Comparing and ! via polarities Sonia Marin Inria, LIX, Ecole - PowerPoint PPT Presentation

Comparing ✷ and ! via polarities Sonia Marin Inria, LIX, ´ Ecole Polytechnique ITU Copenhagen May 19, 2017

The answer

The answer “from a proof-theoretical point of view exponentials behave exactly like S4 modalities” [Martini & Masini, 1994]

Are ! and ✷ interchangeable?

Are ! and ✷ interchangeable? Modal logic S4 : A ::= x | x ⊥ | A ∧ A | ⊤ | A ∨ A | ⊥

Are ! and ✷ interchangeable? Modal logic S4 : A ::= x | x ⊥ | A ∧ A | ⊤ | A ∨ A | ⊥ | ✷ A | ✸ A

Are ! and ✷ interchangeable? Modal logic S4 : A ::= x | x ⊥ | A ∧ A | ⊤ | A ∨ A | ⊥ | ✷ A | ✸ A Linear logic LL : A ::= x | x ⊥ | A ⊗ A | 1 | A ` A | ⊥ | A ⊕ A | 0 | A & A | ⊤

Are ! and ✷ interchangeable? Modal logic S4 : A ::= x | x ⊥ | A ∧ A | ⊤ | A ∨ A | ⊥ | ✷ A | ✸ A Linear logic LL : A ::= x | x ⊥ | A ⊗ A | 1 | A ` A | ⊥ | A ⊕ A | 0 | A & A | ⊤ | ! A | ? A

Are ! and ✷ interchangeable? Modal logic S4 : A ::= x | x ⊥ | A ∧ A | ⊤ | A ∨ A | ⊥ | ✷ A | ✸ A ⊢ ✸ Γ , A ⊢ Γ , ✸ A , A ✷ − ✸ − − − − − − − − − − − − − − − − − − − − − − − − − ⊢ ✸ Γ , ✷ A , ∆ ⊢ Γ , ✸ A Linear logic LL : A ::= x | x ⊥ | A ⊗ A | 1 | A ` A | ⊥ | A ⊕ A | 0 | A & A | ⊤ | ! A | ? A ⊢ ?Γ , A ⊢ Γ , A ! − ? − − − − − − − − − − − − − − − − ⊢ ?Γ , ! A ⊢ Γ , ? A

Are ! and ✷ interchangeable? Modal logic S4 : A ::= x | x ⊥ | A ∧ A | ⊤ | A ∨ A | ⊥ | ✷ A | ✸ A ⊢ ✸ Γ , A ⊢ Γ , ✸ A , A ✷ − ✸ − − − − − − − − − − − − − − − − − − − − − − − − − ⊢ ✸ Γ , ✷ A , ∆ ⊢ Γ , ✸ A Linear logic LL : A ::= x | x ⊥ | A ⊗ A | 1 | A ` A | ⊥ | A ⊕ A | 0 | A & A | ⊤ | ! A | ? A ⊢ ?Γ , A ⊢ Γ , A ! − ? − − − − − − − − − − − − − − − − ⊢ ?Γ , ! A ⊢ Γ , ? A

Are ! and ✷ interchangeable? Theorem: [Martini & Masini, 1994] Γ + provable in LL Γ provable in S4 ⇔ Modal logic S4 : A ::= x | x ⊥ | A ∧ A | ⊤ | A ∨ A | ⊥ | ✷ A | ✸ A ⊢ ✸ Γ , A ⊢ Γ , ✸ A , A ✷ − ✸ − − − − − − − − − − − − − − − − − − − − − − − − − ⊢ ✸ Γ , ✷ A , ∆ ⊢ Γ , ✸ A Linear logic LL : A ::= x | x ⊥ | A ⊗ A | 1 | A ` A | ⊥ | A ⊕ A | 0 | A & A | ⊤ | ! A | ? A ⊢ ?Γ , A ⊢ Γ , A ! − ? − − − − − − − − − − − − − − − − ⊢ ?Γ , ! A ⊢ Γ , ? A

Are ! and ✷ interchangeable? Theorem: [Martini & Masini, 1994] Γ + provable in LL Γ provable in S4 ⇔

Are ! and ✷ interchangeable? Theorem: [Martini & Masini, 1994] Γ + provable in LL Γ provable in S4 ⇔ Their answer: cut-free proof of an S 4 sequent � cut-free proof of its LL translation

Are ! and ✷ interchangeable? Theorem: [Martini & Masini, 1994] Γ + provable in LL Γ provable in S4 ⇔ Our question: focused polarised cut-free proof of an S 4 sequent � focused polarised cut-free proof of its LL translation ?

