Alpha Conversion

Alpha conversion (also written \(\alpha\)-conversion) is a way of removing name clashes in expressions. A name clash arises when a \(\beta\)-reduction places an expression with a free variable in the scope of a bound variable with the same name as the free variable. — Greg Michaelson, An Introduction to Functional Programming Through Lambda Calculus When we are performing a \(beta\)-reduction it is possible that the variable name in an inner expression is the same as a variable name in an outer expression. [Read More]

Beta Reduction

Formally, beta reduction (also written \(\beta\)-reduction) is the replacement of a bound variable in a function body with a function argument. The purpose of \(\beta\)-reduction is to compute the result of a function by function application using specific rules. More formally, the beta reduction rule states that a function application of the form \((\lambda x.t)s\) reduces to the term \(t[x := s]\). The term \(t[x := s]\) means that all instances of \(x\) in \(t\) are replaced with \(s\). [Read More]

Eta Reduction

The purpose of eta reduction (also written \(\eta\)-reduction) is to drop an abstraction over a function to simplify it. This is possible when there is nothing more that a function can do to its argument. For example, imagine that we have a simple function \( f\ x = g\ x \). Both \(g\) and \(f\) take the same argument, \(x\), and the function application function results in the same value (specified by the equality symbol). [Read More]

Normal, Applicative and Lazy Evaluation

A lambda expression is said to be in normal form if it cannot be reduced any further, meaning that the expression no longer contains any function applications. More formally, a reducible expression is called a redex, and a lambda expression is in normal form when it contains no more redexes. Redex A reducible function expression. Normal Form A lambda expression that contains no redexes. Given a lambda expression, there are two primary strategies for reducing it to normal form: normal-order evaluation, or applicative-order evaluation. [Read More]

Representing Pairs and Lists in Lambda Calculus

Having covered types, let’s now turn our attention to lists. Lists are general purpose data structures for storing sequences of items. In lambda calculus, lists are represented using pairs, with the first item of the pair representing the head of the list, and the second item representing the rest of the list. A special value, nil, at as the second item of the pair terminates the list. Pairs Let’s start by focusing on pairs (or tuples). [Read More]

Typed Lambda Calculus

Lambda calculus is a very simple language. If you take away any syntactic sugar, all you are left with is functions that take arguments and return results. You can use these simple building blocks to construct functions that represent numbers and arithmetic, but there is no way to restrict, for example, arithmetic functions to require numeric operands. This is where types come in. Generally speaking, types allow you to control the use of functions so that only meaningful combinations of inputs and outputs are used. [Read More]

Recursive Lambda Functions the Y-Combinator

In a purely functional language — like lambda calculus — programs are expressed as nested function calls. Repetition in such an environment requires that nesting of function calls continues until some condition is met. During the repetition, each function passes its result to the next function in the nested chain and this repetition is completed when a test for some condition passes. The repetitive behaviour I’ve just described is recursion: [Read More]

Simplifying Lambda Syntax

Evaluating lambda functions requires using lots of brackets, which can be tedious and are a major source of error for evaluating expressions by hand. To simplify expressions, you can omit brackets when it is clear what the intention of the function is. Particularly, a function application can omit the brackets surrounding each individual parameter and assume the function is applied to the nearest argument. So, instead of expressing a function of three arguments as [Read More]

Introducing Lambda Calculus

Lambda calculus provides some of the foundational structures that functional programming is built from. It therefore seems fitting to start my journey through functional programming with a thorough examination of lambda calculus. This first post will introduce Lambda calculus with references to the book An Introduction to Functional Programming Through Lambda Calculus. Syntax The complete syntax for Lambda calculus is surprisingly small and comprehensible. This section lists the syntax in its entirety, while following sections break the syntax down to explain each part. [Read More]