Nested sequents for modal logics and beyond Sonia Marin - PowerPoint PPT Presentation

Nested sequents for modal logics and beyond Sonia Marin IT-University of Copenhagen July 7, 2018 This presentation was made possible by grant NPRP 097-988-1-178 , from the Qatar National Research Fund (a member of the Qatar Foundation). The statements made herein are solely the responsibility of the author. 1 / 6

Survey talk on nested sequents 1 / 6

Survey talk on nested sequents What are nested sequents? 1 / 6

Survey talk on nested sequents What are nested sequents? What can they achieve? 1 / 6

Survey talk on nested sequents What are nested sequents? What can they achieve? 1. for logics without a sequent system ◮ intuitionistic modal logic IK 1 / 6

Survey talk on nested sequents What are nested sequents? What can they achieve? 1. for logics without a sequent system ◮ intuitionistic modal logic IK 2. for sequent systems without a cut-free version ◮ classical modal logic S5 1 / 6

Survey talk on nested sequents What are nested sequents? What can they achieve? 1. for logics without a sequent system ◮ intuitionistic modal logic IK 2. for sequent systems without a cut-free version ◮ classical modal logic S5 3. for cut-free systems without a syntactic cut-elimination procedure ◮ modal fixed-point logic 1 / 6

Survey talk on nested sequents What are nested sequents? What can they achieve? 1. for logics without a sequent system ◮ intuitionistic modal logic IK 2. for sequent systems without a cut-free version ◮ classical modal logic S5 3. for cut-free systems without a syntactic cut-elimination procedure ◮ modal fixed-point logic Where will they take you? 1 / 6

What are nested sequents? 2 / 6

From semantics to syntax Syntactical term encoding of semantical (tree) structure w 6 w 7 w 4 w 3 w 5 w 1 w 2 w 0 [Br¨ unnler, 2009] [Poggiolesi, 2009] 2 / 6

From semantics to syntax Syntactical term encoding of semantical (tree) structure w 6 w 7 w 4 w 3 w 5 w 1 w 2 w 0 Nested sequents: � 0 � [Br¨ unnler, 2009] [Poggiolesi, 2009] 2 / 6

From semantics to syntax Syntactical term encoding of semantical (tree) structure w 6 w 7 w 4 w 3 w 5 w 1 w 2 p s ¯ w 0 Nested sequents: � 0 ¯ � p , s , . . . [Br¨ unnler, 2009] [Poggiolesi, 2009] 2 / 6

From semantics to syntax Syntactical term encoding of semantical (tree) structure w 6 w 7 w 4 w 3 w 5 w 1 w 2 p s ¯ w 0 Nested sequents: � 0 ¯ � 1 � � p , s , . . ., [Br¨ unnler, 2009] [Poggiolesi, 2009] 2 / 6

From semantics to syntax Syntactical term encoding of semantical (tree) structure w 6 w 7 w 4 w 3 w 5 w 1 p ¯ s w 2 p s ¯ w 0 Nested sequents: � 0 ¯ � 1 p , ¯ � � p , s , . . ., s , . . . [Br¨ unnler, 2009] [Poggiolesi, 2009] 2 / 6

From semantics to syntax Syntactical term encoding of semantical (tree) structure w 6 w 7 w 4 w 3 w 5 w 1 p ¯ s w 2 p s ¯ w 0 Nested sequents: � 0 ¯ � 1 p , ¯ � � 2 � � p , s , . . ., s , . . . , [Br¨ unnler, 2009] [Poggiolesi, 2009] 2 / 6

From semantics to syntax Syntactical term encoding of semantical (tree) structure w 6 w 7 w 4 w 3 w 5 w 2 p s w 1 p ¯ s p s ¯ w 0 Nested sequents: � 0 ¯ � 1 p , ¯ � � 2 p , s , . . . � � p , s , . . ., s , . . . , [Br¨ unnler, 2009] [Poggiolesi, 2009] 2 / 6