Polarity and focusing non-invertible rules : positive connectives Polarities: invertible rules : negative connectives [Andreoli, 1990] [Laurent, 2004]

Polarity and focusing non-invertible rules : positive connectives Polarities: invertible rules : negative connectives � � � Inversion: in π � the last rule is negative. � � � ⊢ N , Γ [Andreoli, 1990] [Laurent, 2004]

Polarity and focusing non-invertible rules : positive connectives Polarities: invertible rules : negative connectives � � � Inversion: in π � the last rule is negative. � � � ⊢ N , Γ Focus on a positive formula: � � � in π � only rules decomposing P between two rules decomposing P � � � ⊢ P , Γ [Andreoli, 1990] [Laurent, 2004]

Polarity and focusing non-invertible rules : positive connectives Polarities: invertible rules : negative connectives � � � Inversion: in π � the last rule is negative. � � � ⊢ N , Γ Focus on a positive formula: � � � in π � only rules decomposing P between two rules decomposing P � � � ⊢ P , Γ Completeness of focusing: if a formula F is provable then F has a focused proof [Andreoli, 1990] [Laurent, 2004]

Polarity and connectives non-invertible rules : positive connectives Polarities: invertible rules : negative connectives [Andreoli, 1990] [Laurent, 2004]

Polarity and connectives non-invertible rules : positive connectives Polarities: invertible rules : negative connectives Modal logic S4 : A ::= x | x ⊥ | A ∧ A | ⊤ | A ∨ A | ⊥ | ✷ A | ✸ A Linear logic LL : A ::= x | x ⊥ | A ⊗ A | 1 | A ` A | ⊥ | A ⊕ A | 0 | A & A | ⊤ | ! A | ? A [Andreoli, 1990] [Laurent, 2004]

Polarity and connectives non-invertible rules : positive connectives Polarities: invertible rules : negative connectives Modal logic S4 : A ::= x | x ⊥ | A ∧ A | ⊤ | A ∨ A | ⊥ | ✷ A | ✸ A Linear logic LL : A ::= x | x ⊥ | A ⊗ A | 1 | A ` A | ⊥ | A ⊕ A | 0 | A & A | ⊤ | ! A | ? A P ::= x | A ⊗ A | 1 | A ⊕ A | 0 x ⊥ | A ` A | ⊥ | A & A | ⊤ N ::= [Andreoli, 1990] [Laurent, 2004]

Polarity and connectives non-invertible rules : positive connectives Polarities: invertible rules : negative connectives Modal logic S4 : A ::= x | x ⊥ | A ∧ A | ⊤ | A ∨ A | ⊥ | ✷ A | ✸ A Linear logic LL : A ::= x | x ⊥ | A ⊗ A | 1 | A ` A | ⊥ | A ⊕ A | 0 | A & A | ⊤ | ! A | ? A P ::= x | A ⊗ A | 1 | A ⊕ A | 0 | ! A x ⊥ | A ` A | ⊥ | A & A | ⊤ | ? A N ::= [Andreoli, 1990] [Laurent, 2004]

Polarity and connectives non-invertible rules : positive connectives Polarities: invertible rules : negative connectives Modal logic S4 : A ::= x | x ⊥ | A ∧ A | ⊤ | A ∨ A | ⊥ | ✷ A | ✸ A + + + + ::= | A ∧ A | ⊤ | A ∨ A | P x ⊥ x ⊥ | A ::= − − N ∨ A | − ⊥ | A ∧ A | − ⊤ Linear logic LL : A ::= x | x ⊥ | A ⊗ A | 1 | A ` A | ⊥ | A ⊕ A | 0 | A & A | ⊤ | ! A | ? A P ::= x | A ⊗ A | 1 | A ⊕ A | 0 | ! A x ⊥ | A ` A | ⊥ | A & A | ⊤ | ? A N ::= [Andreoli, 1990] [Laurent, 2004]

