The expressive power of modal logic with inclusion atoms Johanna - PowerPoint PPT Presentation

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation The expressive power of modal logic with inclusion atoms Johanna Stumpf Dagstuhl Seminar 24. Juni 2015

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics Definition Let Φ be a set of proposition symbols. The set of formulas of ML (Φ) is generated by the following grammar ϕ := p | ¬ p | ( ϕ ∧ ϕ ) | ( ϕ ∨ ϕ ) | ♦ ϕ | � ϕ, where p ∈ Φ .

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics Definition A Kripke model K is a triple ( W , R , V ) , where W is a set, R ⊆ W × W is the accessibility relation, V : Φ → P ( W ) is the valuation.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics Definition Let K = ( W , R , V ) be a Kripke model.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics Definition Let K = ( W , R , V ) be a Kripke model. Any subset T of W is called a team of K .

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics Definition Let K = ( W , R , V ) be a Kripke model. Any subset T of W is called a team of K . For any T ⊆ W we write R [ T ] = { v ∈ W | ∃ w ∈ T : wRv } and R − 1 [ T ] = { w ∈ W | ∃ v ∈ T : wRv } .

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics Definition Let K = ( W , R , V ) be a Kripke model. Any subset T of W is called a team of K . For any T ⊆ W we write R [ T ] = { v ∈ W | ∃ w ∈ T : wRv } and R − 1 [ T ] = { w ∈ W | ∃ v ∈ T : wRv } . For teams T , S ⊆ W we write T [ R ] S if S ⊆ R [ T ] and T ⊆ R − 1 [ S ] .

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics Definition The team semantics for ML is defined as follows: K , T | = p ⇐ ⇒ T ⊆ V ( p ) K , T | = ¬ p ⇐ ⇒ T ∩ V ( p ) = ∅ K , T | = ϕ ∧ ψ ⇐ ⇒ K , T | = ϕ and K , T | = ψ K , T | = ϕ ∨ ψ ⇐ ⇒ K , T 1 | = ϕ and K , T 2 | = ψ for some T 1 , T 2 such that T 1 ∪ T 2 = T . K , T ′ | = ϕ for some T ′ such that T [ R ] T ′ . K , T | = ♦ ϕ ⇐ ⇒ K , T ′ | = ϕ where T ′ = R [ T ] . K , T | = � ϕ ⇐ ⇒

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics Example Let T = { w , v } and T ′ = { w ′ , v ′ } be in a Kripke model K , such that K , T ′ | = ϕ K , w ′ | = ψ Then K , T | = � ϕ and K , T | = ♦ ψ .

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics Definition Assume that Φ is a finite set of proposition symbols. Let k ∈ N and let ( K , w ) be a pointed Φ -model. The k -th Hintikka formula χ k K , w of ( K , w ) is defined recursively as follows:

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics Definition Assume that Φ is a finite set of proposition symbols. Let k ∈ N and let ( K , w ) be a pointed Φ -model. The k -th Hintikka formula χ k K , w of ( K , w ) is defined recursively as follows: χ 0 K , w := � { p | p ∈ Φ , w ∈ V ( p ) } ∧ � {¬ p | p ∈ Φ , w / ∈ V ( p ) }

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics Definition Assume that Φ is a finite set of proposition symbols. Let k ∈ N and let ( K , w ) be a pointed Φ -model. The k -th Hintikka formula χ k K , w of ( K , w ) is defined recursively as follows: χ 0 K , w := � { p | p ∈ Φ , w ∈ V ( p ) } ∧ � {¬ p | p ∈ Φ , w / ∈ V ( p ) } χ k + 1 K , w := χ k v ∈ R [ w ] ♦ χ k v ∈ R [ w ] χ k K , w ∧ � K , v ∧ � � K , v .

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics Proposition Let Φ be a finite set of proposition symbols, k ∈ N , and ( K , w ) and ( K , ′ w ′ ) pointed Φ -models. Then the following holds: K , w ⇄ k K ′ , w ′ ⇐ ⇒ K ′ , w ′ | = χ k K , w .

