Applications of entailments: de Groot duality Tatsuji Kawai Japan - PowerPoint PPT Presentation

Applications of entailments: de Groot duality Tatsuji Kawai Japan Advanced Institute of Science and Technology Workshop DOMAINS 8 July 2018, Oxford 1

De Groot duality A topological space is stably compact if it is sober, locally compact, and finite intersections of compact saturated subsets are compact. De Groot dual X d of a stably compact space X is a set X equipped with the cocompact topology (i.e. the topology generated by the complements of compact saturated subsets). The space X d is stably d = X . compact and ( X d ) Goubault-Larrecq (2010) showed that the de Groot duality induces a family of dualities on various powerdomain constructions: ◮ The dual of the Smyth powerdomain is the Hoare powerdomain of the dual, i.e. P U ( X ) d ∼ = P L ( X d ) , and vice versa . ◮ The Plotkin powerdomain construction commutes with duality, i.e. P V ( X ) d ∼ = P V ( X d ) . ◮ So does the probabilistic powerdomain construction. ◮ ... 2

Aim We give a point-free (and constructive) account of de Groot duality. A locale X is stably compact if the frame Ω( X ) is a continuous lattice and the set x ′ ∈ X | x ≪ x ′ � def � ։ = x is a filter for each x ∈ X . Stably compact locales are the Stone dual of stably compact spaces through the equivalence SoberSpa ∼ = SpatialLoc . The de Groot dual X △ of a stably compact locale X is the frame of Scott open filters on Ω( X ) (cf. Escardó 2000). 3

I tried to reconstruct the dualities due to Goubault-Larrecq in the point-free setting, and got a couple of results, e.g. P U ( X ) d ∼ = P L ( X d ) . But I got stuck . . . 4

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Plan 1. Strong proximity lattices 2. De Groot duality 3. Applications 7

Stably compact locales Proposition A locale is stably compact if and only if it is a retract of a spectral locale (i.e. the frame of ideals of a distributive lattice). Proof. A stably compact locale X is a retract of the frame of ideals Idl ( X ) . Since every idempotent splits in the category of locales: Corollary The category of stably compact locales is equivalent to the splitting of idempotents Split ( Spec ) of the category Spec of spectral locales. ◮ An object of Split ( Spec ) is an idempotent (i.e. f : X → X s.t. f ◦ f = f ) in Spec . ◮ A morphism g : ( f : X → X ) → ( f ′ : X ′ → X ′ ) in Split ( Spec ) is a continuous map g : X → X ′ in Spec such that f ′ ◦ g = g = g ◦ f . 8

Spectral locales A relation r ⊆ D × D ′ between distributive lattices D and D ′ is approximable if = { b ∈ D ′ | a r b } is a filter for each a ∈ D , def 1. ra def 2. r − b = { a ∈ D | a r b } is an ideal of D for each b ∈ D ′ , 3. a r 0 ′ = ⇒ a = 0 , ⇒ ( ∃ b ′ , c ′ ∈ D ) a ≤ b ′ ∨ c ′ & b ′ r b & c ′ r c . 4. a r b ∨ ′ c = Distributive lattices and approximable relations form a category DL AP with identities ≤ D and relational compositions. Proposition The category DL AP is equivalent to the category of spectral locales. 9

Strong proximity lattices (Jung & Sünderhauf) A strong proximity lattice is an object of Split ( DL AP ) , i.e. a distributive lattice D equipped with an idempotent relation ≺ such that def 1. ↓ a = { b ∈ D | b ≺ a } is an ideal def 2. ↑ a = { b ∈ D | b ≻ a } is a filter 3. a ≺ 0 = ⇒ a = 0 ⇒ ( ∃ b ′ ≺ b ) ( ∃ c ′ ≺ c ) a ≤ b ′ ∨ c ′ 4. a ≺ b ∨ c = 5. 1 ≺ a = ⇒ a = 1 ⇒ ( ∃ a ′ ≻ a ) ( ∃ b ′ ≻ b ) a ′ ∧ b ′ ≤ c . 6. a ∧ b ≺ c = Morphisms of strong proximity lattices are approximable relations. Remark A strong proximity lattice ( D , ≺ ) represents a stably compact locale X such that Ω( X ) ∼ = Rounded ideals of ( D , ≺ ) . An ideal I ⊆ D is rounded if a ∈ I ⇐ ⇒ ( ∃ b ≻ a ) b ∈ I . 10

Continuous entailment relations (Coquand & Zhang) An entailment relation on a set S is a binary relation ⊢ on the finite subsets of S such that a ∈ S A ⊢ B A ⊢ B , a a , A ⊢ B a ⊢ a A ′ , A ⊢ B , B ′ A ⊢ B where “,” denotes a union. Remark An entailment relation ( S , ⊢ ) presents a distributive lattice generated with generators S and relations � A ≤ � B for A ⊢ B . An entailment relation ( S , ⊢ ) is continuous if it is equipped with an idempotent relation ≺ on S such that ( ∃ C ) A ≺ U C ⊢ B ⇐ ⇒ ( ∃ D ) A ⊢ D ≺ L B where def A ≺ U B ⇐ ⇒ ( ∀ b ∈ B ) ( ∃ a ∈ A ) a ≺ b def A ≺ L B ⇐ ⇒ ( ∀ a ∈ A ) ( ∃ b ∈ B ) a ≺ b . 11

