Type theoretical approaches to opetopes Journes Logique Homotopie - PowerPoint PPT Presentation

point x System Opt ! : the point rule The first rule of Opt ! states that we may create points without any prior assumption: point . 14

System Opt ! : the point rule The first rule of Opt ! states that we may create points without any prior assumption: point point . x ∶ ∅ 14

x T degen-fill x x T System Opt ! : the degen-fill rule This rule takes an opetope and produces a degenerate opetope from it: . degen-fill . ⇓ 15

System Opt ! : the degen-fill rule This rule takes an opetope and produces a degenerate opetope from it: . x ∶ T degen-fill . degen-fill δ ∶ x ⊷ x ⊷ T ⇓ 15

t T fill t T System Opt ! : the fill rule This rule takes a pasting diagram (that is, a term), and creates an opetope by “filling” it: f g h i . . . . . fill g h . . . f i ⇓ µ . . 16

System Opt ! : the fill rule This rule takes a pasting diagram (that is, a term), and creates an opetope by “filling” it: f g h i . . . . . fill g h . . . f i ⇓ µ . . t ∶ T fill µ ∶ t ⊷ T 16

t s T x y U graft- a t a x s y a T System Opt ! : the graft rule This rules glues an opetope to a pasting diagram of the same dimension: . . . ⇓ ⇓ . . . . graft- a . . . ⇓ ⇓ . . 17

System Opt ! : the graft rule This rules glues an opetope to a pasting diagram of the same dimension: . . . ⇓ ⇓ . . . . graft- a . . . ⇓ ⇓ . . t ∶ s ⊷ T x ∶ y ⊷ U graft- a t ( a ← x ) ∶ s [ y / a ] ⊷ T 17

Example 1 Let’s derive g g c A c b b ⇓ α ⇓ γ f i h ⇛ f h ⇓ β a d a d j j 18

Example 1 Let’s derive g g b c A b c ⇓ α ⇓ γ f i h ⇛ f h ⇓ β a a d d j j Derivation of α point point b ∶ ∅ a ∶ ∅ fill fill g ∶ b ⊷ ∅ f ∶ a ⊷ ∅ graft- b g ( b ← f ) ∶ b [ a / b ] ⊷ ∅ �ÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜ� ≡ a α ∶ g ( b ← f ) ⊷ a ⊷ ∅ fill . 18

Example 1 Let’s derive g g c A c b b ⇓ α ⇓ γ ⇛ f i h f h ⇓ β a d a d j j Derivation of β point point c ∶ ∅ a ∶ ∅ fill fill h ∶ c ⊷ ∅ i ∶ a ⊷ ∅ graft- c h ( c ← i ) ∶ c [ a / c ] ⊷ ∅ �ÜÜÜÜÜ�ÜÜÜÜÜ� ≡ a β ∶ h ( c ← i ) ⊷ a ⊷ ∅ fill 18

Example 1 Let’s derive g g c c b A b ⇓ α ⇓ γ f i h ⇛ f h ⇓ β a a d d j j And we assemble to get A ⋮ ⋮ β ∶ h ( c ← i ) ⊷ a ⊷ ∅ α ∶ g ( b ← f ) ⊷ a ⊷ ∅ graft- i β ( i ← α ) ∶ h ( c ← i )[ g ( b ← f )/ i ] ⊷ a ⊷ ∅ �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� ≡ h ( c ← g ( b ← f )) A ∶ β ( i ← α ) ⊷ h ( c ← g ( b ← f )) ⊷ a ⊷ ∅ fill 18

Example 2 Let’s derive . . . . ⇓ ⇓ ⇓ ⇛ 19

Example 2 Let’s derive . . . . ⇓ ⇓ ⇓ ⇛ Top left part point a ∶ ∅ degen-fill α ∶ a ⊷ a ⊷ ∅ 19

Example 2 Let’s derive . . . . ⇓ ⇓ ⇓ ⇛ Bottom part point point b ∶ ∅ a ∶ ∅ fill fill g ∶ b ⊷ ∅ f ∶ a ⊷ ∅ graft- b g ( b ← f ) ∶ a ⊷ ∅ β ∶ g ( b ← f ) ⊷ a ⊷ ∅ fill 19

