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Lambda calculus consists of constructing lambda terms and performing reduction operations on them. Applying a function to an argument. M and N are lambda terms. Parentheses can be dropped if the expression is unambiguous. For some applications, terms for logical and mathematical constants and operations may be included.
In typed lambda calculus, functions can be applied only if they are capable of accepting the given input’s “type” of data. Subsequently, in 1936 Church isolated and published just the portion relevant to computation, what is now called the untyped lambda calculus. Until the 1960s when its relation to programming languages was clarified, the λ-calculus was only a formalism. The λ-calculus incorporates two simplifications that make this semantics simple.
The first simplification is that the λ-calculus treats functions “anonymously”, without giving them explicit names. The second simplification is that the λ-calculus only uses functions of a single input. As described above, all functions in the lambda calculus are anonymous functions, having no names. They only accept one input variable, with currying used to implement functions with several variables. A valid lambda calculus expression is called a “lambda term”. 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. However, some parentheses can be omitted according to certain rules. For example, the outermost parentheses are usually not written. The definition of a function with a lambda abstraction merely “sets up” the function but does not invoke it. There is no concept in lambda calculus of variable declaration.
Bracketing may be used and may be needed to disambiguate terms. There are several notions of “equivalence” and “reduction” that allow lambda terms to be “reduced” to “equivalent” lambda terms. A basic form of equivalence, definable on lambda terms, is alpha equivalence. In many presentations, it is usual to identify alpha-equivalent lambda terms.
In general, failure to meet the freshness condition can be remedied by alpha-renaming with a suitable fresh variable. Under this view, beta reduction corresponds to a computational step. This step can be repeated by additional beta conversions until there are no more applications left to reduce. In the untyped lambda calculus, as presented here, this reduction process may not terminate. That is, the term reduces to itself in a single beta reduction, and therefore the reduction process will never terminate. Another aspect of the untyped lambda calculus is that it does not distinguish between different kinds of data. For instance, it may be desirable to write a function that only operates on numbers.