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A Tutorial Introduction to the Lambda Calculus. Raul Rojas FU Berlin, WS-97/98. Abstract This paper is a short and painless introduction to the calculus. Originally developed in order to study some mathematical properties of e ectively com- putable functions, this formalism has provided a strong theoretical foundation for the family of ...
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A Tutorial Introduction to the Lambda Calculus Raul Rojas (Edited by Norman Ramsey) Freie Universit at Berlin Version 2.0, 2015 Edit version 2.0-1, March 2018 Editor’s note This tutorial has been lightly edited in three ways. The original uses a nonstandard abbreviation for Curried functions of multiple arguments; this abbreviation has been ...
Lambda Abstraction The only other thing in the lambda calculus is lambda abstraction: a notation for defining unnamed functions. (λx. + x 1) (λ x. + x 1 ) ↑ ↑ ↑ ↑ ↑ ↑ That function of x that adds x to 1 Replace the λ with fun and the dot with an arrow to get a lambda expression in Ocaml: fun x -> (+) x 1
the lambda calculus • Originally, the lambda calculus was developed as a logic by Alonzo Church in 1932 at Princeton –Church says: “There may, indeed, be other applications of the system than its use as a logic.” –Dave says: “There sure are!” • The lambda calculus is a language of pure functions
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- Today, we will learn about
- Pure vs Applied Lambda Calculus
- Abstractions
- Lambda expressions extend as far to the right as possible
- Variable binding
- y : y a) x ) y
- Well-behaved nondeterminism
- Normal Order Evaluation
- CBV vs CBN
- CBV and CBN
Lambda calculus Full -reduction Call-by-value semantics Call-by-name semantics
The pure -calculus contains just function de nitions (called abstractions), variables, and function applications. If we add additional data types and operations (such as integers and addition), we have an applied -calculus.
Abstractions I An abstraction x: e is a function Variable x is the argument Expression e is the body of the function. The expression y : y y is a function that takes an argument y and returns square of y .
x: x y : y is the same as x: (x ( y : y )), and is not the same as ( x: x) ( y : y ).
An occurrence of a variable x in a term is bound if there is an enclosing x: e; otherwise, it is free. A closed term is one in which all identi ers are bound. Variable binding: (x : x (
Variable binding: (x : x ( y : y a) x ) y Both occurrences of x are bound The rst occurrence of y is bound The a is free The last y is also free, since it is outside the scope of the y .
( x: y : y ) ( z: z) Well-behaved nondeterminism ( x: y : y ) ( z: z) This term has two redexes in it, the one with abstraction x, and the one inside . Well-behaved nondeterminism The full -reduction strategy is non-deterministic. When a term has a normal form, however, it never has more than one. Full -reduction is con uent
Normal order evaluation uses the full -reduction rules, except the left-most redex is always reduced rst. Will eventually yield the normal form, if one exists. Allows reducing redexes inside abstractions
One way in which CBV and CBN di er is when arguments to functions have no normal forms. ( x:( y:y)) Under CBV semantics, this term does not have a normal form. If we use CBN semantics, then we have ( x:( y:y)) ! CBN y:y
CBV and CBN are common evaluation orders Many programming languages use CBV semantics \Lazy" languages, such as Haskell, typically use CBN semantics, a more e cient semantics similar to CBN in that it does not evaluate actual arguments unless necessary However, Call-by-value semantics ensures that arguments are evaluated at most once.
A Tutorial Introduction to the Lambda Calculus Raul Rojas Freie Universit at Berlin Version 2.0, 2015 Abstract This paper is a concise and painless introduction to the -calculus. This formalism was developed by Alonzo Church as a tool for study-ing the mathematical properties of e ectively computable functions.
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lambda calculus in the area of computation theory, it has contributed im-portant results to the formal semantics of programming languages: • Although the lambda calculus has the power to represent all computable functions, its uncomplicated syntax and semantics provide an excellent vehicle for studying the meaning of programming language ...