Example 2 Let’s derive . . . . ⇓ ⇓ ⇓ ⇛ And we assemble ⋮ ⋮ β ∶ g ( b ← f ) ⊷ a ⊷ ∅ α ∶ a ⊷ a ⊷ ∅ graft- f a = b ⊢ β ( f ← α ) ∶ g ( b ← f )[ a / f ] ⊷ a ⊷ ∅ �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� ≡ g a = b ⊢ A ∶ β ( f ← α ) ⊷ g ⊷ a ⊷ ∅ fill 19

The “unnamed” approach

Then a cell in a pasting diagram no longer needs to have a name, it can be identified by its address in that tree. Idea Since opetopes are pasting diagrams whose cells are many-to-one , they can be represented as trees: . . ∎ ∎ ∎ 2 1 ⇓ . . . . ⟿ ⇛ ∎ ∎ ∎ ⇓ ⇓ 3 ⇓ . . . . ∎ 20

Idea Since opetopes are pasting diagrams whose cells are many-to-one , they can be represented as trees: . . ∎ ∎ ∎ 2 1 ⇓ . . . . ⟿ ⇛ ∎ ∎ ∎ ⇓ ⇓ 3 ⇓ . . . . ∎ Then a cell in a pasting diagram no longer needs to have a name, it can be identified by its address in that tree. 20

and by the unique 1-opetope, a.k.a. the arrow: . . We can represent as a node of a tree as follows: Let us add address information. Idea: dimension 0 and 1 Denote by ⧫ the unique 0-opetope, a.k.a. the point: . 21

We can represent as a node of a tree as follows: Let us add address information. Idea: dimension 0 and 1 Denote by ⧫ the unique 0-opetope, a.k.a. the point: . and by ∎ the unique 1-opetope, a.k.a. the arrow: . . 21

Let us add address information. Idea: dimension 0 and 1 Denote by ⧫ the unique 0-opetope, a.k.a. the point: . and by ∎ the unique 1-opetope, a.k.a. the arrow: . . We can represent ∎ as a node of a tree as follows: ⧫ ∎ ⧫ 21

Idea: dimension 0 and 1 Denote by ⧫ the unique 0-opetope, a.k.a. the point: . and by ∎ the unique 1-opetope, a.k.a. the arrow: . . We can represent ∎ as a node of a tree as follows: ⧫ ∗ [ ϵ ] ∎ ⧫ Let us add address information. 21

2. consider that tree like a node, where the input edges are the nodes of said tree 3. be convinced that this is a good representation of some 2-opetope! Idea: dimension 2 Then we can: 1. create a tree with that node representing ∎ ⧫ ∗ [ ∗∗ ] ∎ ⧫ ∗ [ ∗ ] ∎ ⧫ ∗ [ ϵ ] ∎ ⧫ 22

3. be convinced that this is a good representation of some 2-opetope! Idea: dimension 2 Then we can: 1. create a tree with that node representing ∎ ⧫ ∗ [ ∗∗ ] ∎ [ ϵ ] [ ∗ ] ⧫ ∗ [ ∗∗ ] [ ∗ ] ∎ ∎ ∎ [ ϵ ] 3 ∎ ⧫ ∗ [ ϵ ] ∎ ∎ ⧫ 2. consider that tree like a node, where the input edges are the nodes of said tree 22

Idea: dimension 2 Then we can: 1. create a tree with that node representing ∎ ⧫ ∗ [ ∗∗ ] ∎ [ ϵ ] . . [ ∗ ] ⧫ ∗ [ ∗∗ ] [ ∗ ] ∎ ∎ ∎ [ ϵ ] ⇓ 3 ∎ . . ⧫ ∗ [ ϵ ] ∎ ∎ ⧫ 2. consider that tree like a node, where the input edges are the nodes of said tree 3. be convinced that this is a good representation of some 2-opetope! 22

Idea: dimension 2 Depending on the original tree, we obtain different 2-opetopes: ⧫ ∗ [ ∗ ] [ ϵ ] . [ ∗ ] ∎ ∎ ∎ [ ϵ ] ⟿ ⟿ ⧫ 2 ∗ ⇓ [ ϵ ] . . ∎ ∎ ⧫ 23

Idea: dimension 2 Depending on the original tree, we obtain different 2-opetopes: ⧫ ∗ [ ∗∗⋯∗ ] ∎ ⧫ ∗ ⋯ [ ϵ ] [ ∗ ] ⋮ [ ∗∗⋯∗ ] ( 2 ) . ∎ . . ∎ ∎ [ ϵ ] n ⟿ ⟿ ( 1 ) ⇓ ( n ) . . ⧫ ∗ [ ∗ ] ∎ ∎ ⧫ ∗ [ ϵ ] [ ϵ ] ∎ ⧫ 23