From semantics to syntax Syntactical term encoding of semantical (tree) structure w 6 w 7 w 4 w 3 w 5 w 2 p s w 1 p ¯ s p s ¯ w 0 Nested sequents: � 0 ¯ � 1 p , ¯ � � 2 p , s , . . ., � 3 . . ., � 6 . . . � � 7 . . . �� � 4 . . . � � 5 . . . �� � p , s , . . ., s , . . . , , , , [Br¨ unnler, 2009] [Poggiolesi, 2009] 2 / 6

What can they achieve? 3 / 6

Intuitionistic modal logic IK Intuitionistic modal logic IK is obtained from intuitionistic propositional logic ◮ by adding the necessitation rule : ◻ A is a theorem if A is a theorem; ◮ and the following five variants of the k axiom. k 1 : ◻ ( A ⊃ B ) ⊃ ( ◻ A ⊃ ◻ B ) k 3 : ◇ ( A ∨ B ) ⊃ ( ◇ A ∨ ◇ B ) k 2 : ◻ ( A ⊃ B ) ⊃ ( ◇ A ⊃ ◇ B ) k 4 : ( ◇ A ⊃ ◻ B ) ⊃ ◻ ( A ⊃ B ) k 5 : ◇ ⊥ ⊃ ⊥ 3 / 6

Intuitionistic modal logic IK Intuitionistic modal logic IK is obtained from intuitionistic propositional logic ◮ by adding the necessitation rule : ◻ A is a theorem if A is a theorem; ◮ and the following five variants of the k axiom. k 1 : ◻ ( A ⊃ B ) ⊃ ( ◻ A ⊃ ◻ B ) k 3 : ◇ ( A ∨ B ) ⊃ ( ◇ A ∨ ◇ B ) k 2 : ◻ ( A ⊃ B ) ⊃ ( ◇ A ⊃ ◇ B ) k 4 : ( ◇ A ⊃ ◻ B ) ⊃ ◻ ( A ⊃ B ) k 5 : ◇ ⊥ ⊃ ⊥ Sequent system: Λ ⇒ A Λ , A ⇒ B ◻ o ◇ o k − − − − − − − − − − − k − − − − − − − − − − − − − − − − − ◻ Λ ⇒ ◻ A ◻ Λ , ◇ A ⇒ ◇ B 3 / 6

Intuitionistic modal logic IK Intuitionistic modal logic IK is obtained from intuitionistic propositional logic ◮ by adding the necessitation rule : ◻ A is a theorem if A is a theorem; ◮ and the following five variants of the k axiom. k 1 : ◻ ( A ⊃ B ) ⊃ ( ◻ A ⊃ ◻ B ) k 3 : ◇ ( A ∨ B ) ⊃ ( ◇ A ∨ ◇ B ) k 2 : ◻ ( A ⊃ B ) ⊃ ( ◇ A ⊃ ◇ B ) k 4 : ( ◇ A ⊃ ◻ B ) ⊃ ◻ ( A ⊃ B ) k 5 : ◇ ⊥ ⊃ ⊥ Sequent system: Λ ⇒ A Λ , A ⇒ B ? ◻ o ◇ o k − − − − − − − − − − − k − − − − − − − − − − − − − − − − − ◻ Λ ⇒ ◻ A ◻ Λ , ◇ A ⇒ ◇ B Theorem: LJ p + ◻ o k + ◇ o k is sound and complete for IK − { k 3 , k 4 , k 5 } . 3 / 6

Intuitionistic modal logic IK Intuitionistic modal logic IK is obtained from intuitionistic propositional logic ◮ by adding the necessitation rule : ◻ A is a theorem if A is a theorem; ◮ and the following five variants of the k axiom. k 1 : ◻ ( A ⊃ B ) ⊃ ( ◻ A ⊃ ◻ B ) k 3 : ◇ ( A ∨ B ) ⊃ ( ◇ A ∨ ◇ B ) k 2 : ◻ ( A ⊃ B ) ⊃ ( ◇ A ⊃ ◇ B ) k 4 : ( ◇ A ⊃ ◻ B ) ⊃ ◻ ( A ⊃ B ) k 5 : ◇ ⊥ ⊃ ⊥ Nested sequent system: Λ 1 { [Λ 2 , A ] } Π { [ A ] } Λ { [ A ] } ∆ 1 { ◻ A , [ A , ∆ 2 ] } ◇ n ◇ n ◻ n ◻ n Rk − − − − − − − − − − − − − − − L − − − − − − − − − R − − − − − − − − Lk − − − − − − − − − − − − − − − − − − − Λ 1 { [Λ 2 ] , ◇ A } Π { ◇ A } Λ { ◻ A } ∆ 1 { ◻ A , [∆ 2 ] } 3 / 6

