A Duality for Distributive Unimodal Logic Adam Pˇ renosil Institute of Computer Science, Academy of Sciences of the Czech Republic 17 October 2014, Groningen Advances in Modal Logic 2014 Adam Pˇ renosil A Duality for Distributive Unimodal Logic 1 / 23
Introduction Semantics Expanding the language Duality Introduction Distributive unimodal logic is the modal logic of posets with an arbitrary binary relation: ( W , ≤ , R ). It is a non-classical modal logic over the following signature: distributive lattice connectives ∧ , ∨ , ⊤ , ⊥ unary normal modal operators � + , � − . ♦ + , ♦ − possibly also intuitionistic implication → and its dual � − In most of this talk, we restrict to {∧ , ∨ , ⊤ , ⊥ , � , ♦ } + {→ , � −} . Why is this logic interesting? Several reasons. Adam Pˇ renosil A Duality for Distributive Unimodal Logic 2 / 23
Introduction Semantics Expanding the language Duality Related logics intuitionistic modal logic (IML): ∧ , ∨ , ⊤ , ⊥ + � , ♦ + → positive modal logic (PML): ∧ , ∨ , ⊤ , ⊥ + � , ♦ distributive modal logic (DML): ∧ , ∨ , ⊤ , ⊥ + � ± , ♦ ± Adam Pˇ renosil A Duality for Distributive Unimodal Logic 3 / 23
Introduction Semantics Expanding the language Duality Related logics Intuitionistic modal logic (IML) was introduced by Fischer-Servi (1977) as the intuitionistic counterpart of classical modal logic. completeness due to Fischer-Servi (1977) duality due to Palmigiano (2004) Positive modal logic (PML) was introduced by Dunn (1995) as the negation-free fragment of classical modal logic. completeness due to Dunn (1995) and Celani and Jansana (1997) duality due to Celani and Jansana (1999) Distributive modal logic (DML) was introduced by Gehrke, Nagahashi and Venema (2004) as an expansion of distributive logic by four mutually unrelated normal modalities � , ♦ , ⊳ , ⊲ . duality framed in terms of canonical extensions and perfect algebras Adam Pˇ renosil A Duality for Distributive Unimodal Logic 4 / 23
Introduction Semantics Expanding the language Duality Motivation #1 We have a completeness and a duality theorem for IML: ( W , ≤ , R ) + ( R ◦ ≤ ⊆ ≤ ◦ R ) + ( ≥ ◦ R ⊆ R ◦ ≥ ) . Independently, we have a completeness and a duality theorem for PML: ( W , ≤ , R ) + ( ≤ ◦ R ⊆ R ◦ ≤ ) + ( ≥ ◦ R ⊆ R ◦ ≥ ) . What we want to have: a completeness and a duality theorem for ( W , ≤ , R ) plus correspondence and canonicity theorems for the side conditions. Adam Pˇ renosil A Duality for Distributive Unimodal Logic 5 / 23
Introduction Semantics Expanding the language Duality Motivation #2 DML provides a different kind of modal semantics in terms of ( W , ≤ , R � , R ♦ ), where ≤ ◦ R � ◦ ≤ ⊆ R � and ≥ ◦ R ♦ ◦ ≥ ⊆ R ♦ . What is the relationship between this semantics and the semantics of ( W , ≤ , R )? Can we define in the modal language the class of frames such that R � = ≤ ◦ R ◦ ≤ R ♦ = ≥ ◦ R ◦ ≥ for some R ? What if we add the side conditions? Adam Pˇ renosil A Duality for Distributive Unimodal Logic 6 / 23
Introduction Semantics Expanding the language Duality Motivation #3 (my motivation) There is a natural way of extending a set-based relational semantics to a poset-based semantics: restrict the interpretation to upsets: if u � p & u ≤ v ⇒ v � p compose each � -like operation ◦ with the upper interior: u � ϕ ◦ + ψ iff ∀ v ≥ u v � ϕ ◦ ψ compose each ♦ -like operation ◦ with the upper closure: u � ϕ ◦ − ψ iff ∃ v ≤ u v � ϕ ◦ ψ What logic do we get when we apply this general procedure to the relational semantics of classical modal logic? Adam Pˇ renosil A Duality for Distributive Unimodal Logic 7 / 23
Introduction Semantics Expanding the language Duality Unirelational semantics Unirelational frames take the form ( W , ≤ , R ) for arbitrary binary R . The semantic clauses are: u � ϕ ∧ ψ iff u � ϕ and u � ψ u � ⊤ u � ϕ ∨ ψ iff u � ϕ or u � ψ u � ⊥ u � � ϕ iff ∀ v ≥ u ∀ w ( vRw ⇒ w � ϕ ) u � ♦ ϕ iff ∃ v ≤ u ∀ w ( vRw & w � ϕ ) and possibly also: u � ϕ → ψ iff ∀ v ≥ u ( v � ϕ ⇒ v � ψ ) u � ϕ � − ψ iff ∃ v ≤ u ( v � ϕ & v � ψ ) Adam Pˇ renosil A Duality for Distributive Unimodal Logic 8 / 23
