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Programming Language Theory

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Applicative/Functional Programming

Expressions compose functions rather than values. Backus proposed three tiers of complexity in his Turing Award lecture:

  • Simply functional language (fp): No state, limited names, finitely many functional forms, simple substitution semantics, algebraic laws
  • Formal functional system (ffp): Extensible functional forms, functions represented by objects, translation of object representation to applicable form, formal semantics
  • Applicative state transition system (ast): ffp plus mutable state and coarse-grained operations thereupon

Untyped λ-calculus

Two operators (function definition and application) upon one operand type (λ-expression).

  • Function definition: formalparam . body)
  • Function application: function(actualparam)

The integers (or any countably infinite set) can be represented via the Church encoding:

  • n ≡ λf . λx . fnx
  • plus ≡ λm . λn . λf . λx . m f (n f x) (from f(m + n)(x) = fm(fn(x)))

Common syntactic sugar:

  • Left-associative application as implicit parentheses
  • Use of definitions (allowing identifiers to stand in as λ-expressions)
  • Currying: (λx, y . x + y) rather than (λx . (λy . x + y))
  • Numeric literals rather than Church encoding