A Duality for Distributive Unimodal Logic Adam P renosil Institute - - PowerPoint PPT Presentation

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

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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.

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Introduction Semantics Expanding the language Duality

Related logics

intuitionistic modal logic (IML): ∧, ∨, ⊤, ⊥ + , ♦ + → positive modal logic (PML): ∧, ∨, ⊤, ⊥ + , ♦ distributive modal logic (DML): ∧, ∨, ⊤, ⊥ + ±, ♦±

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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

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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.

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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?

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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?

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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)

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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)

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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 (uRv ⇒ 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.

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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 (uRv ⇒ 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.

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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.

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Introduction Semantics Expanding the language Duality

Canonical frames

The canonical frame A• of a multimodal algebra A is the poset of prime filters Prime(A) equipped with the relations R

A and R♦ A such that

UR

A V iff a ∈ U ⇒ a ∈ V

UR♦

AV iff a ∈ V ⇒ ♦a ∈ U

Let ηA : 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.

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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 (, ♦)

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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

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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

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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 v ♦a w a

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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 v ♦a v c w a

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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 v ♦a v c v b w a

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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 v ♦a v c v b w a w b

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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 v ♦a v c v b w a w b w a ∧ b

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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 u ♦(a ∧ b) v ♦a v c v b w a w b w a ∧ b

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Introduction Semantics Expanding the language Duality

Correspondence

Proposition The positive modal law corresponds to the following property (♦, ): For all u, v as on the left, there are u′, v′ as on the right. u v ♦ u u′ v′ v ♦ ♦

• Adam Pˇ

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Introduction Semantics Expanding the language Duality

Correspondence

Not what we want. We want to capture the condition R♦ ⊆ ≥ ◦ (R♦ ∩ R) ◦ ≥. Let (u, v) ⊑ (u′, v′) iff u ≤ v and v′ ≤ u′. Call a multimodal frame minimally generated if below each pair (u, v) ∈ R♦ there is some pair (u′, v′) ∈ R♦ which is minimal in R♦ with respect to ⊑. For each minimally generated frame F we have that F satisfies (♦, ) iff R♦ ⊆ ≥ ◦ (R ∩ R♦) ◦ ≥ Each canonical frame is minimally generated.

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Introduction Semantics Expanding the language Duality

Completeness and undefinability

Theorem (Completeness) Each unimodal algebra embeds in the complex algebra of a unimodal frame via η. Corollary (Undefinability) The class of unimodal frames is not definable by means of universal formulas in the language of multimodal (bi-Heyting) algebras.

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Introduction Semantics Expanding the language Duality

Locality conditions

We have the following correspondences: ≥ ◦ R ⊆ R ◦ ≥ a ∧ ♦b ≤ ♦(a ∧ b) ≤ ◦ R ⊆ R ◦ ≤ (a ∨ b) ≤ a ∨ ♦b R ◦ ≤ ⊆ ≤ ◦ R ♦a → b ≤ (a → b) R ◦ ≥ ⊆ ≥ ◦ R ♦(a −b) ≤ a −♦b These equations are all canonical. Each implies either the positive or the negative modal law. We thus obtain the existing completeness theorems for IML and PML. The class of unimodal frames remains undefinable even if we add some combination of these locality conditions.

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Introduction Semantics Expanding the language Duality

Full signature (without adjoints)

The completeness theorem extends to (any fragment of) the full language: u +ϕ iff ∀v≥u ∀w (vRw ⇒ w ϕ) u −ϕ iff ∀v≥u ∀w (vRw ⇒ w ϕ) u ♦+ϕ iff ∃v≤u ∀w (vRw & w ϕ) u ♦−ϕ iff ∃v≤u ∀w (vRw & w ϕ) The undefinability theorem extends precisely to all fragments of the full language which include {+, ♦+} or {−, ♦−}.