Idea: dimension 2 Depending on the original tree, we obtain different 2-opetopes: ⧫ [ ϵ ] ∗ ∎ [ ϵ ] [ ϵ ] 1 ⟿ ⟿ ⇓ ∎ . . ⧫ ∎ 23

Idea: dimension 2 Depending on the original tree, we obtain different 2-opetopes: [ ϵ ] . 0 ⟿ ⟿ ⧫ ⇓ ∎ 23

Idea: dimension 3 From there, repeat the process! [ ϵ ] [ ∗ ] ∎ ∎ [ ϵ ] [[ ∗ ]] [[ ∗ ]] 2 2 2 [ ϵ ] [ ϵ ] ⟿ A ∎ ∎ ] [ ϵ ] ∗ [ 2 3 ∎ . . . . ⟿ ⇓ ⇓ ⇛ ⇓ . . . . 24

Idea: dimension 3 From there, repeat the process! [[ ϵ ]] 0 [ ϵ ] [[ ϵ ]] 1 0 [ ϵ ] ∎ [ ϵ ] [ ϵ ] ⟿ 1 B 0 ∎ . . ⇓ ⟿ ⇛ ⇓ ⇓ 24

Idea: dimension 3 From there, repeat the process! [[ ∗ ]] 0 [ ϵ ] [ ϵ ] [[ ∗ ]] 2 0 ∎ ∎ ] [ ϵ ] [ ϵ ] ∗ [ ⟿ 2 C 0 ∎ . . . . ⇓ ⟿ ⇓ ⇓ ⇛ 24

Idea: dimension 3 From there, repeat the process! [ ϵ ] [ ∗ ] [ ϵ ] [ ϵ ] ∎ ∎ ∎ [[ ∗ ]] [[ ∗∗ ]] [[ ∗ ]] 2 1 3 2 1 [[ ∗∗ ]] [ ϵ ] [ ϵ ] ⟿ D [ ∗ ] ∎ ∎ ∎ [ ϵ ] [ ∗∗ ] 3 4 ∎ . . ⇓ . . . . ⟿ ⇛ ⇓ ⇓ ⇓ . . . . 24

Solution In an n -opetope, every node is decorated by n 1 -opetope, but n 1 -opetope does not uniquely identify a node. But addresses do! So we just need to describe a partial map n 1 Syntax We now want a syntactical description of such trees. 25

but n 1 -opetope does not uniquely identify a node. But addresses do! So we just need to describe a partial map n 1 Syntax We now want a syntactical description of such trees. Solution ∎ ∎ 2 . . . . ⟿ ⇓ ∎ ∎ ⇓ ⇛ ⇓ 2 . . . . ∎ In an n -opetope, every node is decorated by ( n − 1 ) -opetope, 25

But addresses do! So we just need to describe a partial map n 1 Syntax We now want a syntactical description of such trees. Solution ∎ ∎ 2 . . . . ⟿ ⇓ ∎ ∎ ⇓ ⇛ ⇓ 2 . . . . ∎ In an n -opetope, every node is decorated by ( n − 1 ) -opetope, but ( n − 1 ) -opetope does not uniquely identify a node. 25

Syntax We now want a syntactical description of such trees. Solution [ ϵ ] [ ∗ ] ∎ ∎ [[ ∗ ]] 2 . . . . [ ϵ ] ⟿ ⇓ ⇓ ⇛ ∎ ∎ ] ⇓ [ ϵ ] ∗ [ . . . . 2 ∎ In an n -opetope, every node is decorated by ( n − 1 ) -opetope, but ( n − 1 ) -opetope does not uniquely identify a node. But addresses do! So we just need to describe a partial map 25 A � → O n − 1 .

Syntax We encode opetopes recursively as follows: [ ϵ ] [ ∗ ] ⎧ ∎ ∎ ⎪ [[ ∗ ]] ⎪ [ ϵ ] ← 2 2 ⎨ [ ϵ ] ⎪ [[ ϵ ]] ← 2 ⎪ ⟿ ⎩ ∎ ∎ ] [ ϵ ] ∗ [ 2 ∎ 26

Syntax We encode opetopes recursively as follows: [ ϵ ] [ ∗ ] ⎧ ∎ ∎ ⎪ [[ ∗ ]] ⎪ [ ϵ ] ← 2 2 ⎨ [ ϵ ] ⎪ [[ ϵ ]] ← 2 ⎪ ⟿ ⎩ ∎ ∎ ] [ ϵ ] ∗ [ 2 ∎ Reminder ⧫ ∗ [ ∗ ] ∎ 2 = ⧫ ∗ [ ϵ ] ∎ 26 ⧫

