Discrete duality for downset lattices and their Positive Modal - PowerPoint PPT Presentation

Duality and Residuation Sam van Gool Discrete Duality Discrete duality for downset lattices and their Positive Modal Logic residuated operations Residuation Research Question Sam van Gool 30 October 2010 The categorical flow of information University of Oxford 1 / 13

Duality and Residuation Motivation Sam van Gool Discrete Duality Positive Modal Logic Residuation • Labelled transition systems: Research Question 2 / 13

Duality and Residuation Motivation Sam van Gool Discrete Duality Positive Modal Logic Residuation • Labelled transition systems: Research Question States ( S , ≤ ) , 2 / 13

Duality and Residuation Motivation Sam van Gool Discrete Duality Positive Modal Logic Residuation • Labelled transition systems: Research Question Actions A ∗ , States ( S , ≤ ) , 2 / 13

Duality and Residuation Motivation Sam van Gool Discrete Duality Positive Modal Logic Residuation • Labelled transition systems: Research Question Transitions R ⊆ S × A ∗ × S . Actions A ∗ , States ( S , ≤ ) , 2 / 13

Duality and Residuation Motivation Sam van Gool Discrete Duality Positive Modal Logic Residuation • Labelled transition systems: Research Question Transitions R ⊆ S × A ∗ × S . Actions A ∗ , States ( S , ≤ ) , • Logical description: 2 / 13

Duality and Residuation Motivation Sam van Gool Discrete Duality Positive Modal Logic Residuation • Labelled transition systems: Research Question Transitions R ⊆ S × A ∗ × S . Actions A ∗ , States ( S , ≤ ) , • Logical description: 1 Without negation, 2 / 13

Duality and Residuation Motivation Sam van Gool Discrete Duality Positive Modal Logic Residuation • Labelled transition systems: Research Question Transitions R ⊆ S × A ∗ × S . Actions A ∗ , States ( S , ≤ ) , • Logical description: 1 Without negation, 2 With converse, 2 / 13

Duality and Residuation Motivation Sam van Gool Discrete Duality Positive Modal Logic Residuation • Labelled transition systems: Research Question Transitions R ⊆ S × A ∗ × S . Actions A ∗ , States ( S , ≤ ) , • Logical description: 1 Without negation, 2 With converse, 3 With • as primitive operation. 2 / 13

Duality and Residuation Presentation outline Sam van Gool Discrete Duality Positive Modal Logic 1 Discrete Duality Residuation Research Question 3 / 13

Duality and Residuation Presentation outline Sam van Gool Discrete Duality Positive Modal Logic 1 Discrete Duality Residuation Research Question Positive Modal Logic 2 3 / 13

Duality and Residuation Presentation outline Sam van Gool Discrete Duality Positive Modal Logic 1 Discrete Duality Residuation Research Question Positive Modal Logic 2 Residuation 3 3 / 13

Duality and Residuation Presentation outline Sam van Gool Discrete Duality Positive Modal Logic 1 Discrete Duality Residuation Research Question Positive Modal Logic 2 Residuation 3 4 Research Question 3 / 13

Duality and Residuation Discrete Duality Sam van Gool Kripke semantics for modal logic Discrete Duality • Discrete duality for complete atomic Boolean algebras Positive Modal Logic Residuation Research Question 4 / 13

Duality and Residuation Discrete Duality Sam van Gool Kripke semantics for modal logic Discrete Duality • Discrete duality for complete atomic Boolean algebras Positive Modal Logic CABA ⇆ Set Residuation B �→ At ( B ) Research Question P ( X ) �→ X 4 / 13

Duality and Residuation Discrete Duality Sam van Gool Kripke semantics for modal logic Discrete Duality • Discrete duality for complete atomic Boolean algebras Positive Modal Logic CABA ⇆ Set Residuation B �→ At ( B ) Research Question P ( X ) �→ X • Complete operator � on B ↔ relation R on At ( B ) : xR � y ⇐⇒ x ≤ � y 4 / 13

Duality and Residuation Discrete Duality Sam van Gool Kripke semantics for modal logic Discrete Duality • Discrete duality for complete atomic Boolean algebras Positive Modal Logic CABA ⇆ Set Residuation B �→ At ( B ) Research Question P ( X ) �→ X • Complete operator � on B ↔ relation R on At ( B ) : xR � y ⇐⇒ x ≤ � y • Canonical extension: every BA with operator embeds in a CABA with complete operator 4 / 13

Duality and Residuation Discrete Duality Sam van Gool Kripke semantics for modal logic Discrete Duality • Discrete duality for complete atomic Boolean algebras Positive Modal Logic CABA ⇆ Set Residuation B �→ At ( B ) Research Question P ( X ) �→ X • Complete operator � on B ↔ relation R on At ( B ) : xR � y ⇐⇒ x ≤ � y • Canonical extension: every BA with operator embeds in a CABA with complete operator “Canonical Extension + Discrete Duality = Frame Completeness” 4 / 13