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Introduction Semantics Expanding the language Duality

Full signature (without adjoints)

The completeness theorem extends to (any fragment of) the full language: u +ϕ iff ∀v (uR

+v ⇒ v ϕ)

where ≤ ◦ R

+ ◦ ≤ ⊆ R +

u −ϕ iff ∀v (uR

−v ⇒ v ϕ)

where ≤ ◦ R

− ◦ ≥ ⊆ R −

u ♦+ϕ iff ∃v (uR♦

+v & v ϕ)

where ≥ ◦ R♦

+ ◦ ≥ ⊆ R♦ +

u ♦−ϕ iff ∃v (uR♦

−v & v ϕ)

where ≥ ◦ R♦

− ◦ ≤ ⊆ R♦ −

The undefinability theorem extends precisely to all fragments of the full language which include {+, ♦+} or {−, ♦−}.

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Introduction Semantics Expanding the language Duality

Full signature (without adjoints)

−a ∧ +(a ∨ b) ≤ +b ♦−a ∧ c ≤ +a ⇒ c ≤ +a +a ∧ −(a ∧ b) ≤ −b ♦+a ∧ c ≤ −a ⇒ c ≤ −a ♦+b ≤ ♦+(a ∧ b) ∨ ♦−a ♦+a ≤ −a ∨ c ⇒ ♦+a ≤ c ♦−b ≤ ♦−(a ∨ b) ∨ ♦+a ♦−a ≤ +a ∨ c ⇒ ♦−a ≤ c ♦+b ∧ c ≤ +a ⇒ +(a ∨ b) ∧ c ≤ +a ♦−b ∧ c ≤ −a ⇒ −(a ∧ b) ∧ c ≤ −a ♦+a ≤ +b ∨ c ⇒ ♦+a ≤ ♦+(a ∧ b) ∨ c ♦−a ≤ −b ∨ c ⇒ ♦−a ≤ ♦−(a ∨ b) ∨ c

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Introduction Semantics Expanding the language Duality

Duality for multimodal algebras

The completeness theorem can be extended to a duality based on the Priestley duality for distributive lattices. but . . . we want to get a Hennessy-Milner theorem for free which means we cannot assume compactness which means we need to have a dual adjunction a bitopological framework is suitable for this

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Introduction Semantics Expanding the language Duality

Duality for multimodal algebras

The completeness theorem can be extended to a duality based on the Priestley duality for distributive lattices. but . . . we want to get a Hennessy-Milner theorem for free which means we cannot assume compactness which means we need to have a dual adjunction a bitopological framework is suitable for this

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Introduction Semantics Expanding the language Duality

Duality for multimodal algebras

The completeness theorem can be extended to a duality based on the Priestley duality for distributive lattices. but . . . we want to get a Hennessy-Milner theorem for free which means we cannot assume compactness which means we need to have a dual adjunction a bitopological framework is suitable for this

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Introduction Semantics Expanding the language Duality

Duality for multimodal algebras

A bitopological poset is a poset equipped with a pair of topologies (τ+, τ−) such that each τ+-open is an upset and each τ−-open is a downset. We can define what it means to be compact to be Hausdorff to have a basis of clopens to be compactly branching (image-compact): for each u, {v | uR♦v} is compact in the lower topology (= is the downclosure of a finite set in the discrete case) and for each u, {v | uRv} is compact in the upper topology (= is the upclosure of a finite set in the discrete case) For compact spaces, τ+ and τ− can be recovered from τ+ ∨ τ−.

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Introduction Semantics Expanding the language Duality

Duality for multimodal algebras

Theorem (Dual adjunction) The complex algebra – canonical frame assignment extends to a dual adjunction between the category of multimodal algebras and a suitably defined category of compactly branching modal spaces. Restricting to compact modal space yields a dual equivalence. Corollary (Hennessy-Milner theorem) Any pair of non-bisimilar states is two unimodal finitely branching models can be distinguished by a formula of bi-intuitionistic unimodal logic.

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Introduction Semantics Expanding the language Duality

Conclusion

We succeeded in unifying the completeness and duality proofs for IML and PML relating the unirelational semantics of IML & PML and the multirelational semantics of DML It would be nice to better understand canonicity for quasiequations the assignment of an intuitionistic counterpart to a classical logic

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Introduction Semantics Expanding the language Duality

Conclusion

We succeeded in unifying the completeness and duality proofs for IML and PML relating the unirelational semantics of IML & PML and the multirelational semantics of DML It would be nice to better understand canonicity for quasiequations the assignment of an intuitionistic counterpart to a classical logic

Thank you for your attention.

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