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. First-order support of functions provides composition of functions at runtime.

Combinatory Logic

Built from only variables, a set of primitive combinators (closed λ-expressions with their own reduction rules), and function application, combinatory logic provides no abstraction mechanism. A Turing-complete kernel of primitive combinators is complete. The SKI calculus requires only the K and S combinators; Chris Barker's Iota and Jot require only one each.

SKI Calculus

• I ≡ λx. x (identity)
• K ≡ λx, y. x (generation of constant functions)
• S ≡ λx, y, z. (x z (y z)) (generalized function application)

Grammar

C-term ::= C-app | C-prim | var
C-app ::= C-term C-term

Fixed-Point Combinators

For a function F, its set of fixed points are those inputs which map to themselves. Provided algebraic functions, for instance, fixed points of a function F(x) would be found by setting F(x) = x and solving for the roots. The fixed-point combinators compute the fixed points of their functional 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. Distinct free variables specify a λ-expression's distinct substitution classes.

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 via Substitution

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
  • An expression is in β-normal form if no β-reductions are possible. It is otherwise a β-redex.
• η-reduction eliminates bindings unused within a scope, preserving η-equivalence:
  • (λx. (f x)) →_η f
  • An expression is in β-η-normal form ("Beta-eta-normal form") if neither β- nor η-reductions are possible
    • If a η-reduction is applicable, the expression is a η-redex.

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 leftmost call-by-value ("applicative order")
• leftmost-outermost application substitutes prior to evaluation, and is equivalent to lazy call-by-name (call-by-necessity, "normal order")

Call-by-name terminates if any order does, but might require more total reductions. Call-by-value only β-reduces abstractions, not applications. Call-by-need memoizes the results of evaluated functions, and is equivalent to call-by-name if side-effects are disallowed. Call-by-name is text substitution augmented by capture avoidance. Wikipedia's evaluation strategy page is pretty thorough.

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)}(x) = (f^m(x))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

Sources

• Professor Spencer Rugaber's CS6390 "Theory and Design of Programming Languages". Georgia Tech, Fall 2009.
• Goldberg. "On the Recursive Enumerability of Fixed-Point Combinators". BRICS Report RS-05-1, January 2005.
• Van Roy and Haridi. Concepts, Techniques and Models of Computer Programming. MIT Press, 2004.
• Pierce. Types and Programming Languages. MIT Press, 2002.
• Felleisen. "A Lecture on the Why of Y". 1997.
• Barendregt. The Lambda Calculus: Its Syntax and Semantics, Second Edition. North Holland, 1985.