Programming Language Theory

The Church-Turing Thesis equates a vaguely-defined set of "computable" functions with the partial recursive functions. Several systems are only as powerful as the partial recursives (Turing-complete): Turing machines and the λ-calculus are two. Programming languages provide further syntaxes and semantics, but all you really need in life is combinatory logic.

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

Higher-order functions map one or more functions to a function.

Combinatory Logic

SKI Calculus

• I ≡ λx. x
• K ≡ λx, y. x
• S ≡ λx, y, z. (x z (y z))

Fixed-Point Combinators

Higher-order functions which compute the fixed points of their inputs. Curry's Y-combinator was the first:

• Y ≡ λf. (λx. f (x x)) (λx. f (x x)) (untyped λ-calculus)
• Y ≡ S (K (S I I)) (S (S (K S) K) (K (S I I))) (SKI calculus)

Divergence-free evaluation of the Y-combinator requires call-by-name semantics. Call-by-value semantics can make use of the Θ_v (Turing) or Z-combinators:

• Θ_v ≡ (λx. λy. (y (λz. x x y z))) (λx. λy. (y (λz. x x y z)))
• Z ≡ λf. (λx. f (λy. x x y)) (λx. f (λy. x x y)) (via η-expansion on Y)

The infinitely many fixed-point combinators of untyped λ-calculus are recursively enumerable.

Untyped λ-calculus

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

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

The body is made up of free and bound variables. Those not present in the λ's list of bound variables are free. A λ-expression with no free variables is closed (closed expressions are equivalent in power to combinatory logic). A λ-expression enclosed in scopes of its free variables is meaningful.

Grammar

λ-term ::= λ-app | λ-abst | var
λ-app ::= λ-term λ-term
λ-abst ::= 'λ'var'.' λ-term

This abstract grammar can be augmented with associativity rules and grouping syntax (parentheses) to provide a concrete grammar. If verbosity is no issue, no associativity rules need be specified for the following grammar:

λ-term ::= λ-app | λ-abst | var
λ-app ::= '(' λ-term λ-term ')'
λ-abst ::= '(' 'λ'var'.' λ-term ')'

Evaluation

We'll use a substitution syntax to rewrite λ-calculus strings. {N / X}M substitutes expression N for free instances of X in expression M:

• If M is a variable ({N / X}V), replace M with X if and only if M is N.
• If M is a λ-app ({N / X}(A B)), {N / X}M = (({N / X}A) ({N / X}B))
• Otherwise, M is a λ-abst ({N / X}(λV. B)).
  • If X is V, M is unchanged (X is bound in M).
  • If X does not appear in B, M is unchanged (no free X in M)
  • If V occurs free in N, substitute N for X throughout B.
  • Otherwise, the substitution in this form is invalid (but it can be made valid...)

With this syntax, we define evaluation rules. Application of the a rule preserves its equivalence:

• ɑ-conversion renames bound variables with any variable y which is not free, preserving ɑ-equivalence:
  • λx. m →_ɑ λy. {y / x}m
• β-reduction replaces a formal parameter with an actual parameter, preserving β-equivalence:
  • (λx. m) n →_β {n / x}m
• η-reduction eliminates bindings unused within a scope, preserving η-equivalence:
  • (λx. (f x)) →_η f

According to the Church-Rosser Theorem, all terminating evaluations will compute the same function. Certain application orders might not terminate for a given expression, though, despite termination via other orders. This gives rise to the core difference between call-by-value and call-by-name semantics:

• leftmost-innermost application evaluates arguments as soon as possible, and is equivalent to call-by-value
• leftmost-outermost application substitutes prior to evaluation, and is equivalent to lazy call-by-name (call-by-necessity)
• call-by-name terminates if any order does, but might require more total reductions

Numerics

The integers (or any countably infinite set) can be represented via the Church encoding (or Mogensen-Scott, or others):

• 0 ≡ λf. λx. x
• 1 ≡ λf. λx. f x
• 2 ≡ λf. λx. f (f x)
• 3 ≡ λf. λx. f (f (f x))
• n ≡ λf. λx. fⁿx

Some basic arithmetic operations:

• plus ≡ λm. λn. λf. λx. m f (n f x) (from f^{(m + n)}(x) = f^m(fⁿ(x)))
• succ ≡ λn. λf. λx. f (n f x) (β-equivalent to (plus 1) for our 1 from above)
• mult ≡ λm. λn. λf. n (m f) (from f^{(m * n)} = (f^m)ⁿ)
• pow ≡ λm. λn. n m
• pred ≡ λn. λf. λx. n (λg. λh. h (g f)) (λu. x) (λu. u) (returns 0 when applied to 0)
• sub ≡ λm. λn. (n pred) m (using pred from above)

Logic

The Church booleans evaluate to one of their two arguments:

• true ≡ λa. λb . a
• false ≡ λa. λb . b

Using them, we implement basic conditional logic:

• if[A, B, C] ≡ A B C (use true/false for A. Evaluates to B on true, C on false)
• not ≡ λt. t false true (β-equivalent to if[A, false, true])

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