Introduction Semantics Expanding the language Duality Unirelational semantics Unirelational frames take the form ( W , ≤ , R ) for arbitrary binary R . The semantic clauses are: u � ϕ ∧ ψ iff u � ϕ and u � ψ u � ⊤ u � ϕ ∨ ψ iff u � ϕ or u � ψ u � ⊥ u � � ϕ iff ∀ v ≥ u ∀ w ( vRw ⇒ w � ϕ ) u � ♦ ϕ iff ∃ v ≤ u ∀ w ( vRw & w � ϕ ) and possibly also: u � ϕ → ψ iff ∀ v ≥ u ( v � ϕ ⇒ v � ψ ) u � ϕ � − ψ iff ∃ v ≤ u ( v � ϕ & v � ψ ) Adam Pˇ renosil A Duality for Distributive Unimodal Logic 8 / 23
Introduction Semantics Expanding the language Duality Multirelational semantics Multirelational frames take the form ( W , ≤ , R � , R ♦ ), where ≤ ◦ R � ◦ ≤ ⊆ R � and ≥ ◦ R ♦ ◦ ≥ ⊆ R ♦ . The semantic clauses are: u � � ϕ iff ∀ v ( uR � v ⇒ v � ϕ ) u � ♦ ϕ iff ∃ v ( uR ♦ v & v � ϕ ) Unirelational frames can be viewed as multirelational frames such that R � = ≤ ◦ R ◦ ≤ & R ♦ = ≥ ◦ R ◦ ≥ for some R Conversely, the logic of multirelational frames can be viewed as a fragment ( � 1 , ♦ 2 ) of the bimodal logic ( � 1 , ♦ 1 , � 2 , ♦ 2 ) of unirelational frames. Adam Pˇ renosil A Duality for Distributive Unimodal Logic 9 / 23
Introduction Semantics Expanding the language Duality Multirelational semantics Multirelational frames take the form ( W , ≤ , R � , R ♦ ), where ≤ ◦ R � ◦ ≤ ⊆ R � and ≥ ◦ R ♦ ◦ ≥ ⊆ R ♦ . The semantic clauses are: u � � ϕ iff ∀ v ( uR � v ⇒ v � ϕ ) u � ♦ ϕ iff ∃ v ( uR ♦ v & v � ϕ ) Unirelational frames can be viewed as multirelational frames such that R � = ≤ ◦ ( R � ∩ R ♦ ) ◦ ≤ & R ♦ = ≥ ◦ ( R � ∩ R ♦ ) ◦ ≥ Conversely, the logic of multirelational frames can be viewed as a fragment ( � 1 , ♦ 2 ) of the bimodal logic ( � 1 , ♦ 1 , � 2 , ♦ 2 ) of unirelational frames. Adam Pˇ renosil A Duality for Distributive Unimodal Logic 9 / 23
Introduction Semantics Expanding the language Duality Multimodal algebras A multimodal algebra (distributive modal algebra) is a distributive lattice ( ∧ , ∨ , ⊤ , ⊥ ) with a box operator ( � ) and a diamond operator ( ♦ ): � ( a ∧ b ) = � a ∧ � b � ⊤ = ⊤ ♦ ( a ∨ b ) = ♦ a ∨ ♦ b ♦ ⊥ = ⊥ A multimodal Heyting (bi-Heyting) algebra is a multimodal algebra with a Heyting implication → (and a Heyting co-implication � − ). Proposition (Soundness) The complex algebra F + of a multimodal frame F is a multimodal (Heyting, bi-Heyting) algebra. Adam Pˇ renosil A Duality for Distributive Unimodal Logic 10 / 23
Introduction Semantics Expanding the language Duality Canonical frames The canonical frame A • of a multimodal algebra A is the poset of prime A and R ♦ filters Prime ( A ) equipped with the relations R � A such that U R � A V iff � a ∈ U ⇒ a ∈ V U R ♦ A V iff a ∈ V ⇒ ♦ a ∈ U Let η A : A → ( A • ) + be the function a �→ {U ∈ Prime ( A ) | a ∈ U} Theorem (Completeness) Each multimodal algebra embeds in the complex algebra of its canonical frame via the homomorphism η . Adam Pˇ renosil A Duality for Distributive Unimodal Logic 11 / 23
Introduction Semantics Expanding the language Duality Unimodal algebras A unimodal algebra is a multimodal algebra which satisfies ♦ a ≤ � b ∨ c ⇒ ♦ a ≤ ♦ ( a ∧ b ) ∨ c ( ♦ , � ) ♦ b ∧ c ≤ � a ⇒ � ( a ∨ b ) ∧ c ≤ � a ( � , ♦ ) We shall call these the positive and the negative modal law. Notice that these are dual: swapping ∧ with ∨ and � with ♦ transforms ( � , ♦ ) into ( ♦ , � ). It therefore suffices to deal with the positive modal law. With the help of → and � − , they can be expressed equationally: ♦ ( a ∧ b ) � − ♦ a ≤ � b � − ♦ a ( ♦ , � ) ♦ b → � a ≤ � ( a ∨ b ) → � a ( � , ♦ ) Adam Pˇ renosil A Duality for Distributive Unimodal Logic 12 / 23
Introduction Semantics Expanding the language Duality Soundness Proposition (Soundness) The complex algebra of a unimodal frame is a unimodal algebra. We show that ♦ a ≤ � b ∨ c ⇒ ♦ a ≤ ♦ ( a ∧ b ) ∨ c holds in the complex algebra. Suppose that ♦ a ≤ � b ∨ c and u � ♦ a , u � c . u � ♦ a u � c w � a Adam Pˇ renosil A Duality for Distributive Unimodal Logic 13 / 23
Introduction Semantics Expanding the language Duality Soundness Proposition (Soundness) The complex algebra of a unimodal frame is a unimodal algebra. We show that ♦ a ≤ � b ∨ c ⇒ ♦ a ≤ ♦ ( a ∧ b ) ∨ c holds in the complex algebra. Suppose that ♦ a ≤ � b ∨ c and u � ♦ a , u � c . u � ♦ a u � c w � a Adam Pˇ renosil A Duality for Distributive Unimodal Logic 13 / 23
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