Duality and Residuation Discrete Duality Sam van Gool Distributive lattices Discrete Duality Positive Modal Logic • Discrete duality for doubly algebraic distributive lattices Residuation (Birkhoff 1940) Research Question 5 / 13

Duality and Residuation Discrete Duality Sam van Gool Distributive lattices Discrete Duality Positive Modal Logic • Discrete duality for doubly algebraic distributive lattices Residuation (Birkhoff 1940) Research Question DADLat ⇆ Poset D �→ ( J ( D ) , ≤ D ) D ( P ) �→ ( P , ≤ ) 5 / 13

Duality and Residuation Discrete Duality Sam van Gool Distributive lattices Discrete Duality Positive Modal Logic • Discrete duality for doubly algebraic distributive lattices Residuation (Birkhoff 1940) Research Question DADLat ⇆ Poset D �→ ( J ( D ) , ≤ D ) D ( P ) �→ ( P , ≤ ) • Complete operator � on D ↔ relation R on J ( D ) with ≤ ◦ R ◦ ≤ = R ( order-compatible ) 5 / 13

Duality and Residuation Discrete Duality Sam van Gool Distributive lattices Discrete Duality Positive Modal Logic • Discrete duality for doubly algebraic distributive lattices Residuation (Birkhoff 1940) Research Question DADLat ⇆ Poset D �→ ( J ( D ) , ≤ D ) D ( P ) �→ ( P , ≤ ) • Complete operator � on D ↔ relation R on J ( D ) with ≤ ◦ R ◦ ≤ = R ( order-compatible ) • Canonical extension: every DLat with operator embeds in a DADLat with complete operator 5 / 13

Duality and Residuation Positive Modal Logic Sam van Gool Condensed Chronology Discrete Duality Positive Modal Logic Residuation Research Question • Dunn (1995): modal logic in the absence of negation 6 / 13

Duality and Residuation Positive Modal Logic Sam van Gool Condensed Chronology Discrete Duality Positive Modal Logic Residuation Research Question • Dunn (1995): modal logic in the absence of negation • Celani, Jansana (1997): simple Kripke-style semantics 6 / 13

Duality and Residuation Positive Modal Logic Sam van Gool Condensed Chronology Discrete Duality Positive Modal Logic Residuation Research Question • Dunn (1995): modal logic in the absence of negation • Celani, Jansana (1997): simple Kripke-style semantics • Gehrke, Nagahashi, Venema (2004): general framework for distributive-lattice-based modal logic 6 / 13

Duality and Residuation Positive Modal Logic Sam van Gool Algebraic definition Discrete Duality Positive Modal Logic • A positive modal algebra (PMA) is a distributive lattice D Residuation equipped with two unary operations � and � , such that Research Question 7 / 13

Duality and Residuation Positive Modal Logic Sam van Gool Algebraic definition Discrete Duality Positive Modal Logic • A positive modal algebra (PMA) is a distributive lattice D Residuation equipped with two unary operations � and � , such that Research Question � ⊤ = ⊤ , � ( a ∧ b ) = � a ∧ � b , 7 / 13

Duality and Residuation Positive Modal Logic Sam van Gool Algebraic definition Discrete Duality Positive Modal Logic • A positive modal algebra (PMA) is a distributive lattice D Residuation equipped with two unary operations � and � , such that Research Question � ⊤ = ⊤ , � ⊥ = ⊥ , � ( a ∧ b ) = � a ∧ � b , � ( a ∨ b ) = � a ∨ � b , 7 / 13

Duality and Residuation Positive Modal Logic Sam van Gool Algebraic definition Discrete Duality Positive Modal Logic • A positive modal algebra (PMA) is a distributive lattice D Residuation equipped with two unary operations � and � , such that Research Question � ⊤ = ⊤ , � ⊥ = ⊥ , � ( a ∧ b ) = � a ∧ � b , � ( a ∨ b ) = � a ∨ � b , � ( a ∨ b ) ≤ � a ∨ � b , 7 / 13

Duality and Residuation Positive Modal Logic Sam van Gool Algebraic definition Discrete Duality Positive Modal Logic • A positive modal algebra (PMA) is a distributive lattice D Residuation equipped with two unary operations � and � , such that Research Question � ⊤ = ⊤ , � ⊥ = ⊥ , � ( a ∧ b ) = � a ∧ � b , � ( a ∨ b ) = � a ∨ � b , � ( a ∨ b ) ≤ � a ∨ � b , � a ∧ � b ≤ � ( a ∧ b ) . 7 / 13