Lambda calculus beta reduction examples software#
Lambda calculus beta reduction examples free#

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Lambda calculus beta reduction examples free#

Copyright 2019 by Robert Kovacsics // // Permission is hereby granted, free of charge, to any person // obtaining a copy of this software and associated documentation // files (the 'Software'), to deal in the Software without // restriction, including without limitation the rights to use, copy, // modify, merge, publish, distribute, sublicense, and/or sell copies // of the Software, and to permit persons to whom the Software is // furnished to do so, subject to the following conditions: // // The above copyright notice and this permission notice shall be // included in all copies or substantial portions of the Software. Let _loop_ (fn loop => fn n => if (eq0 n) To completion isn't as useful as just iterating a couple of steps, Will end up evaluating (pred 2) and (pred (pred 2)) rather than Chosing the non-strict strategyĪlways reduces to beta-normal form, but you might want to eagerlyĮvaluate the predecessor function when you see it, otherwise you Here you have to be careful about the order of the evaluation, Let pred2 (fn n f x => n (fn g h => h (g f)) (fn u => x) (fn u => u)) Let pred (fn n => snd (n (fn p => pair (succ (fst p)) (fst p)) (pair 0 0)))Īnd here is a faster predecessor, try it out to see how it works: Let add (fn n m => fn f x => n f (m f x)) Re-computation of the parse tables which is laborious, and it is notĪ high priority problem, as Poly/ML and GHC have the same parsing Yet, as it would require a change to the grammar, and hence a

pred is fn n f x => n (fn g h => h (g f)) (fn u => x) (fn u => u)Īnd the special built-in let variable value.Įven though both of them are equally unambiguous.

map is fn f l => fn g init => l (compose g f) init.

cons is fn x y => fn f init => f x (y f init).

Y is fn f => (fn x => f (x x)) (fn x => f (x x)).

Note, it uses space-based lexing (though you don't need to put spacesĪround parentheses), and only implements the following prefix x fn x => x Maximum single steps Evaluation strategy Strict/Eager Non-strict/Call-by-name Also have a look at the examples section below, where you canĬlick on an application to reduce it (e.g. “Normal-order reduction will always produce a normal form, if one exists.Type an expression into the following text area (using the fn x =>īody synatx), click parse, then click on applications to evaluate If E1 → E2, and E2 is normal form, then there is a normal-order reduction of E1 to E2. “Reduction in any way can eventually produce the same result.” If E1 ↔ E2, then there exists an E such that E1 → E and E2 → E. The Church-Rosser Theorem states the following − For Example − Alpha ReductionĪlpha reduction is very simple and it can be done without changing the meaning of a lambda expression. E) y replaces every x that occurs free in E with y. In an expression, each appearance of a variable is either 'free' (to λ) or 'bound' (to a λ). The inner x belongs to the inner λ and the outer x belongs to the outer one.

When there are multiple terms, we can handle them as follows − The formal parameter may be used several times − There are two reducible expressions: (* 5 6) and (* 8 3). Here, we can’t start with '+' because it only operates on numbers.

Let us evaluate the following expression − Pure lambda calculus has no built-in functions. Where λx.E is called Lambda abstraction and E is known as λ-expressions. Lamdba calculus includes three different types of expressions, i.e., And all the functions are on single argument. These functions can be of without names and single arguments.įunction application − Church used the notation E 1.E 2 to denote the application of function E 1 to actual argument E 2.

Lambda calculus is a framework developed by Alonzo Church in 1930s to study computations with functions.įunction creation − Church introduced the notation λx.E to denote a function in which ‘x’ is a formal argument and ‘E’ is the functional body.