EPPL

Exogenous Combination of Logics Definition (Satisfaction system) Let L be a set of formulas , M a class of models and � ⊆ M × L a satisfaction relation. The tuple S = �L , M , � � is a satisfaction system .

Exogenous Combination of Logics Definition (Satisfaction system) Let L be a set of formulas , M a class of models and � ⊆ M × L a satisfaction relation. The tuple S = �L , M , � � is a satisfaction system . Definition (Morphism and weak morphism) A morphism h : S → S ′ is a pair � h, h � , with h : M ′ → 2 M h : L → L ′ and for all m ∈ h ( m ′ ) , m � ϕ iff m ′ � ′ h ( ϕ ) morphism:

Exogenous Combination of Logics Definition (Satisfaction system) Let L be a set of formulas , M a class of models and � ⊆ M × L a satisfaction relation. The tuple S = �L , M , � � is a satisfaction system . Definition (Morphism and weak morphism) A morphism h : S → S ′ is a pair � h, h � , with h : M ′ → 2 M h : L → L ′ and for all m ∈ h ( m ′ ) , m � ϕ iff m ′ � ′ h ( ϕ ) morphism: exists m ∈ h ( m ′ ) , m � ϕ iff m ′ � ′ h ( ϕ ) weak morphism: for all ϕ ∈ L and for all m ′ ∈ M h def = { m ′ ∈ M ′ : h ( m ′ ) � = ∅} .

1 - Exogenous Combination of Logics Definition ((Weak) equivalent systems) S and S ′ are (resp. weak) equivalent if there are (resp. weak) total morphisms h : S → S ′ and h ′ : S ′ → S such that � ′ h ′ ( h ( ϕ )) ′ ( ψ )) , for ϕ ∈ L , ψ ∈ L ′ . ϕ and ψ � h ( h � � Denoted by equivalent, S 1 ≅ S S 2 weak equivalent, S 1 ≅ w S S 2

1 - Exogenous Combination of Logics Definition ((Weak) equivalent systems) S and S ′ are (resp. weak) equivalent if there are (resp. weak) total morphisms h : S → S ′ and h ′ : S ′ → S such that � ′ h ′ ( h ( ϕ )) ′ ( ψ )) , for ϕ ∈ L , ψ ∈ L ′ . ϕ and ψ � h ( h � � Denoted by equivalent, S 1 ≅ S S 2 weak equivalent, S 1 ≅ w S S 2 Proposition ( �L , M 1 , � 1 � ≅ S �L , M 2 , � 2 � ) Γ � 1 ϕ iff Γ � 2 ϕ . �L , M 1 , � 1 � ≅ w Proposition ( S �L , M 2 , � 2 � ) iff � 2 ϕ . � 1 ϕ

� Exogenous Combination of Logics Let h 1 : S → S 1 and h 2 : S → S 2 be morphisms. S 1 h 1 h 2 � S 2 S

� Exogenous Combination of Logics Let h 1 : S → S 1 and h 2 : S → S 2 be morphisms. S 1 h 1 h 2 � S 2 S Idea: S 1 ⊗ S 2 = �L 1 ⊗ L 2 , M ′ , � ′ � , with M ′ ⊆ M 1 × M 2 Example (Parametrization) S ( h 1 ⇒ h 2 ) = �L 1 , M ( h 1 ⇒ h 2 ) , � 1 � , where M ( h 1 ⇒ h 2 ) = { m ∈ M h 1 : h 1 ( m ) ⊆ h 2 ( M 2 ) } .

2 - Exogenous Probabilization of Logics Definition (probabilization + globalization) The probabilization + globalization operator transforms �L , M , � � into the system S ( p + g ) = �L ( p + g ) , M ( p + g ) , � ( p + g ) � : L ( p + g ) is (with β ∈ L and r ∈ Alg ( R ) ) t ::= r � � β � ( t + t ) � ( t.t ) ϕ ::= [ β ] � ( t < t ) � ( ∼ ϕ ) � ( ϕ ❂ ϕ );

2 - Exogenous Probabilization of Logics Definition (probabilization + globalization) The probabilization + globalization operator transforms �L , M , � � into the system S ( p + g ) = �L ( p + g ) , M ( p + g ) , � ( p + g ) � : L ( p + g ) is (with β ∈ L and r ∈ Alg ( R ) ) t ::= r � � β � ( t + t ) � ( t.t ) ϕ ::= [ β ] � ( t < t ) � ( ∼ ϕ ) � ( ϕ ❂ ϕ ); M ( p + g ) is the class of all m = � S, F , P , V � , where � S, F , P � is a probability space, and V : S → M is a measurable def valuation , i.e. V − 1 [ β ] = { s ∈ S : V ( s ) � β } ∈ F ;

2 - Exogenous Probabilization of Logics Definition (probabilization + globalization) The probabilization + globalization operator transforms �L , M , � � into the system S ( p + g ) = �L ( p + g ) , M ( p + g ) , � ( p + g ) � : L ( p + g ) is (with β ∈ L and r ∈ Alg ( R ) ) t ::= r � � β � ( t + t ) � ( t.t ) ϕ ::= [ β ] � ( t < t ) � ( ∼ ϕ ) � ( ϕ ❂ ϕ ); M ( p + g ) is the class of all m = � S, F , P , V � , where � S, F , P � is a probability space, and V : S → M is a measurable def valuation , i.e. V − 1 [ β ] = { s ∈ S : V ( s ) � β } ∈ F ; the satisfaction relation � ( p + g ) is given by ] m = P ( V − 1 [ β ]) � [ [ β ] m � ( p + g ) [ β ] iff V ( S ) � β ; (. . . )

2 - Exogenous Probabilization of Logics weak morphism h p : S p → S RCF ( { x β : β ∈ L} ∪ X alg ∪ X ) ∆ p S - probabilistic (sub)theory of S in RCF

2 - Exogenous Probabilization of Logics weak morphism h p : S p → S RCF ( { x β : β ∈ L} ∪ X alg ∪ X ) ∆ p S - probabilistic (sub)theory of S in RCF ϕ ⊆ L RCF , such that ∆ p S � RCF ϕ iff ∆ ϕ finite ∆ Σ Σ � RCF ϕ

2 - Exogenous Probabilization of Logics weak morphism h p : S p → S RCF ( { x β : β ∈ L} ∪ X alg ∪ X ) ∆ p S - probabilistic (sub)theory of S in RCF ϕ ⊆ L RCF , such that ∆ p S � RCF ϕ iff ∆ ϕ finite ∆ Σ Σ � RCF ϕ

2 - Exogenous Probabilization of Logics weak morphism h p : S p → S RCF ( { x β : β ∈ L} ∪ X alg ∪ X ) ∆ p S - probabilistic (sub)theory of S in RCF ϕ ⊆ L RCF , such that ∆ p S � RCF ϕ iff ∆ ϕ finite ∆ Σ Σ � RCF ϕ Proposition (Transference of SAT) ϕ has a model in M p h p ( ϕ ) ∧ ∆ Σ ϕ has a model in R X . iff