Continuous entailment relations Proposition The category of continuous entailment relations is equivalent to that of strong proximity lattices. Proof. ◮ If ( D , ≺ ) is a strong proximity lattice, then ( D , ⊢ D ) defined by def � � A ⊢ D B ⇐ ⇒ A ≤ D B together with ≺ is a continuous entailment relation. ◮ If ( S , ⊢ , ≺ ) is a continuous entailment relation, then the lattice D S generated by ( S , ⊢ ) together with the relation ≪ on D S defined by def � � � � A i ≪ ⇐ ⇒ ∀ i < N ∀ j < M ∃ C [ A i ≺ U C ⊢ B j ] B j i < N j < M is a strong proximity lattice. 12

Generated continuous entailment relations Let R be a set of pairs of finite subsets of a set S (called R a set of axioms ). An entailment relation ( S , ⊢ ) is generated by R if it is the smallest entailment relation on S that contains R , i.e. ⊢ is generated by the following rules: ( A , B ) ∈ R a ∈ S A ⊢ B A ⊢ B , a a , A ⊢ B A ⊢ B a ⊢ a A ′ , A ⊢ B , B ′ A ⊢ B Lemma Let ( S , ⊢ ) be the entailment relation generated by a set R of axioms. Then the dual ⊣ is generated by R op def = { ( B , A ) | ( A , B ) ∈ R } . Proposition Let ( S , ⊢ ) be the entailment relation generated by a set R of axioms, and let ≺ be an idempotent relation on S . Then ( S , ⊢ , ≺ ) is a continuous entailment if and only if for each A and B ⇒ ( ∃ D ′ ) A ⊢ D ′ ≺ L D 1. A ≺ U C & ( C , D ) ∈ R = ⇒ ( ∃ C ′ ) C ≺ U C ′ ⊢ B 2. ( C , D ) ∈ R & D ≺ L B = 13

Continuous entailment relation as the space its models Definition A model of a continuous entailment relation ( S , ⊢ , ≺ ) is a subset α ⊆ S such that 1. A ⊆ α = ⇒ ( ∃ b ∈ B ) b ∈ α for each A ⊢ B , 2. a ∈ α = ⇒ ( ∃ b ≺ a ) b ∈ α . Example If X is a locale presented by a strong proximity lattice ( D , ≺ ) , the Scott topology Σ( X ) can be defined as the space of its rounded ideals. Σ( X ) can be presented by an entailment relation on def = { ⊠ a | a ∈ D } generated by ⊠ D ∅ ⊢ ⊠ 0 ⊠ a , ⊠ b ⊢ ⊠ ( a ∨ b ) ⊠ a ⊢ ⊠ b ( a ≥ b ) def together with the idempotent relation ⊠ a ≺ ⊠ ⊠ b ⇐ ⇒ a ≻ b . 14

Plan 1. Strong proximity lattices 2. De Groot duality 3. Applications 15

Duality in strong proximity lattices Definition ◮ The dual D d of a strong proximity lattice ( D , ≺ ) is � D d , ≻ � , where D d is the dual lattice of D . ◮ The dual S d of a continuous entailment relation ( S , ⊢ , ≺ ) is ( S , ⊣ , ≻ ) . Proposition The equivalence between continuous entailment relations and strong proximity relations commutes with the dualities. Question If X is the stably compact locale presented by ( D , ≺ ) , does ( D d , ≻ ) present the de Groot dual of X ? 16

De Groot duality The de Groot dual X △ of a stably compact locale X is the frame of Scott open filters on Ω( X ) . Definition Let X be a stably compact locale. The upper powerlocale P U ( X ) is a locale whose points (i.e. models) are Scott open filters on Ω( X ) . The de Groot dual of X can be characterized by P U ( X ) ∼ = Σ( X d ) . 17

De Groot duality in strong proximity lattices Proposition Given a strong proximity lattice ( D , ≺ ) , let X and Y be stably compact locales presented by ( D , ≺ ) and ( D d , ≻ ) respectively. Then P U ( X ) ∼ = Σ( Y ) . Proof. The upper powerlocale P U ( X ) is presented by an entailment def = { ✷ a | a ∈ D } generated by relation on ✷ D ∅ ⊢ ✷ 1 ✷ a , ✷ b ⊢ ✷ ( a ∧ b ) ✷ a ⊢ ✷ b ( a ≤ b ) def together with the idempotent relation ✷ a ≺ ✷ ✷ b ⇐ ⇒ a ≺ b . The Scott topology Σ( Y ) is presented by an entailment relation on def = { ⊠ a | a ∈ D } generated by ⊠ D ∅ ⊢ ⊠ 1 ⊠ a , ⊠ b ⊢ ⊠ ( a ∧ b ) ⊠ a ⊢ ⊠ b ( a ≤ b ) def together with the idempotent relation ⊠ a ≺ ⊠ ⊠ b ⇐ ⇒ a ≺ b . � 18

Plan 1. Strong proximity lattices 2. De Groot duality 3. Applications 19

The lower and upper powerlocales Let X be a stably compact locale presented by a strong proximity lattice ( D , ≺ ) . ◮ The lower powerlocale P L ( X ) is presented by an entailment def = { ✸ a | a ∈ D } generated by relation on ✸ D ✸ 0 ⊢ ∅ ✸ ( a ∨ b ) ⊢ ✸ a , ✸ b ✸ a ⊢ ✸ b ( a ≤ b ) def together with the idempotent relation ✸ a ≺ ✸ ✸ b ⇐ ⇒ a ≺ b . ◮ The upper powerlocale P U ( X ) is presented by an entailment def = { ✷ a | a ∈ D } generated by relation on ✷ D ∅ ⊢ ✷ 1 ✷ a , ✷ b ⊢ ✷ ( a ∧ b ) ✷ a ⊢ ✷ b ( a ≤ b ) def together with the idempotent relation ✷ a ≺ ✷ ✷ b ⇐ ⇒ a ≺ b . 20