![]() A scope is a section of text in which variables need to be understood as to whether they are bound or unbound. The abstraction construct introduces a lexical scope for the parameter variable. (Unlike C where a global must be declared, though, unbound variables are not declared, they are merely used.) This equates to the notion of global variable in programming languages, like C, and to static in languages like Java and C#. This makes lambda calculus a higher order system, and this where its power to study functions comes from.Ī variable seen alone is called an unbound variable. However, they are first class in the sense that they can both be applied to (passed as) parameters as well as be the result of (return value) of functions. We have "abstracted" our original printf statement!Įxpressions, including variables, have no particular type, or rather are simply typed: all variables and expressions are all taken as the following type: function - taking one argument - and returning an expression (whose type is also function taking one argument).Ībstractions (i.e.functions) are anonymous - they are not named and thus cannot be referred to or invoked by name. Now, to abstract this, we might replace the 100 with x, and then wrap that in a function: For the equivalent in the C programming language, let's imagine the statement We can define the verb "to abstract" with the following: take an expression, and replace some fixed part of it with a variable - then wrap it in a function that takes the variable as a parameter. Technically, the application construct does not even require the first of the two lambda expressions to be an abstraction construct - however, if it is, we can perform □-reduction, which allows us to remove the both the application and the abstraction, thus, eliminating the abstraction's parameter by substitution of the application-provided parameter value within the abstraction body, and that as a result we have two equivalent expressions: the former application to abstraction, and the latter, what remains after an application and an abstraction pair are eliminated. DiscussionsĪpplication, on its own, brings together two lamba expressions, a function and a parameter value - it does not automatically invoke or execute the function. In order to have the equivalence, the latter must have the same M except with N (the supplied parameter) substituted for parameter variable x in M, written as M. Thus, we have two equivalent expressions: one with application to an abstraction, and one where the application and abstraction pair were eliminated. In some sense, an application can cancel out an abstraction (by providing a value for the parameter). This operation is used to simplify application applied to abstraction. y, though otherwise with the exact same M (e.g. some M in terms of variable x, written as M), and another another using another variable name, e.g. It says that two abstractions are equivalent: one using some variable name, e.g. This operation is used to remove variable name conflicts. ![]() So that we can talk about the steps of transformation in working with lambda calculus, the tranformations themselves have a written form - but to be clear, the written forms of these operations are not themselves lambda calculus syntax, though once a described (written) transformation is actually performed (using textual replacement), the result is once again a legal lambda calculus construction. There are two manipulation operations provided by lambda calculus they manipulate lambda expressions These are not themselves syntactic constructs, but rather transformations of equivalence - meaning that if you properly use an operation on a given expression, the resulting expression is equivalent! Application just provides a parameter value it does not automatically "invoke" the function as we might expect in C (that is done later). In application, we provide a value for a parameter of a function - the value can be any lambda expression we can write. The application form goes to the opposite of abstraction. M is any possible expression in lambda calculus. In lambda calculus, an abstraction (a function) takes one parameter - the x in ( λ x. The abstraction form lets us define functions. Variables can have any name, but are typically single letters like x, y and sometimes ⨍ to indicate a variable that is taken as a function. In the above descriptions, we take M and N as sub expresssions, which can be any of the above forms. Here are the three syntatic constructs of lambda calculus: Name of Syntatic Construct It is less a programming language in the sense we know today, and more of a mathematical form: it is based on written text, and, it provides for several manipulations of the text. Lambda calculus is a simple language for expressions that supports the focused study of functions and their invocations.
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