Initial Algebra Semantics for Cyclic Sharing Structures Makoto - PowerPoint PPT Presentation

Initial Algebra Semantics for Cyclic Sharing Structures Makoto Hamana Department of Computer Science, Gunma University, Japan July, 2009, TLCA’09 http://www.cs.gunma-u.ac.jp/ ˜ hamana/ 1

This Work ⊲ How to inductively capture cylces and sharing ⊲ Intended to apply it to functional programming 2

Introduction ⊲ Terms are a representation of tree structures (i) Reasoning: structural induction (ii) Functional programming: pattern matching, structural recursion (iii) Type: inductive type (iv) Initial algebra property 3

Introduction ⊲ What about tree-like structures? ⊲ How can we represent this data in functional programming? ⊲ Maybe: vertices and edges set, adjacency lists, etc. ⊲ Give up to use pattern matching, structural induction ⊲ Not inductive 4

Introduction Are really no inductive structures in tree-like structures? 5

This Work ⊲ Gives an initial algebra characterisation of cyclic sharing structures Aim ⊲ To derive the following from ↑ : [I] A simple term syntax that admits structural induction / recursion [II] To give an inductive type that represents cyclic sharing structures uniquely in functional languages and proof assistants 6

Variations of Initial Algebra Semantics ⊲ Various computational structures are formulated as initial algebras by varying the base category Abstract syntax Set ADJ 1975 Set S S -sorted abstract syntax Robinson 1994 Set F Abstract syntax with binding Fiore,Plotkin,Turi 1999 Recursive path ordring LO R. Hasegawa 2002 ↓ S ) S ( Set F S -sorted 2nd-order abs. syn. Fiore 2003 Pre F 2nd-order rewriting systems Hamana 2005 Explicit substitutions [ Set , Set ] f 2006 Ghani,Uustalu,Hamana ( Set T ∗ ) T Cyclic sharing structures Hamana 2009 7

Basic Idea 8

Basic Idea: Graph Algorithmic View ⊲ Traverse a graph in a depth-first search manner: Depth-First Search tree ⊲ DFS tree consists of 3 kinds of edges: (i) Tree edge (ii) Back edge (iii) Right-to-left cross edge ⊲ Characterise pointers for back and cross edges 9

Formulation: Cycles by µ -terms Idea ⊲ Binders as pointers ⊲ Back edges = bound variables Cycles µx. bin ( µy 1 . bin ( lf (5) , lf (6)) , µy 2 . bin ( x, lf (7) )) 10

Formulation: Sharing via ? Idea ⊲ Binders as pointers ⊲ Back edges = bound variables ⊲ Right-to-left Cross edges = ? Sharing µx. bin ( µy 1 . bin ( lf (5) , lf (6)) , µy 2 . bin ( , lf (7))) . Can we fill the blank to refer the node 5 by a bound variable? 11

Formulation: Sharing via Pointer ⊲ Cross edges = pointers by a new notation µx. bin ( µy 1 . bin ( lf (5) , lf (6)) , µy 2 . bin ( ւ 11 ↑ x , lf (7))) Pointer ւ 11 ↑ x means ⊲ going back to the node x , then ⊲ going down through the left child twice (by position 11 ) 12

Formulation: Sharing via Pointer ⊲ Cross edges = pointers by a new notation µx. bin ( µy 1 . bin ( lf (5) , lf (6)) , µy 2 . bin ( ւ 11 ↑ x , lf (7))) Pointer ւ 11 ↑ x means Need to ensure a correct pointer only!! ⊲ going back to the node x , then ⊲ going down through the left child twice (by position 11 ) 13

Typed Abstract Syntax for Cyclic Sharing Structures 14

Shape Trees ⊲ Skeltons of cyclic sharing trees Shape trees τ ::= E | P | L | B( τ 1 , τ 2 ) ⊲ Used as types ⊲ Blue nodes represent possible positions for sharing pointers. 15

Syntax and Types Typing rules p ∈ P os ( σ ) k ∈ Z Γ , x : σ, Γ ′ ⊢ ւ p ↑ x : P Γ ⊢ lf ( k ) : L x : B(E , E) , Γ ⊢ ℓ : σ x : B( σ, E) , Γ ⊢ r : τ Γ ⊢ µx. bin ( ℓ, r ) : B( σ, τ ) ⊲ A type declaration x : σ means: “ σ is the shape of a subtree headed by µx ”. ⊲ Taking a position p ∈ P os ( σ ) safely refers to a position in the subtree. 16