Syntax We encode opetopes recursively as follows: [ ϵ ] [ ∗ ] ⎧ ∎ ∎ ⎪ [[ ∗ ]] ⎪ [ ϵ ] ← 2 2 ⎨ [ ϵ ] ⎪ [[ ϵ ]] ← 2 ⎪ ⟿ ⎩ ∎ ∎ ] [ ϵ ] ∗ [ 2 ∎ Reminder ⧫ ∗ ⎧ [ ∗ ] ⎪ ⎪ [ ϵ ] ← ∎ ⎨ ∎ 2 ⎪ [ ∗ ] ← ∎ ⎪ = = ⧫ ⎩ ∗ [ ϵ ] ∎ 26 ⧫

Syntax We encode opetopes recursively as follows: ⎧ ⎧ ⎪ ⎪ [ ϵ ] [ ϵ ] ← ∎ ⎪ ⎪ ⎪ ⎪ [ ϵ ] ← ⎨ [ ∗ ] ⎪ ⎪ ∎ ∎ ⎪ ⎪ [ ∗ ] ← ∎ [[ ∗ ]] ⎪ ⎪ ⎩ 2 ⎨ [ ϵ ] ⎧ ⎪ ⎪ [ ϵ ] ← ∎ ⎪ ⎪ ⟿ ⎪ ⎪ [[ ϵ ]] ← ⎨ ∎ ⎪ ∎ ] [ ϵ ] ⎪ ∗ ⎪ ⎪ [ [ ∗ ] ← ∎ ⎪ ⎪ 2 ⎩ ⎩ ∎ Reminder ⧫ ∗ ⎧ [ ∗ ] ⎪ ⎪ [ ϵ ] ← ∎ ⎨ ∎ 2 ⎪ [ ∗ ] ← ∎ ⎪ = = ⧫ ⎩ ∗ [ ϵ ] ∎ 26 ⧫

Syntax We encode opetopes recursively as follows: ⎧ ⎧ ⎪ ⎪ [ ϵ ] [ ϵ ] ← ∎ ⎪ ⎪ ⎪ ⎪ [ ϵ ] ← ⎨ [ ∗ ] ⎪ ⎪ ∎ ∎ ⎪ ⎪ [ ∗ ] ← ∎ [[ ∗ ]] ⎪ ⎪ ⎩ 2 ⎨ [ ϵ ] ⎧ ⎪ ⎪ [ ϵ ] ← ∎ ⎪ ⎪ ⟿ ⎪ ⎪ [[ ϵ ]] ← ⎨ ∎ ⎪ ∎ ] [ ϵ ] ⎪ ∗ ⎪ ⎪ [ [ ∗ ] ← ∎ ⎪ ⎪ 2 ⎩ ⎩ ∎ Convention { ∗ ← ⧫ = ∎ 26

Syntax We encode opetopes recursively as follows: ⎧ ⎧ ⎪ ⎪ [ ϵ ] ← { ∗ ← ⧫ ⎪ [ ϵ ] ⎪ ⎪ ⎪ ⎪ [ ϵ ] ← ⎨ [ ∗ ] ⎪ ⎪ ⎪ ∎ ∎ ⎪ [ ∗ ] ← { ∗ ← ⧫ ⎪ [[ ∗ ]] ⎪ ⎩ 2 ⎨ [ ϵ ] ⎧ ⎪ ⎪ [ ϵ ] ← { ∗ ← ⧫ ⎪ ⎪ ⟿ ⎪ ⎪ ∎ ⎪ [[ ϵ ]] ← ⎨ ∎ ] [ ϵ ] ⎪ ∗ ⎪ [ ⎪ ⎪ 2 [ ∗ ] ← { ∗ ← ⧫ ⎪ ⎪ ⎩ ⎩ ∎ Convention { ∗ ← ⧫ = ∎ 26

1 0 Syntax: examples [[ ϵ ]] 0 [ ϵ ] ∎ [ ϵ ] 1 ∎ 27

Syntax: examples [[ ϵ ]] 0 ⎧ ⎪ ⎪ [ ϵ ] ← 1 ⎨ [ ϵ ] ∎ [ ϵ ] ⎪ [[ ϵ ]] ← 0 ⎪ ⟿ 1 ⎩ ∎ 27

