Programming Languages and Compilers (CS 421)

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Programming Languages and Compilers (CS 421)

• Munawar Hafiz
• 2219 SC, UIUC
• http//www.cs.illinois.edu/class/cs421/

Based in part on slides by Mattox Beckman, as
updated by Vikram Adve and Gul Agha
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Two Problems

• Type checking
• Question Does exp. e have type ? in env ??
• Answer Yes / No
• Method Type derivation
• Typability
• Question Does exp. e have some type in env. ?? If
  so, what is it?
• Answer Type ? / error
• Method Type inference

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Type Inference - Outline

• Begin by assigning a type variable as the type of
  the whole expression
• Decompose the expression into component
  expressions
• Use typing rules to generate constraints on
  components and whole
• Recursively gather additional constraints to
  guarantee a solution for components
• Solve system of constraints to generate a
  substitution
• Apply substitution to orig. type var. to get
  answer

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Type Inference - Example

• What type can we give to
• fun x -gt fun f -gt f x?
• Start with a type variable and then look at the
  way the term is constructed

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Type Inference - Example

• First approximate
• - (fun x -gt fun f -gt f x) ?
• Second approximate use fun rule
• x ? - (fun f -gt f x) ?
• - (fun x -gt fun f -gt f x) ?
• ? ? (? ? ?)

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Type Inference - Example

• Third approximate use fun rule
• f ? x ? - (f x) ?
• x ? - (fun f -gt f x) ?
• - (fun x -gt fun f -gt f x) ?
• ? ? (? ? ?) ? ? (? ? ?)

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Type Inference - Example

• Fourth approximate use app rule
• f ? x ? - f ? ? ? f ? x ? - x
  ?
• f ? x ? - (f x) ?
• x ? - (fun f -gt f x) ?
• - (fun x -gt fun f -gt f x) ?
• ? ? (? ? ?) ? ? (? ? ?)

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Type Inference - Example

• Fifth approximate use var rule
• f ? x ? - f ? ? ? f ? x ? -
  x ?
• f ? x ? - (f x) ?
• x ? - (fun f -gt f x) ?
• - (fun x -gt fun f -gt f x) ?
• ? ? (? ? ?) ? ? (? ? ?) ? ? (? ? ?) .

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Type Inference - Example

• Sixth approximate use var rule
• f ? x ? - f ? ? ? f ? x ? -
  x ?
• f ? x ? - (f x) ?
• x ? - (fun f -gt f x) ?
• - (fun x -gt fun f -gt f x) ?
• ? ? (? ? ?) ? ? (? ? ?) ? ? (? ? ?) ? ? ?

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Type Inference - Example

• Done building proof tree now solve!
• f ? x ? - f ? ? ? f ? x ? -
  x ?
• f ? x ? - (f x) ?
• x ? - (fun f -gt f x) ?
• - (fun x -gt fun f -gt f x) ?
• ? ? (? ? ?) ? ? (? ? ?) ? ? (? ? ?) ? ? ?

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Type Inference - Example

• Type unification solve like linear equations
• f ? x ? - f ? ? ? f ? x ? -
  x ?
• f ? x ? - (f x) ?
• x ? - (fun f -gt f x) ?
• - (fun x -gt fun f -gt f x) ?
• ? ? (? ? ?) ? ? (? ? ?) ? ? (? ? ?) ? ? ?

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Type Inference - Example

• Eliminate ?
• f ? x ? - f ? ? ? f ? x ? -
  x ?
• f ? x ? - (f x) ?
• x ? - (fun f -gt f x) ?
• - (fun x -gt fun f -gt f x) ?
• ? ? (? ? ?) ? ? (? ? ?) ? ? (? ? ?) ? ? ?

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Type Inference - Example

• Next eliminate ?
• f ? ? ? x ? - f ? ? ? f ? ? ? x
  ? - x ?
• f ? ? ? x ? - (f x) ?
• x ? - (fun f -gt f x) ?
• - (fun x -gt fun f -gt f x) ?
• ? ? (? ? ?) ? ? ((? ? ?) ? ?) ? ? (? ? ?)

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Type Inference - Example

• Next eliminate ?
• f? ? ? x? - f? ? ? f? ? ? x? - x?
• f ? ? ? x ? - (f x) ?
• x ? - (fun f -gt f x) ((? ? ?) ? ?)
• - (fun x -gt fun f -gt f x) ?
• ? ? (? ? ((? ? ?) ? ?)) ? ? ((? ? ?) ? ?)

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Type Inference - Example

• Next eliminate ?
• f ? ? ? x? - f ? ? ? f? ? ? x? -
  x?
• f ? ? ? x ? - (f x) ?
• x ? - (fun f -gt f x) ((? ? ?) ? ?)
• - (fun x -gt fun f -gt f x) (? ? ((? ? ?) ?
  ?))
• ? ? (? ? ((? ? ?) ? ?))

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Type Inference - Example

• No more equations to solve we are done
• f? ? ? x? - f? ? ? f? ? ? x? -
  x?
• f ? ? ? x ? - (f x) ?
• x ? - (fun f -gt f x) ((? ? ?) ? ?)
• - (fun x -gt fun f -gt f x) (? ? ((? ? ?) ?
  ?))
• Any instance of (? ? ((? ? ?) ? ?)) is a valid
  type

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Type Inference Algorithm

• Let has_type (?, e, ?) S
• ? is a typing environment
• e is an expression
• ? is a (generalized) type,
• S is a set of equations between generalized
  types
• Idea S is the constraints on type variables
  necessary for ? - e ?
• Let Unif(S) be a substitution of generalized
  types for type variables solving S
• Solution Unif(S)(?) - e Unif(S)(?)

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Type Inference Algorithm

• has_type (?, exp, ?)
• Case exp of
• Var v --gt return ? ? ?(v)
• Const c --gt return ? ? ? where ? - c ? by
  the constant rules
• fun x -gt e --gt
• Let ?, ? be fresh variables
• Let S has_type (x ? ?, e, ?)
• Return ? ? ? ? ? ? S

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Type Inference Algorithm (cont)

• Case exp of
• App (e1 e2) --gt
• Let ? be a fresh variable
• Let S1 has_type(?, e1, ? ? ?)
• Let S2 has_type(?, e2, ?)
• Return S1 ? S2

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Type Inference Algorithm (cont)

• Case exp of
• If e1 then e2 else e3 --gt
• Let S1 has_type(?, e1, bool)
• Let S2 has_type(?, e2, ?)
• Let S2 has_type(?, e2, ?)
• Return S1 ? S2? S3

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Type Inference Algorithm (cont)

• Case exp of
• let x e1 in e2 --gt
• Let ? be a fresh variable
• Let S1 has_type(?, e1, ?)
• Let S2
• has_type(x ? ?, e2, ?)
• Return S1 ? S2

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Type Inference Algorithm (cont)

• Case exp of
• let rec x e1 in e2 --gt
• Let ? be a fresh variable
• Let S1 has_type(x ? ?, e1, ?)
• Let S2 has_type(x ? ?, e2, ?)
• Return S1 ? S2

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Type Inference Algorithm (cont)

• To infer a type, introduce type_of
• Let ? be a fresh variable
• type_of (?, e)
• Let S has_type (?, e, ?)
• Return Unif(S)(?)
• Need an algorithm for Unif