Polymorphism, Subtyping, and Type Inference in MLsub Stephen Dolan - - PowerPoint PPT Presentation
Polymorphism, Subtyping, and Type Inference in MLsub Stephen Dolan and Alan Mycroft January 18, 2017 Computer Laboratory University of Cambridge The select function select p v d = if ( p v ) then v else d In ML, select has type scheme . (
Polymorphism, Subtyping, and Type Inference in MLsub
Stephen Dolan and Alan Mycroft January 18, 2017
Computer Laboratory University of Cambridge
The select function
select p v d = if (p v) then v else d In ML, select has type scheme ∀α. (α → bool) → α → α → α
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Data flow in select
select p v d = if (p v) then v else d v argument to p result d
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Data flow in select
select p v d = if (p v) then v else d v argument to p result d In MLsub, select has this type scheme: ∀α, β. (α → bool) → α → β → (α ⊔ β)
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The MLsub Type System
Expressions of MLsub
We have functions x λx.e e1 e2 ... and records {ℓ1 = e1, . . . , ℓn = en} e.ℓ ... and booleans true false if e1 then e2 else e3 ... and let ˆ x let ˆ x = e1 in e2
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Typing rules of MLsub
MLsub is ML + (Sub) Γ ⊢ e : τ1 Γ ⊢ e : τ2 τ1 ≤ τ2
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Constructing Types
The standard definition of types looks like: τ ::= ⊥ | τ → τ | ⊤ (ignoring records and booleans for now)
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Constructing Types
The standard definition of types looks like: τ ::= ⊥ | τ → τ | ⊤ (ignoring records and booleans for now) with a subtyping relation like: ⊥ ≤ τ τ ≤ ⊤ τ ′
1 ≤ τ1
τ2 ≤ τ ′
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τ1 → τ2 ≤ τ ′
1 → τ ′ 2
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Lattices
These types form a lattice:
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Lattices
These types form a lattice:
e1 : τ1 e2 : τ2 if rand () then e1 else e2 : τ 1 ⊔ τ 2
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Bizzarely difficult questions
Is this true, for all α? α → α ≤ ⊥ → ⊤
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Bizzarely difficult questions
Is this true, for all α? α → α ≤ ⊥ → ⊤ How about this? (⊥ → ⊤) → ⊥ ≤ (α → ⊥) ⊔ α
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Bizzarely difficult questions
Is this true, for all α? α → α ≤ ⊥ → ⊤ How about this? (⊥ → ⊤) → ⊥ ≤ (α → ⊥) ⊔ α Yes, it turns out, by case analysis on α.
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Bizzarely difficult questions
Is this true, for all α? α → α ≤ ⊥ → ⊤ How about this? (⊥ → ⊤) → ⊥ ≤ (α → ⊥) ⊔ α Yes, it turns out, by case analysis on α. And only by case analysis.
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Extensibility
Let’s add a new type of function τ1
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Extensibility
Let’s add a new type of function τ1
It’s a supertype of τ1 → τ2 “function that may have side effects”
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Extensibility
Let’s add a new type of function τ1
It’s a supertype of τ1 → τ2 “function that may have side effects” Now we have a counterexample: α = (⊤
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Extensibility and vacuous reasoning
τr = some record type τf = some function type Is τr ⊓ τf ≤ bool?
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Extensibility and vacuous reasoning
τr = some record type τf = some function type Is τr ⊓ τf ≤ bool? Yes, vacuously.
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Extensible type systems
Two techniques give us an extensible system:
gets rid of case analysis
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Extensible type systems
Two techniques give us an extensible system:
gets rid of case analysis
gets rid of vacuous reasoning
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Resulting types
We end up with all the standard types
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Resulting types
We end up with all the standard types ... with the same subtyping order
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Resulting types
We end up with all the standard types ... with the same subtyping order ... but we identify fewer of the weird types {foo : bool} ⊓ (⊤ → ⊤) ≤ bool
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Type Inference with Polar Types
Input and output types
τ ⊔ τ ′: produces a value which is a τ or a τ ′ τ ⊓ τ ′: requires a value which is a τ and a τ ′ ⊔ is for outputs, and ⊓ is for inputs.
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Input and output types
τ ⊔ τ ′: produces a value which is a τ or a τ ′ τ ⊓ τ ′: requires a value which is a τ and a τ ′ ⊔ is for outputs, and ⊓ is for inputs. Divide types into
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Polar types
τ + ::= bool | τ −
1 → τ + 2 | {ℓ1 : τ + 1 , . . . , ℓn : τ + n } |
α | τ +
1 ⊔ τ + 2 | ⊥ | µα.τ +
τ − ::= bool | τ +
1 → τ − 2 | {ℓ1 : τ − 1 , . . . , ℓn : τ − n } |
α | τ −
1 ⊓ τ − 2 | ⊤ | µα.τ −
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Cases of unification
In HM inference, unification happens in three situations:
another
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Cases of unification
In HM inference, unification happens in three situations:
Introduce ⊓
Introduce ⊔
another τ + ≤ τ − constraint
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Eliminating variables, ML-style
Suppose we have an identity function α → α
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Eliminating variables, ML-style
Suppose we have an identity function, which uses its argument as a τ α → α | α = τ
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Eliminating variables, ML-style
Suppose we have an identity function, which uses its argument as a τ α → α | α = τ ≡∀ τ → τ
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Eliminating variables, ML-style
Suppose we have an identity function, which uses its argument as a τ α → α | α = τ ≡∀ τ → τ The substitution [τ/α] solves the constraint α = τ
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Eliminating variables, MLsub-style
Suppose we have an identity function, which uses its argument as a τ −. α → α | α ≤ τ −
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Eliminating variables, MLsub-style
Suppose we have an identity function, which uses its argument as a τ −. α → α | α ≤ τ − ≡∀ (α ⊓ τ −) → α
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Eliminating variables, MLsub-style
Suppose we have an identity function, which uses its argument as a τ −. α → α | α ≤ τ − ≡∀ (α ⊓ τ −) → α The bisubstitution [α ⊓ τ −/α−] solves α ≤ τ −
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Decomposing constraints
We only need to decompose constraints of the form τ + ≤ τ −. τ1 ⊔ τ2 ≤ τ3 ≡ τ1 ≤ τ3, τ2 ≤ τ3 τ1 ≤ τ2 ⊓ τ3 ≡ τ1 ≤ τ2, τ1 ≤ τ3 Thanks to the input/output type distinction, the hard cases of τ1 ⊓ τ2 ≤ τ3 and τ1 ≤ τ2 ⊔ τ3 can never come up.
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Combining solutions
We solve a system of multiple constraints C1, C2 by:
Then ξ ◦ ζ solves the system C1, C2.
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Summary
MLsub infers types by walking the syntax of the program, but must deal with subtyping constraints rather than just equalities. Thanks to:
we can always handle these constraints, producing a principal type.
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Future work {-# LANGUAGE ?? #-}
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Questions? http://www.cl.cam.ac.uk/~sd601/mlsub stephen.dolan@cl.cam.ac.uk
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Mutable references
References are generally considered “invariant”. Instead, consider ref a two-argument constructor (α, β) ref with operations: make : (α, α) ref get : (⊥, β) ref → β set : (α, ⊤) ref → α → unit
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