Syntax: examples [[ ϵ ]] 0 ⎧ ⎪ ⎪ [ ϵ ] ← 1 ⎨ [ ϵ ] ∎ [ ϵ ] ⎪ [[ ϵ ]] ← 0 ⎪ ⟿ 1 ⎩ ∎ Reminder ⧫ ∗ {[ ϵ ] ← ∎ [ ϵ ] 1 = = ∎ ⧫ 27

Syntax: examples [[ ϵ ]] 0 ⎧ ⎪ [ ϵ ] ← {[ ϵ ] ← ∎ ⎪ ⎨ [ ϵ ] ∎ [ ϵ ] ⎪ [[ ϵ ]] ← 0 ⎪ ⟿ 1 ⎩ ∎ Reminder ⧫ ∗ {[ ϵ ] ← ∎ [ ϵ ] 1 = = ∎ ⧫ 27

Syntax: examples [[ ϵ ]] 0 ⎧ ⎪ [ ϵ ] ← {[ ϵ ] ← ∎ ⎪ ⎨ [ ϵ ] ∎ [ ϵ ] ⎪ [[ ϵ ]] ← 0 ⎪ ⟿ 1 ⎩ ∎ Reminder { ∗ ← ⧫ = ∎ 27

Syntax: examples [[ ϵ ]] 0 ⎧ ⎪ [ ϵ ] ← {[ ϵ ] ← { ∗ ← ⧫ ⎪ ⎨ [ ϵ ] ∎ [ ϵ ] ⎪ [[ ϵ ]] ← 0 ⎪ ⟿ 1 ⎩ ∎ Reminder { ∗ ← ⧫ = ∎ 27

Syntax: examples [[ ϵ ]] 0 ⎧ ⎪ [ ϵ ] ← {[ ϵ ] ← { ∗ ← ⧫ ⎪ ⎨ [ ϵ ] ∎ [ ϵ ] ⎪ [[ ϵ ]] ← 0 ⎪ ⟿ 1 ⎩ ∎ Reminder + convention { { ⧫ 0 = = ⧫ 27

Syntax: examples [[ ϵ ]] 0 ⎧ ⎪ [ ϵ ] ← {[ ϵ ] ← { ∗ ← ⧫ ⎪ ⎨ [ ϵ ] ∎ [ ϵ ] ⎪ [[ ϵ ]] ← { { ⧫ ⎪ ⟿ 1 ⎩ ∎ Reminder + convention { { ⧫ 0 = = ⧫ 27

2 0 Syntax: examples [[ ∗ ]] 0 [ ϵ ] ∎ ∎ ] [ ϵ ] ∗ [ 2 ∎ 28

Syntax: examples [[ ∗ ]] 0 ⎧ [ ϵ ] ⎪ ⎪ [ ϵ ] ← 2 ⎨ ∎ ∎ ] [ ϵ ] ⎪ [[ ϵ ]] ← 0 ∗ ⎪ [ ⟿ 2 ⎩ ∎ 28

Syntax: examples [[ ∗ ]] 0 ⎧ [ ϵ ] ⎪ ⎪ [ ϵ ] ← 2 ⎨ ∎ ∎ ] [ ϵ ] ⎪ [[ ϵ ]] ← 0 ∗ ⎪ [ ⟿ 2 ⎩ ∎ Reminder ⧫ ∗ ⎧ [ ∗ ] ⎪ [ ϵ ] ← ∎ ⎪ ∎ ⎨ 2 ⎪ [ ∗ ] ← ∎ ⎪ = = ⧫ ⎩ ∗ [ ϵ ] ∎ ⧫ 28

Syntax: examples ⎧ [[ ∗ ]] ⎧ ⎪ 0 ⎪ ⎪ [ ϵ ] ← ∎ ⎪ [ ϵ ] ⎪ [ ϵ ] ← ⎨ ⎪ ⎪ ⎨ [ ∗ ] ← ∎ ⎪ ∎ ∎ ] ⎩ [ ϵ ] ⎪ ∗ ⎪ [ ⟿ ⎪ 2 ⎪ [[ ϵ ]] ← 0 ⎩ ∎ Reminder ⧫ ∗ ⎧ [ ∗ ] ⎪ [ ϵ ] ← ∎ ⎪ ∎ ⎨ 2 ⎪ [ ∗ ] ← ∎ ⎪ = = ⧫ ⎩ ∗ [ ϵ ] ∎ ⧫ 28

