Category: Untyped lambda calculus repl

Previous chapters consider intrinsically-typed calculi; here we consider one that is untyped but intrinsically scoped. Previous chapters consider weak head normal formwhere reduction stops at a lambda abstraction; here we consider full normalisationwhere reduction continues underneath a lambda.

Previous chapters consider deterministic reduction, where there is at most one redex in a given term; here we consider non-deterministic reduction where a term may contain many redexes and any one of them may reduce.

Previous chapters consider reduction of closed terms, those with no free variables; here we consider open terms, those which may have free variables. Previous chapters consider lambda calculus extended with natural numbers and fixpoints; here we consider a tiny calculus with just variables, abstraction, and application, in which the other constructs may be encoded.

Untyped: Untyped lambda calculus with full normalisation

In general, one may mix and match these features, save that full normalisation requires open terms and encoding naturals and fixpoints requires being untyped. The aim of this chapter is to give some appreciation for the range of different lambda calculi one may encounter.

One consequence of this approach is that constructs which previously had to be given separately such as natural numbers and fixpoints can now be defined in the language itself. But it does ensure that all variables are in scope.

For instance, we cannot use S S Z in a context that only binds two variables.

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Now we have a tiny calculus, with only variables, abstraction, and application. Below we will see how to encode naturals and fixpoints into this calculus. It is convenient to define a term to represent four as a Church numeral, as well as two.

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The reduction rules are altered to switch from call-by-value to call-by-name and to enable full normalisation:. One can choose to reduce a term inside either L or M. How would the rules change if we want call-by-value where terms normalise completely?

How would the rules change if we want call-by-value where terms do not reduce underneath lambda? Progress adapts. Instead of claiming that every term either is a value or takes a reduction step, we claim that every term is either in normal form or takes a reduction step.

Previously, progress only applied to closed, well-typed terms. Now we can demonstrate it for open, well-scoped terms. The definition of normal form permits free variables, and we have no terms that are not functions. The final equation for progress uses an at pattern of the form P Qwhich matches only if both pattern P and pattern Q match. In this case, the pattern ensures that L is an application.Join Stack Overflow to learn, share knowledge, and build your career.

Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. I've been reading about the lambda calculus, and love the ideas proposed by it, but there are some things I just can't explain.

Yes, you can define numbers and indeed, arbitrary data types inside the lambda calculus. Here's the idea. First, let's pick what numbers we're going to define. The simplest numbers to work with are the natural numbers : 0, 1, 2, 3, and so on. How do we define these? The usual approach is to use the Peano axioms :.

Lambda calculus

Now, in the lambda calculus, we can represent function application, so we can represent S nbut we don't know how to represent 0 and S themselves. But luckily, the lambda calculus offers us a way of deferring that choice: we can take them as arguments, and let someone else decide! Let's write z for the 0 we're given, and s for the S we're given. Just as a natural number n is n applications of S to 0, a lambda-calculus representation of n is an application of n copies of any successor function s to any zero z.

We can define successor, too:. Here, we see that the successor applies one extra copy of s to nafter making sure n uses the same z and s. Yes, that gets dense and hard to read quickly.

Working through it is a pretty good exercise if you feel like you need more practice — it led to me catching an error in what I'd originally written! Now, we've defined 0 and S, so that's a good start, but we want a principle of induction, too.

That's what makes the natural numbers what they are, after all! So, how will that work? Well, it turns out we're basically set. When thinking about our principle of induction programmatically, we want a function that takes as input a base case and an inductive case, and produces a function from natural numbers to some sort of output.Previous chapters consider intrinsically-typed calculi; here we consider one that is untyped but intrinsically scoped.

Previous chapters consider weak head normal formwhere reduction stops at a lambda abstraction; here we consider full normalisationwhere reduction continues underneath a lambda. Previous chapters consider deterministic reduction, where there is at most one redex in a given term; here we consider non-deterministic reduction where a term may contain many redexes and any one of them may reduce.

Previous chapters consider reduction of closed terms, those with no free variables; here we consider open terms, those which may have free variables.

Previous chapters consider lambda calculus extended with natural numbers and fixpoints; here we consider a tiny calculus with just variables, abstraction, and application, in which the other constructs may be encoded.

In general, one may mix and match these features, save that full normalisation requires open terms and encoding naturals and fixpoints requires being untyped. The aim of this chapter is to give some appreciation for the range of different lambda calculi one may encounter.

