Computable dyadic subbases Arno Pauly and Hideki Tsuiki Second - PowerPoint PPT Presentation

Computable dyadic subbases Arno Pauly and Hideki Tsuiki Second Workshop on Mathematical Logic and its Applications 5-9 March 2018, Kanazawa Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 1 / 24

What is atomic information? Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 2 / 24

What is atomic information? Yes No Boolean Values { 0 , 1 } = 0 1 Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 2 / 24

What is atomic information? Yes No Boolean Values { 0 , 1 } = 0 1 ◮ ψ : ⊆ { 0 , 1 } ω → X : { 0 , 1 } ω -representation (partial surjective map from { 0 , 1 } ω to X ). Foundation of TTE theory [Weihrauch,...]. Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 2 / 24

What is atomic information? Yes No Boolean Values { 0 , 1 } = 0 1 ◮ ψ : ⊆ { 0 , 1 } ω → X : { 0 , 1 } ω -representation (partial surjective map from { 0 , 1 } ω to X ). Foundation of TTE theory [Weihrauch,...]. 1 Yes = Sierpinski Space S ⊥ Undefined Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 2 / 24

What is atomic information? Yes No Boolean Values { 0 , 1 } = 0 1 ◮ ψ : ⊆ { 0 , 1 } ω → X : { 0 , 1 } ω -representation (partial surjective map from { 0 , 1 } ω to X ). Foundation of TTE theory [Weihrauch,...]. 1 Yes = Sierpinski Space S ⊥ Undefined → S ω : embedding into S ω (= P ω ). ◮ ϕ : X ֒ ⋆ Every second countable space ( X , ( B n ) n ∈ N ) can be embedded into S ω by ϕ ( x )( n ) = 1 iff x ∈ B n . Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 2 / 24

What is atomic information? Yes No Boolean Values { 0 , 1 } = 0 1 ◮ ψ : ⊆ { 0 , 1 } ω → X : { 0 , 1 } ω -representation (partial surjective map from { 0 , 1 } ω to X ). Foundation of TTE theory [Weihrauch,...]. 1 Yes = Sierpinski Space S ⊥ Undefined → S ω : embedding into S ω (= P ω ). ◮ ϕ : X ֒ ⋆ Every second countable space ( X , ( B n ) n ∈ N ) can be embedded into S ω by ϕ ( x )( n ) = 1 iff x ∈ B n . ◮ ψ : ⊆ S ω → X : S ω -representation. Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 2 / 24

What is atomic information? Yes No Boolean Values { 0 , 1 } = 0 1 ◮ ψ : ⊆ { 0 , 1 } ω → X : { 0 , 1 } ω -representation (partial surjective map from { 0 , 1 } ω to X ). Foundation of TTE theory [Weihrauch,...]. 1 Yes = Sierpinski Space S ⊥ Undefined → S ω : embedding into S ω (= P ω ). ◮ ϕ : X ֒ ⋆ Every second countable space ( X , ( B n ) n ∈ N ) can be embedded into S ω by ϕ ( x )( n ) = 1 iff x ∈ B n . ◮ ψ : ⊆ S ω → X : S ω -representation. ⋆ A S ω -embedding ϕ induces a S ω -representation ϕ − 1 . Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 2 / 24

What is atomic information? Yes No Boolean Values { 0 , 1 } = 0 1 ◮ ψ : ⊆ { 0 , 1 } ω → X : { 0 , 1 } ω -representation (partial surjective map from { 0 , 1 } ω to X ). Foundation of TTE theory [Weihrauch,...]. 1 Yes = Sierpinski Space S ⊥ Undefined → S ω : embedding into S ω (= P ω ). ◮ ϕ : X ֒ ⋆ Every second countable space ( X , ( B n ) n ∈ N ) can be embedded into S ω by ϕ ( x )( n ) = 1 iff x ∈ B n . ◮ ψ : ⊆ S ω → X : S ω -representation. ⋆ A S ω -embedding ϕ induces a S ω -representation ϕ − 1 . ⋆ Enumeration-based { 0 , 1 } ω -representation ψ S ω : ⊆ { 0 , 1 } ω → S ω . A S ω -representation ψ induces a { 0 , 1 } ω -representation ψ S ω ◦ ψ . Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 2 / 24

Yes No 0 1 = Plotkin’s T ⊥ Undefined Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 3 / 24

Yes No 0 1 = Plotkin’s T ⊥ Undefined → T ω : embedding into T ω . ◮ ϕ : X ֒ Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 3 / 24

Yes No 0 1 = Plotkin’s T ⊥ Undefined → T ω : embedding into T ω . ◮ ϕ : X ֒ ◮ ψ : ⊆ T ω → X : T ω -representation. ⋆ A T ω -embedding ϕ induces a T ω -representation ϕ − 1 . Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 3 / 24