Syntax: examples ⎧ [[ ∗ ]] ⎧ ⎪ 0 ⎪ ⎪ [ ϵ ] ← ∎ ⎪ [ ϵ ] ⎪ [ ϵ ] ← ⎨ ⎪ ⎪ ⎨ [ ∗ ] ← ∎ ⎪ ∎ ∎ ] ⎩ [ ϵ ] ⎪ ∗ ⎪ [ ⟿ ⎪ 2 ⎪ [[ ϵ ]] ← 0 ⎩ ∎ Reminder { ∗ ← ⧫ = ∎ 28

Syntax: examples ⎧ ⎧ [[ ∗ ]] ⎪ ⎪ [ ϵ ] ← { ∗ ← ⧫ 0 ⎪ ⎪ ⎪ [ ϵ ] ⎪ [ ϵ ] ← ⎨ ⎪ ⎪ ⎨ [ ∗ ] ← { ∗ ← ⧫ ⎪ ∎ ∎ ] ⎩ [ ϵ ] ⎪ ∗ ⎪ [ ⟿ ⎪ 2 ⎪ ⎪ [[ ϵ ]] ← 0 ⎩ ∎ Reminder { ∗ ← ⧫ = ∎ 28

Syntax: examples ⎧ ⎧ [[ ∗ ]] ⎪ ⎪ [ ϵ ] ← { ∗ ← ⧫ 0 ⎪ ⎪ ⎪ [ ϵ ] ⎪ [ ϵ ] ← ⎨ ⎪ ⎪ ⎨ [ ∗ ] ← { ∗ ← ⧫ ⎪ ∎ ∎ ] ⎩ [ ϵ ] ⎪ ∗ ⎪ [ ⟿ ⎪ 2 ⎪ ⎪ [[ ϵ ]] ← 0 ⎩ ∎ Reminder { { ⧫ 0 = = ⧫ 28

Syntax: examples ⎧ ⎧ [[ ∗ ]] ⎪ ⎪ [ ϵ ] ← { ∗ ← ⧫ ⎪ 0 ⎪ ⎪ [ ϵ ] ⎪ [ ϵ ] ← ⎨ ⎪ ⎪ ⎨ [ ∗ ] ← { ∗ ← ⧫ ⎪ ∎ ∎ ⎩ ] [ ϵ ] ⎪ ∗ ⎪ [ ⟿ ⎪ 2 ⎪ ⎪ [[ ϵ ]] ← { { ⧫ ⎩ ∎ Reminder { { ⧫ 0 = = ⧫ 28

3 1 2 Syntax: examples [ ϵ ] [ ∗ ] [ ϵ ] ∎ ∎ ∎ [[ ∗ ]] [[ ∗∗ ]] 2 1 [ ϵ ] ∎ [ ∗ ] ∎ ∎ [ ϵ ] [ ∗∗ ] 3 ∎ 29

Syntax: examples [ ϵ ] ⎧ [ ∗ ] ⎪ [ ϵ ] ← 3 [ ϵ ] ⎪ ∎ ∎ ∎ ⎪ [[ ∗ ]] [[ ∗∗ ]] ⎪ 2 1 ⎨ [[ ϵ ]] ← 1 [ ϵ ] ⎪ ⎪ ⟿ ⎪ ⎪ [[ ∗ ]] ← 2 ∎ [ ∗ ] ∎ ∎ ⎩ [ ϵ ] [ ∗∗ ] 3 ∎ 29

Syntax: examples [ ϵ ] ⎧ [ ∗ ] ⎪ [ ϵ ] ← 3 [ ϵ ] ⎪ ∎ ∎ ∎ ⎪ [[ ∗ ]] [[ ∗∗ ]] ⎪ 2 1 ⎨ [[ ϵ ]] ← 1 [ ϵ ] ⎪ ⎪ ⟿ ⎪ ⎪ [[ ∗ ]] ← 2 ∎ [ ∗ ] ∎ ∎ ⎩ [ ϵ ] [ ∗∗ ] 3 ∎ Reminder ⧫ ∗ [ ∗∗ ] ⎧ ⎪ [ ϵ ] ← ∎ ∎ ⎪ ⎪ ⎪ ⧫ ⎨ ∗ [ ∗ ] ← ∎ [ ∗ ] 3 ⎪ ⎪ = = ⎪ ∎ ⎪ [ ∗∗ ] ← ∎ ⎩ ⧫ ∗ [ ϵ ] 29 ∎ ⧫