11/25/2023 0 Comments Should i learn lambda calculusIt’s called β-reduction and it’s defined something like this: (λx.M)N -> M Interestingly enough, through this example, we have arrived at the central principle in lambda calculus. When given such a value, say ‘a’, the value of the expression is a² + 3a + 4. One can intuitively read ‘λx ’ as an expression that is waiting for a value for the variable x. The lambda operator allows us to abstract over x. To do the same thing in lambda calculus, we do this: λx. What is the value of f(x) when x = 3? We compute this by substituting the value of 3 for x in the expression. We are given a simple polynomial such as: f(x) = x² + 3x + 4 Let’s take a simple mathematical example. Lambda calculus is an elegant notation for working with applications of functions to arguments. If you want to apply a function ‘f’ to an input argument ‘y’, you can just write it as shown above. In this particular case, it will just return ‘x’. The lambda abstraction is basically a function that takes ‘x’ as the input and returns the expression on the right hand side of the dot. Here, the variable name ‘x’ is trivial and it can be replaced with anything. A “name”, also called a “variable”, is an identifier which can be any of the letters like a, b, or c. The central concept here is the “expression”. It is an approach more related to software than to hardware. However, the difference here is that lambda calculus emphasizes the use of transformation rules, and it does not care about the actual machine implementing them. Therefore, we can say that it is equivalent to Turing machines. Lambda calculus is universal in the sense that any computable function can be expressed and evaluated using this formalism. The transformation rule in question here is variable substitution. It’s nice to see such a direct link between mathematics and programming, right? Lambda calculus consists of a single transformation rule and a single function definition scheme. Lambda calculus can be called the smallest universal programming language of the world. For example, in Python, you can define a lambda function by using the keyword “lambda”. All the modern programming languages offer this in some form. The concept of anonymous functions comes from the field of lambda calculus. In your life as a programmer, you must have dealt with anonymous functions at some point of time. This particular thing is very useful when you are designing large software systems. Now you may ask, isn’t this a roundabout way of doing things? Well, as it turns out, this currying operation is a very useful concept and it lends itself very nicely when you are composing complex functions. Internally, we have ’n’ functions that take a single argument. In the end, you will have a full function that can take ’n’ arguments. What this means is that you can cascade any number of functions, where each function takes exactly one argument. For example, the abs() function is a good example of a unary function because it just takes a single argument and gives the absolute value of that argument. But what if we want to make multiple arguments? Well, in the world of lambda calculus, we deal with it in the form of sequence of abstractions. Unary functions are those that take exactly one argument. In the study of functions, it is sufficient to focus on unary functions. It was introduced in the 1930s by Alonzo Church as a way of formalizing the concept of effective computability. Lambda calculus arose from the study of functions as rules. So what exactly is lambda calculus? How do we understand it?īefore we proceed further, it is important to understand why we need lambda calculus in the first place. This feature makes is particularly useful in the field of mathematical logic. Even though the syntax of lambda calculus is sparse, it is really flexible and expressive. Here, we get a well-formed theory of functions as rules of computation. The good thing about lambda calculus is that the syntax is quite sparse, which makes it an elegant notation for representing functions. The main idea here is to apply a function to an argument and forming functions by abstraction. It forms the basis for all the modern functional programming languages like Haskell, Scala, Erlang, etc. It has a significant impact in the field of programming language theory. Now what on earth is lambda calculus? Well, lambda calculus is basically a simple notation for functions and applications in mathematics and computer science. Back in school, you must remember studying differential and integral calculus.
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