Yes No 0 1 = Plotkin’s T ⊥ Undefined → T ω : embedding into T ω . ◮ ϕ : X ֒ ◮ ψ : ⊆ T ω → X : T ω -representation. ⋆ A T ω -embedding ϕ induces a T ω -representation ϕ − 1 . However, T can be encoded in S × S ◮ T can be embed into S × S (0 �→ (1 , ⊥ ) , 1 �→ ( ⊥ , 1) , ⊥ �→ ( ⊥ , ⊥ )). ◮ ι : T ω ֒ → S ω . Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 3 / 24

Yes No 0 1 = Plotkin’s T ⊥ Undefined → T ω : embedding into T ω . ◮ ϕ : X ֒ ◮ ψ : ⊆ T ω → X : T ω -representation. ⋆ A T ω -embedding ϕ induces a T ω -representation ϕ − 1 . However, T can be encoded in S × S ◮ T can be embed into S × S (0 �→ (1 , ⊥ ) , 1 �→ ( ⊥ , 1) , ⊥ �→ ( ⊥ , ⊥ )). ◮ ι : T ω ֒ → S ω . Therefore, → T ω induces an embedding ι ◦ ϕ : X ֒ ◮ ϕ : X ֒ → S ω . ◮ ψ : ⊆ T ω → X induces a S ω -representation ψ ◦ ι − 1 : ⊆ S ω → X . ◮ They indue { 0 , 1 } ω -representations. Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 3 / 24

Yes No 0 1 = Plotkin’s T ⊥ Undefined → T ω : embedding into T ω . ◮ ϕ : X ֒ ◮ ψ : ⊆ T ω → X : T ω -representation. ⋆ A T ω -embedding ϕ induces a T ω -representation ϕ − 1 . However, T can be encoded in S × S ◮ T can be embed into S × S (0 �→ (1 , ⊥ ) , 1 �→ ( ⊥ , 1) , ⊥ �→ ( ⊥ , ⊥ )). ◮ ι : T ω ֒ → S ω . Therefore, → T ω induces an embedding ι ◦ ϕ : X ֒ ◮ ϕ : X ֒ → S ω . ◮ ψ : ⊆ T ω → X induces a S ω -representation ψ ◦ ι − 1 : ⊆ S ω → X . ◮ They indue { 0 , 1 } ω -representations. If X is represented over T ω , then X is also represented over { 0 , 1 } ω . Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 3 / 24

Yes No 0 1 = Plotkin’s T ⊥ Undefined → T ω : embedding into T ω . ◮ ϕ : X ֒ ◮ ψ : ⊆ T ω → X : T ω -representation. ⋆ A T ω -embedding ϕ induces a T ω -representation ϕ − 1 . However, T can be encoded in S × S ◮ T can be embed into S × S (0 �→ (1 , ⊥ ) , 1 �→ ( ⊥ , 1) , ⊥ �→ ( ⊥ , ⊥ )). ◮ ι : T ω ֒ → S ω . Therefore, → T ω induces an embedding ι ◦ ϕ : X ֒ ◮ ϕ : X ֒ → S ω . ◮ ψ : ⊆ T ω → X induces a S ω -representation ψ ◦ ι − 1 : ⊆ S ω → X . ◮ They indue { 0 , 1 } ω -representations. If X is represented over T ω , then X is also represented over { 0 , 1 } ω . Why do we study T ω -representation? Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 3 / 24

Why T ω -representation? Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 4 / 24

Why T ω -representation? T ω is more close to the space, so some information of the space can be reflected into the representation. ◮ Every n -dimensional second countable metrizable space can be embed n , which is a subspace of T ω with up to n copies of ⊥ [T 2002]. into T ω Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 4 / 24

Why T ω -representation? T ω is more close to the space, so some information of the space can be reflected into the representation. ◮ Every n -dimensional second countable metrizable space can be embed n , which is a subspace of T ω with up to n copies of ⊥ [T 2002]. into T ω Order structures ( T , � ) and ( T ω , � ). ◮ Natural representation of a space with order. 0 1 ⊥ ⊥ Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 4 / 24

Why T ω -representation? T ω is more close to the space, so some information of the space can be reflected into the representation. ◮ Every n -dimensional second countable metrizable space can be embed n , which is a subspace of T ω with up to n copies of ⊥ [T 2002]. into T ω Order structures ( T , � ) and ( T ω , � ). ◮ Natural representation of a space with order. Contains { 0 , 1 } ω as top elements. ◮ Sub-structure of the space can be represented with { 0 , 1 } ω . 0 1 ⊥ ⊥ Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 4 / 24

Why T ω -representation? T ω is more close to the space, so some information of the space can be reflected into the representation. ◮ Every n -dimensional second countable metrizable space can be embed n , which is a subspace of T ω with up to n copies of ⊥ [T 2002]. into T ω Order structures ( T , � ) and ( T ω , � ). ◮ Natural representation of a space with order. Contains { 0 , 1 } ω as top elements. ◮ Sub-structure of the space can be represented with { 0 , 1 } ω . A bottomed sequence is an unspecified sequence. ◮ 10 ⊥ 10 .. = { 10010 .., 10110 ... } . 0 1 ⊥ ⊥ Arno Pauly and Hideki Tsuiki Computable dyadic subbases Kanazawa, March 2018 4 / 24