Lecture: Lambda the Magnificent

Background Material



The semantics of a pure functional programming language can be mathematically described by a substitution process that mimics our understanding of function application - i.e., substitute all occurrences of the formal parameters of a function in the function body by the arguments in the function call.

In order to formally study function definition and application and computability, logicians (Alonzo Church, Stephen Cole Kleene, and others) in the 1930s developed a formal notation called the lambda calculus and rules defining an equivalence relation on lambda expressions that encodes function definition and application. Church and others were able to prove properties of the lambda calculus that provide deep understanding of functions and provide important insight into programming languages. In particular, they showed the universality of the lambda calculus, the uniqueness of normal forms, and the undecidablity of the equivalence of lambda expressions.


  1. substitution model of function application
    1. applicative order [call be value]
    2. normal order [call by need]
  2. lambda expressions
    1. < exp > &rarr variable
    2. application: < exp > &rarr (< exp > < exp >)
    3. abstraction: < exp > &rarr (lambda variable . < exp >)
  3. free and bound variables
  4. alpha conversion: (lambda x.E) is equivalent to (lambda y,E[x/y]) provided y does not appear free in E and y is not bound by a lambda when substituted for x in E.
  5. beta reduction: ((lambda x.E) F) is equivalent to E[F/x], where F is substituted for all free occurences of the variable x in E, provided all free variables in F remain free when substituted for x in E.
  6. Normal forms and the Church-Rosser theorem.
  7. Universality of the lambda calculus
    1. Church numerals and arithmetic using lambda calculus
    2. boolean logic, predicates, and conditional statements using lambda calculus
    3. Data structures (lists, cons, car, cdr) using lambda calculus
    4. Recursion using lambda calculus
      • (define fact (lambda (n) (if (= n 0) 1 (* n (fact (- n 1))))))
      • (define g (lambda (f n) (if (= n 0) 1 (* n (f (- n 1))))))
      • Y combinator: (define Y (lambda (g) ((lambda (x) (g (x x))) (lambda (x) (g (x x))))))
      • fixed point: (g (Y g)) = (Y g)
      • (g (Y g) n)

Sample Functions



Created: May 22, 2008 (modified April 19, 2012) by jjohnson AT cs DOT drexel DOT DOT edu