Intuitionistic modal logic IK Intuitionistic modal logic IK is obtained from intuitionistic propositional logic ◮ by adding the necessitation rule : ◻ A is a theorem if A is a theorem; ◮ and the following five variants of the k axiom. k 1 : ◻ ( A ⊃ B ) ⊃ ( ◻ A ⊃ ◻ B ) k 3 : ◇ ( A ∨ B ) ⊃ ( ◇ A ∨ ◇ B ) k 2 : ◻ ( A ⊃ B ) ⊃ ( ◇ A ⊃ ◇ B ) k 4 : ( ◇ A ⊃ ◻ B ) ⊃ ◻ ( A ⊃ B ) k 5 : ◇ ⊥ ⊃ ⊥ Nested sequent system: Λ 1 { [Λ 2 , A ] } Π { [ A ] } Λ { [ A ] } ∆ 1 { ◻ A , [ A , ∆ 2 ] } ◇ n ◇ n ◻ n ◻ n Rk − − − − − − − − − − − − − − − L − − − − − − − − − R − − − − − − − − Lk − − − − − − − − − − − − − − − − − − − Λ 1 { [Λ 2 ] , ◇ A } Π { ◇ A } Λ { ◻ A } ∆ 1 { ◻ A , [∆ 2 ] } Theorem: nIK is sound and complete for IK. [Straßburger, 2013] 3 / 6

Intuitionistic modal logic IK Intuitionistic modal logic IK is obtained from intuitionistic propositional logic ◮ by adding the necessitation rule : ◻ A is a theorem if A is a theorem; ◮ and the following five variants of the k axiom. k 1 : ◻ ( A ⊃ B ) ⊃ ( ◻ A ⊃ ◻ B ) k 3 : ◇ ( A ∨ B ) ⊃ ( ◇ A ∨ ◇ B ) k 2 : ◻ ( A ⊃ B ) ⊃ ( ◇ A ⊃ ◇ B ) k 4 : ( ◇ A ⊃ ◻ B ) ⊃ ◻ ( A ⊃ B ) k 5 : ◇ ⊥ ⊃ ⊥ Nested sequent system: Λ 1 { [Λ 2 , A ] } Π { [ A ] } Λ { [ A ] } ∆ 1 { ◻ A , [ A , ∆ 2 ] } ◇ n ◇ n ◻ n ◻ n Rk − − − − − − − − − − − − − − − L − − − − − − − − − R − − − − − − − − Lk − − − − − − − − − − − − − − − − − − − Λ 1 { [Λ 2 ] , ◇ A } Π { ◇ A } Λ { ◻ A } ∆ 1 { ◻ A , [∆ 2 ] } Theorem: nIK is sound and complete for IK. [Straßburger, 2013] Note: A system can also be designed using labelled sequents. [Simpson, 1994] 3 / 6

Classical modal logic S5 Classical modal logic S5 is obtained from classical propositional logic ◮ by adding the necessitation rule : ◻ A is a theorem if A is a theorem; ◮ and the axioms: ◻ ( A ⊃ B ) ⊃ ( ◻ A ⊃ ◻ B ) k : t: A ⊃ ◇ A 4: ◇◇ A ⊃ ◇ A 5: ◇ A ⊃ ◻ ◇ A 4 / 6

Classical modal logic S5 Sequent system: Γ , A ◇ Γ 1 , Γ 1 , ◻ Γ 2 , A ◇ o ◻ o t − − − − − − − k45 − − − − − − − − − − − − − − − − − − − Γ , ◇ A ◇ Γ 1 , ◻ Γ 2 , Γ 3 , ◻ A 4 / 6