![]() 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. The parallel reduction relation is defined as follows.Lambda calculus is a simple language for expressions that supports the focused study of functions and their invocations. _ _ rename ext exts Z S_ subst subst-zero ) Parallel Reduction.Imports open import using ( _≡_ refl ) open import Function using ( _∘_ ) open import Data.Product using ( _×_ Σ Σ-syntax ∃ ∃-syntax proj₁ proj₂ ) renaming ( _,_ to ⟨_,_⟩ ) open import using ( Rename Subst ) open import using ( _-→_ β ξ₁ ξ₂ ζ _-↠_ begin_ _-→⟨_⟩_ _-↠⟨_⟩_ _∎ abs-cong appL-cong appR-cong -↠-trans _⊢_ _∋_ `_ #_ _,_ ★ ƛ_ _ Thus, we can reduce the proof of confluence for beta reduction to confluence for parallel reduction. Furthermore, we show that a parallel reduction sequence exists between any two terms if and only if a beta reduction sequence exists between them. ![]() To side-step this problem, we’ll define an auxiliary reduction relation, called parallel reduction, that can perform many reductions simultaneously and thereby satisfy the diamond property. x) a)īoth terms can reduce to a a, but the second term requires two steps to get there, not one. Unfortunately, reduction in the lambda calculus does not satisfy the diamond property. If we write ⇛* for the reflexive and transitive closure of ⇛, then confluence of ⇛* follows immediately from the diamond property. This is just an instance of the same picture above, where downward lines are now instance of ⇛. Then ⇛ has the diamond property if whenever L ⇛ M₁ and L ⇛ M₂, then there exists an N such that M₁ ⇛ N and M₂ ⇛ N. Where downward lines are instances of -↠.Ĭonfluence is studied in many other kinds of rewrite systems besides the lambda calculus, and it is well known how to prove confluence in rewrite systems that enjoy the diamond property, a single-step version of confluence. That is, if there are reduction sequences from any term L to two different terms M₁ and M₂, then there exist reduction sequences from those two terms to some common term N. In this chapter we prove that beta reduction is confluent, a property also known as Church-Rosser. ![]()
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