Example: making bin-node x : B(E , E) ⊢ x : B(B(L , L) , E) ⊢ µy 1 . bin (5 , 6) : B(L , L) µy 2 . bin ( ւ 11 ↑ x, 7) : B(P , L) ⊢ µx. bin ( µy 1 . bin (5 , 6) , µy 2 . bin ( ւ 11 ↑ x, 7)) : B(B(L , L) , B(P , L)) 17

Syntax and Types Typing rules (de Bruijn version) | Γ | = i − 1 p ∈ P os ( σ ) k ∈ Z Γ , σ, Γ ′ ⊢ ւ p ↑ i : P Γ ⊢ lf ( k ) : L B(E , E) , Γ ⊢ s : σ B( σ, E) , Γ ⊢ t : τ Γ ⊢ bin ( s, t ) : B( σ, τ ) Thm. Given rooted, connected and edge-ordered graph, the term representation in de Bruijn is unique. 18

Initial Algebra Semantics ⊲ Cyclic sharing trees are all well-typed terms: T τ (Γ) = { t | Γ ⊢ t : τ } ⊲ Need: sets indexed by contexts T ∗ and shape trees T Consider algebras in ( Set T ∗ ) T 19

Initial Algebra Semantics ⊲ Σ -algebra ( A, α : Σ A → A ) ✲ ( Set T ∗ ) T for ⊲ Functor Σ : ( Set T ∗ ) T cyclic sharing trees is defined by (Σ A ) E = 0 (Σ A ) P = PO (Σ A ) L = K Z (Σ A ) B( σ,τ ) = δ B(E , E) A σ × δ B( σ, E) A τ 20

Initial Algebra Semantics ⊲ Σ -algebra ( A, α : Σ A → A ) ✲ ( Set T ∗ ) T for ⊲ Functor Σ : ( Set T ∗ ) T cyclic sharing trees is given by ptr A : PO → A P lf A : K Z → A L bin σ,τ A : δ B(E , E) A σ × δ B( σ, E) A τ → A B( σ,τ ) Typing rules (de Bruijn version) | Γ | = i − 1 p ∈ P os ( σ ) k ∈ Z Γ , σ, Γ ′ ⊢ ւ p ↑ i : P Γ ⊢ lf ( k ) : L B(E , E) , Γ ⊢ s : σ B( σ, E) , Γ ⊢ t : τ Γ ⊢ bin ( s, t ) : B( σ, τ ) 21

Initial Algebra ⊲ All cyclic sharing trees T τ (Γ) = { t | Γ ⊢ t : τ } Thm. T forms an initial Σ -algebra. [Proof] ⊲ Smith-Plotkin construction of an initial algebra 22

Principles The initial algebra characterisation derives (i) Structural recursion by the unique homomorphism (ii) Structural induction by [Hermida,Jacobs I&C’98] (iii) Inductive type (in Haskell) 23

Inductive Type for Cyclic Sharing Structures Constructors of the initial algebra T ∈ ( Set T ∗ ) T ptr T (Γ) : PO(Γ) → T P (Γ); ւ p ↑ i �→ ւ p ↑ i. lf T (Γ) : Z → T L (Γ); k �→ lf ( k ) . bin σ,τ T (Γ) : T σ (B(E , E) , Γ) × T τ (B( σ, E) , Γ) → T B( σ,τ ) (Γ) GADT in Haskell data T :: * -> * -> * where Ptr :: Ctx n => n -> T n P Lf :: Ctx n => Int -> T n L Bin :: (Ctx n, Shape s, Shape t) => T (TyCtx (B E E) n) s -> T (TyCtx (B s E) n) t -> T n (B s t) ⊲ Dependent type def. in Agda is more straightforward 24

Summary ⊲ An initial algebra characterisation Goals ⊲ To derive the following from ↑ : [I] A simple term syntax [II] An inductive type for cyclic sharing structures 25

Connections to Other Works There are interpretations: ✲ S ! ✲ Equational Term Graphs T where S is any of (i) Coalgebraic (ii) Domain-theoretic (iii) Categorical semantics: Traced sym. monoidal categories [M. Hasegawa TLCA’97] – (Equational) term graphs [Barendregt et al.’87][Ariola,Klop’96] 26

Connections to Other Works There are interpretations: ✲ S ! ✲ Equational Term Graphs T where S is any of (i) Coalgebraic (ii) Domain-theoretic (iii) Categorical semantics: Traced sym. monoidal categories [M. Hasegawa TLCA’97] Further applications ✲ C (cyclic) (cartesian-center traced) T PPPPPP ∼ “arrows” with loops in Haskell q Haskell (efficient implementation) 27