One consequence of this approach is that constructs which previously had to be given separately such as natural numbers and fixpoints can now be defined in the language itself.

As before, a context is a list of types, with the type of the most recently bound variable on the right:. But it does ensure that all variables are in scope. For instance, we cannot use S S Z in a context that only binds two variables. The result is that we check that terms are well scoped — that is, that all variables they mention are in scope — but not that they are well typed:. Now we have a tiny calculus, with only variables, abstraction, and application.

Below we will see how to encode naturals and fixpoints into this calculus.

Lambda Calculus

As before, we can convert a natural to the corresponding de Bruijn index. We no longer need to lookup the type in the context, since every variable has the same type:. It is convenient to define a term to represent four as a Church numeral, as well as two.

Reduction continues until a term is fully normalised. Hence, instead of values, we are now interested in normal forms. Terms in normal form are defined by mutual recursion with neutral terms:.

untyped lambda calculus repl

Neutral terms arise because we now consider reduction of open terms, which may contain free variables.Evan recommended to build an repl for a simple lambda calculus first so I gave it a shot. You can try the repl online here. I would love your feedback on the UI, design, and functionalities of the repl.

My main question is how to implement the call by name evaluation strategy? Looks beautiful!

clojureD 2018: \

It seems like you have a 10k step limit. Maybe we could combine forces… use your UI and my evaluator? Your implementation seems to use explicit substitutions, which is the clearest way to go and most obviously tied to the theory.

The Krivine machine is a very nice way to implement CBN. Call-by-need is a little bit trickier: you need to be careful to only evaluate things once. Your approach is definitely more practical.

I literally pulled my implementation out of the Types and Programming Languages book… I will be happy to work with you to combine the UI and the evaluator.

Sounds great. As for the typed vs. Take that, halting problem. Sounds good. I tried looking for type theory learning resources for beginners but TAPL is the only one that suits my level. I did find a good collection fo resources on learn-tt but most of them are papers and in-depth books. Do you know any online classes, video lectures, course notes, or any similar resources for beginners?

Ideally they should be at or below TAPL level with a focus on practice. Disclosure: Shriram was my undergraduate advisor, and Benjamin was my PhD advisor.

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Jeremy Siek wrote a nice blog post about notation that might be helpful, too. Those resources are fabulous! Especially the PLAI and the blog post. I can clearly see how to carry on my learning in PL thanks to you. Just wanted to say thank you to mgree for these resources as well. Feel free to ask me questions—here, or mgrnbrg on twitter.

Been trying to learn about programming language theory and practice — some success with TAPL, which I really like. Thank you for these references! Congrats on the release! This topic was automatically closed 10 days after the last reply.

untyped lambda calculus repl

New replies are no longer allowed.It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the s as part of his research into the foundations of mathematics.

Lambda calculus consists of constructing lambda terms and performing reduction operations on them. In the simplest form of lambda calculus, terms are built using only the following rules:. Parentheses can be dropped if the expression is unambiguous. For some applications, terms for logical and mathematical constants and operations may be included. Variable names are not needed if using a universal lambda function, such as Iota and Jotwhich can create any function behavior by calling it on itself in various combinations.

Lambda calculus is Turing completethat is, it is a universal model of computation that can be used to simulate any Turing machine. Lambda calculus may be untyped or typed. In typed lambda calculus, functions can be applied only if they are capable of accepting the given input's "type" of data. Typed lambda calculi are weaker than the untyped lambda calculus, which is the primary subject of this article, in the sense that typed lambda calculi can express less than the untyped calculus can, but on the other hand typed lambda calculi allow more things to be proven; in the simply typed lambda calculus it is, for example, a theorem that every evaluation strategy terminates for every simply typed lambda-term, whereas evaluation of untyped lambda-terms need not terminate.

One reason there are many different typed lambda calculi has been the desire to do more of what the untyped calculus can do without giving up on being able to prove strong theorems about the calculus. Lambda calculus has applications in many different areas in mathematicsphilosophy[2] linguistics[3] [4] and computer science. Functional programming languages implement the lambda calculus.

