Today I Learnt 10

Lambda Functions in Python

• First I looked at the following article from realpython.com

History

• Lambda calculus (or $\lambda$-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution.¹

• It is a universal model of computation. Alternative to a Turing Machine.

• Until the 1960s, the lambda calculus was only a formalism. Since Richard Montague and other linguists’ applied it in understanding the semantics of natural language, the lambda calculus has gained popularity in both linguistics, and computer science.

• Computable functions² are a fundamental concept within computer science and mathematics. The lambda calculus provides a simple semantics for computation, enabling properties of computation to be studied formally.

• Lambda calculus simplifies the semantics in two ways:

  1. Lambda calculus treats functions “anonymously”, without giving them explicit names. For example:

     square_sum$(x,y) = x^2 + y^2$

     can be turned anonymous form by rewriting it as

     $(x,y) \mapsto x^2 + y^2$

     You now think of it as a mapping of the tuple $x$, and $y$ to $x^2+y^2$.

  2. Lambda calculus only uses functions of a single input. For instance, the square_sum function is equivalent to the functions

     $x \mapsto (y \mapsto x^2+ y^2)$

     This method is known as currying. It transforms a function that takes multiple arguments into a chain of functions each with a single argument.

     We understand this approach to functions by calculating square_sum at $(5,2)$.

     $((x,y) \mapsto x^2 + y^2)(5,2)$ $= 5^2 +2^2$ $=29$

     On the other hand, the curried version requires an additional step.

     $\left( \left( x \mapsto (y \mapsto x^2 + y^2) \right)(5) \right)(2)$ $= (y \mapsto 5^2 + y^2)(2)$ $= 5^2 + 2^2$ $= 29$

• The lambda calculus consists of a language of lambda terms, which is defined by a certain formal syntax, and a set of transformation rules, which allow manipulation of the lambda terms.

• Lambda terms: The following three rules provide an inductive definition.

  1. A variable, $x$, is itself a valid lambda term.
  2. If $t$ is a lambda term, and $x$ is a variable, then $( \lambda x, t )$ is a lambda term (called an abstraction).
  3. If $t$ and $s$ are lambda terms, then $(ts)$ is a lambda term (called an application). Nothing else is a lambda term. Thus, a lambda term is valid if and only if it can be obtained by repeated application of these three rules.

• I will look at the mathematical background in detail at some other time. Let’s move on to Python.

Understanding Lambda Functions in Python

Let’s consider the following simple code:

lambda x: x + 1

It can be divided in the following three parts:

1. The keyword: lambda; indicates that a lambda function is being used.
2. A bound variable: x; refers to the variable to which we are applying the lambda function.
3. A body: x+1; indicates what the lambda function does to the bounded variable.

Continuing with Go 101

Introduction to Source Code Elements

https://go101.org/article/basic-code-elements-introduction.html