alphaconversie

Alphaconversie, in the context of lambda calculus, refers to the renaming of bound variables within a lambda expression without changing its meaning. In Dutch mathematical and logical literature the term alphaconversie is used to describe this process, which is the basis for alpha-equivalence: two lambda terms that differ only by the names of their bound variables are considered the same term.

Formally, alphaconversie defines the rule that a lambda-term λx.M can be renamed to λy.(M[x := y]) provided that y is fresh for M, meaning y is not a free variable occurring in M. This prevents variable capture, where a free occurrence of a renamed variable becomes bound in ways that alter the term’s semantics. For example, λx.x is alpha-equivalent to λy.y, and λx.λy.x is alpha-equivalent to λa.λb.a. In general, alpha-conversion allows systematic renaming of bound variables to simplify substitution and comparison of terms.

Alpha-equivalence is an equivalence relation: two terms are α-equivalent if one can be obtained from the other through a finite sequence of alphaconversions. The concept preserves the functional behavior of lambda terms while ignoring the specific names chosen for bound variables. It is distinct from beta-conversion (which changes the structure by applying functions to arguments) and eta-conversion (which expresses extensional equality).

Origins and use: alphaconversie and the related notion of alpha-equivalence arose with the development of lambda calculus in the 1930s by Alonzo Church. Today they are fundamental in formal semantics, type theory, and the implementation of compilers and interpreters to avoid name collisions during substitution. See also beta-conversion, eta-conversion, and lambda calculus.