Polarity and connectives non-invertible rules : positive connectives Polarities: invertible rules : negative connectives Modal logic S4 : A ::= x | x ⊥ | A ∧ A | ⊤ | A ∨ A | ⊥ | ✷ A | ✸ A + + + + ::= | A ∧ A | ⊤ | A ∨ A | ⊥ | ✸ A P x x ⊥ | A ::= − − N ∨ A | − ⊥ | A ∧ A | − ⊤ | ✷ A [Miller, Volpe, 2015] [Chaudhuri, M., Strassburger, 2016] Linear logic LL : A ::= x | x ⊥ | A ⊗ A | 1 | A ` A | ⊥ | A ⊕ A | 0 | A & A | ⊤ | ! A | ? A P ::= x | A ⊗ A | 1 | A ⊕ A | 0 | ! A x ⊥ | A ` A | ⊥ | A & A | ⊤ | ? A N ::= [Andreoli, 1990] [Laurent, 2004]

Polarity and connectives non-invertible rules : positive connectives Polarities: invertible rules : negative connectives Modal logic S4 : A ::= x | x ⊥ | A ∧ A | ⊤ | A ∨ A | ⊥ | ✷ A | ✸ A ::= | ✸ A P This is... ::= N | ✷ A [Miller, Volpe, 2015] [Chaudhuri, M., Strassburger, 2016] Linear logic LL : A ::= x | x ⊥ | A ⊗ A | 1 | A ` A | ⊥ | A ⊕ A | 0 | A & A | ⊤ | ! A | ? A P ::= | ! A ...not the same! N ::= | ? A [Andreoli, 1990] [Laurent, 2004]

Modular focused systems for modal logics

Modular focused systems for modal logics Classical normal modal logics: S4 S5 ◦ ◦ k: ✷ ( A → B ) → ( ✷ A → ✷ B ) T ◦ ◦ TB D4 D45 d: ✷ A → ✸ A (Seriality) ◦ ◦ t: ✷ A → A (Reflexivity) ◦ D5 b: ✸✷ A → A (Symmetry) D ◦ ◦ DB K4 4: ✷ A → ✷✷ A (Transitivity) ◦ ◦ ◦ KB5 K45 5: ✸✷ A → ✷ A (Euclideanness) ◦ K5 ◦ ◦ K KB

Modular focused systems for modal logics Classical normal modal logics: S4 S5 ◦ ◦ k: ✷ ( A → B ) → ( ✷ A → ✷ B ) T ◦ ◦ TB D4 D45 d: ✷ A → ✸ A (Seriality) ◦ ◦ t: ✷ A → A (Reflexivity) ◦ D5 b: ✸✷ A → A (Symmetry) D ◦ ◦ DB K4 4: ✷ A → ✷✷ A (Transitivity) ◦ ◦ ◦ KB5 K45 5: ✸✷ A → ✷ A (Euclideanness) ◦ K5 ◦ ◦ K KB Nested sequent system:

Modular focused systems for modal logics Classical normal modal logics: S4 S5 ◦ ◦ k: ✷ ( A → B ) → ( ✷ A → ✷ B ) T ◦ ◦ TB D4 D45 d: ✷ A → ✸ A (Seriality) ◦ ◦ t: ✷ A → A (Reflexivity) ◦ D5 b: ✸✷ A → A (Symmetry) D ◦ ◦ DB K4 4: ✷ A → ✷✷ A (Transitivity) ◦ ◦ ◦ KB5 K45 5: ✸✷ A → ✷ A (Euclideanness) ◦ K5 ◦ ◦ K KB Nested sequent system: 1. complete and modular F is a theorem of K + axioms iff F is provable in KN + rules [Br¨ unnler, 2009]