![]() Numbers and arithmetic can be made precise using Church numerals. The above syntax uses a few things beyond the plain lambda calculus. \langle x*i, i+1 \rangle) \langle 1,1 \rangle Since $N$ can be represented as a Church numeral, we get \langle x*i, i+1 \rangle$ to the initial pair $\langle 1,1 \rangle$ for $N$ times. This effectively applies the function $\lambda\langle x,i \rangle. Then we put all the variables we use in a tuple (a pair suffices, here) (x,i) := (1,1) Suppose we want to translate a simple factorial FOR program x := 1 ![]() We'll extract the variables from each of them using used and merge them together using merge.Sure! Let me show how to encode FOR using an example. This case will go quite quickly using the same idea as above. Generally, lambda calculus refers to untyped lambda calculus, although, that is shown. It might look a bit silly to use merge to merge a singleton list into another list, but we need to keep the result ordered and merge is the hammer we have so we turned this problem into a nail. lambda calculus is a formalization of logic which describes computation. It's the function we're currently defining: used! So we can write this case as: used (Lambda var term) = merge (used term) It was one of many related systems that were proposed in the late 1920s and 1930s. For term we need a way to turn a Term into a variable or a list of variables. The lambda calculus was invented by Alonzo Church in the 1930s to study the interaction of functional abstraction and function application from an abstract, purely mathematical point of view. For var we can do the same thing as in the Variable case. If we knew all of those variables we could put those results together with merge. So some variables appear in var and some variables appear in term. Imperative programming languages are based on the Turing-machine. Maybe that's why we used those variable names ). Functional programming languages are based on the Lambda-calculus computational model. Lambda Calculus is so cool that many hackers. So again checking the type declaration of Term we see that var is a Var and term is a Term. Even cooler is that those very ideas affect us today: anytime you use a function, you owe a hat tip to Mr. The Lambda case used (Lambda var term) =. Thus, really the only option for the implementation is: used (Varable var) = ![]() ![]() Checking the declaration of Term and the Variable case, we see that var must be a Var. So the right hand side here has to be a list of Var. Lambda-reduction (also called lambda conversion) refers to all three. Three theorems of lambda calculus are -conversion, -conversion, and -conversion. In the lambda calculus, is defined as the abstraction operator. The Variable case: used (Variable var) =. Lambda Calculus A formal logic developed by Alonzo Church and Stephen Kleene to address the computable number problem. Most of that isn't captured in the type of used. While we do so let's keep in mind that used is meant to return an ordered list of Var without repeats. Let's go over how to fill in all the right hand sides of these cases. Finally note that we haven't actually written any of the bodies of the cases for used. Also note that variables in Haskell must start with lowercase characters. Note the parenthesis around each set of data constructor and arguments. It should look like the following: used :: Term -> Next lets look at the syntax of pattern matching. In particular being an empty list ( ) is not a way to be a Term, so that case should be removed. Furthermore, the fact that it is easy to define arithmetic on them streamlines the proof that lambda calculus is equivalent to recursive functions. Looking at the definition of Term we see there are 3 ways to be a Term: by being a variable, a lambda, or an apply. It is a very natural encoding, especially in the context of lambda calculus. And so to write used in a pattern matching style, we'll need to make a case for each of the different ways to be a Term. How do I collect the variable names and add them to a list? If pattern matching and wild cards are not correct, please can you suggest how I could go about this? I'll use this function to merge two ordered lists together: An abstraction consists of a head and a body, separated by the dot (.) symbol. We introduce new abstractions using the lambda symbol. I think I could achieve this using pattern matching and wild cards, is this possible? import Data.CharĮxample = Lambda "a" (Lambda "x" (Apply (Apply (Lambda "y" (Variable "a")) Lambda calculus only defines univariate (single-variable) functions, but we can easily extend it to include multivariate functions too, as we’ll see later. I want to return them in an ordered list. I want to write a function that will collect the variable names that have been used in a lambda calculus term, both as a Variable and in a Lambda abstraction.įor example in the code I would get the following output: *Main> used example
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