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics A Φ -model with a team is a pair ( K , T ) , where K is a Kripke model over Φ and T is a team of K . We denote by KT (Φ) the class of all Φ -models with teams. Definition Let ( K , T ) , ( K ′ , T ′ ) ∈ KT (Φ) and k ∈ N . We say that ( K , T ) and ( K ′ , T ′ ) are team k -bisimilar and denote K , T [ ⇄ k ] K ′ , T ′ if

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics A Φ -model with a team is a pair ( K , T ) , where K is a Kripke model over Φ and T is a team of K . We denote by KT (Φ) the class of all Φ -models with teams. Definition Let ( K , T ) , ( K ′ , T ′ ) ∈ KT (Φ) and k ∈ N . We say that ( K , T ) and ( K ′ , T ′ ) are team k -bisimilar and denote K , T [ ⇄ k ] K ′ , T ′ if for every w ∈ T there exists some w ′ ∈ T ′ such that K , w ⇄ k K ′ , w ′ ,

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics A Φ -model with a team is a pair ( K , T ) , where K is a Kripke model over Φ and T is a team of K . We denote by KT (Φ) the class of all Φ -models with teams. Definition Let ( K , T ) , ( K ′ , T ′ ) ∈ KT (Φ) and k ∈ N . We say that ( K , T ) and ( K ′ , T ′ ) are team k -bisimilar and denote K , T [ ⇄ k ] K ′ , T ′ if for every w ∈ T there exists some w ′ ∈ T ′ such that K , w ⇄ k K ′ , w ′ , for every w ′ ∈ T ′ there exists some w ∈ T such that K , w ⇄ k K ′ , w ′ .

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic and team semantics A class K is closed under team k -bisimulation if ( K , T ) ∈ K and K , T [ ⇄ k ] K ′ , T ′ imply that ( K ′ , T ′ ) ∈ K . A class K is closed under unions if ( K , T i ) ∈ K for i ∈ I implies ( K , � i ∈ I T i ) ∈ K . A class K has the empty team property if ( K , ∅ ) ∈ K for each model K .

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic with nonempty disjunction ⊲ ⊳ Nonempty disjunction: ϕ ∨ ψ Definition ⊲ ⊳ K , T | = ϕ ∨ ψ if and only if T = ∅ or there exist T 1 , T 2 � = ∅ such that T = T 1 ∪ T 2 and K , T 1 | = ϕ and K , T 2 | = ψ .

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic with nonempty disjunction Example Let T in propositional logic be the set of assignments {{ 01 } , { 10 }} . The assignment { 01 } satisfies q and the ⊲ ⊳ assignment { 10 } satisfies p . Thus, T | = p ∨ q .

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic with nonempty disjunction Theorem ⊲ ⊳ ML ( ∨ ) has the empty team property.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic with nonempty disjunction Theorem ⊲ ⊳ ML ( ∨ ) has the empty team property. Theorem ⊲ ⊳ ML ( ∨ ) is closed under unions.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic with nonempty disjunction Theorem ⊲ ⊳ ML ( ∨ ) has the empty team property. Theorem ⊲ ⊳ ML ( ∨ ) is closed under unions. Theorem ML ( ⊲ ⊳ ∨ ) is closed under k-bisimulation.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic with nonempty disjunction Theorem Let Φ be a finite set of propositional symbols and let K ⊆ KT (Φ) . The class K is definable in ML ( ⊲ ∨ ) if and only if K closed under ⊳ unions, closed under k-bisimulation for some k ∈ N and has the empty team property.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic with nonempty disjunction Proof idea: ⇒ � ⇐ Lemma Let Φ := { p 1 , . . . , p n } be a set of propositional symbols. For any ⊲ ⊳ pair ( K , X ) ∈ KT (Φ) there exists a formula φ K , X ∈ ML ( ∨ )(Φ) such that K ′ , T ′ | = φ K , X if and only if K , X [ ⇄ k ] K ′ , T ′ or T ′ = ∅ . X = ∅ , then φ K , X = ⊥ . ⊲ ⊳ s ∈ X χ k Else: φ K , X = � K , s . Then the formula � ( K , X ) ∈K φ K , X defines the class K .

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic with inclusion atoms Inclusion atom: ϕ 1 . . . ϕ k ⊆ ψ 1 . . . ψ k , where ϕ i , ψ i ∈ ML Definition The semantics for MINC is given by the semantics for ML and the following additional clause: K , T | = ϕ 1 . . . ϕ k ⊆ ψ 1 . . . ψ k ⇐ ⇒ n ∀ w ∈ T ∃ w ′ ∈ T : ⇒ K , w ′ | � ( K , w | = ϕ i ⇐ = ψ i ) i = 1

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic with inclusion atoms Example Let T in propositional logic be the set of assignments {{ 01 } , { 10 }} . Then for every value of p in T exists the same value for q in T . Thus, T | = p ⊆ q .

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation Modal logic with inclusion atoms Theorem MINC has the empty team property.