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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