Lambda calculus is also a current research topic in Category theory. The lambda calculus was introduced by mathematician Alonzo Church in the s as part of an investigation into the foundations of mathematics. Rosser developed the Kleene—Rosser paradox. Subsequently, in Church isolated and published just the portion relevant to computation, what is now called the untyped lambda calculus.

untyped lambda calculus repl

Until the s when its relation to programming languages was clarified, the lambda calculus was only a formalism. Thanks to Richard Montague and other linguists' applications in the semantics of natural language, the lambda calculus has begun to enjoy a respectable place in both linguistics [12] and computer science.The main ideas are applying a function to an argument and forming functions by abstraction.

Functions and arguments are on a par with one another. The result is a non-extensional theory of functions as rules of computation, contrasting with an extensional theory of functions as sets of ordered pairs. This entry develops some of the central highlights of the field and prepares the reader for further study of the subject and its applications in philosophy, linguistics, computer science, and logic. Continuing with the example, we get:. The remaining equalities are justified by computing with natural numbers.

But this will be described in Section 2. What about functions of multiple arguments? We find, finally, that hypotenuse-length 3 4—the application of hypotenuse-length to 3 and then to 4—is 5, as expected. In set theory, a function is standardly understood as a set of argument-value pairs. This is the concept of functions-as-sets. Consequently, the notion of equality of functions-as-sets is equality qua sets, which, under the standard principle of extensionality, entails that two functions are equal precisely when they contain the same ordered pairs.

In other words, two functions are identical if and only if they assign the same values to the same arguments. In this sense, functions-as-sets are extensional objects.

This is the conception of functions-as-rules. In this sense, functions-as-rules are non-extensional objects. This terminology is particularly predominant in the community of mathematical logicians and philosophers of mathematics working on the foundations of mathematics.

But from the perspective of the philosophy of language, the terminology can be somewhat misleading, since in this context, the extensional-intensional distinction has a slightly different meaning. In the standard possible-worlds framework of philosophical semantics, we would distinguish between an extensional and an intensional function concept as follows.

Let us say that that two functions are extensionally equivalent at a world if and only if they assign the same values to the same arguments at that world. And let us say that two functions are intensionally equivalent if and only if they assign the same values to the same arguments at every possible-world.

To illustrate, consider the functions highest-mountain-on-earth and highest-mountain-in-the-Himalayaswhere highest-mountain-on-earth assigns the highest mountain on earth as the value to every argument and highest-mountain-in-the-Himalayas assigns the highest mountain in the Himalayas as the value to every argument. The two functions are extensionally equivalent at the actual worldbut not intensionally so. At the actual world, the two functions assign the same value to every argument, namely Mt.

Now consider a world where Mt. Everest is not the highest mountain on earth, but say, Mt. Rushmore is. Suppose further that this is so, just because Mt. Rushmore is Everest, with its roughly At that world, highest-mountain-on-earth now assigns Mt. Rushmore as the value to every argument, while highest-mountain-in-the-Himalayas still assigns Mt.

Everest to every object. In other words, highest-mountain-on-earth and highest-mountain-in-the-Himalayas are extensionally equivalent at the actual world but not intensionally equivalent.This exposition will adopt this convention. The philosophical significance of the calculus comes from the expressive power of a seemingly simple formal system. The variety and expressiveness of these calculi yield results in formal logic, recursive function theory, the foundations of mathematics, and programming language theory.

The intended use of the formal system Church developed was, as mentioned in the introduction, function application.

untyped lambda calculus repl

Later, it will be shown how it can be. For our present purposes, the use of squaring is pedagogical. By limiting the use of free variables and the law of excluded middle in his system in certain ways, Church hoped to escape paradoxes of transfinite set theory [Church,p. The original formal system of — turned out, however, not to be consistent.

In it, Church defined many symbols besides function definition and application: a two-place predicate for extensional equality, an existential quantifier, negation, conjunction, and the unique solution of a function.

InChurch isolated the portion of his formal system dealing solely with functions and proved the consistency of this system. We will discuss these results later. In an unpublished letter, Church writes that he placed the hat in front, not a typesetter.

According to Cardone and Hindley [, p.

Lambda Calculi

In order to proceed properly, we must define the alphabet and syntax of our language and then the rules for forming and manipulating well-formed formulas in this language.

In the process of this exposition, formal definitions will be given along with informal description of the intuitive meaning behind the expressions. These are not symbols in the calculus itself, but rather convenient metalinguistic abbreviations. The terms formulas, expressions; these three will be used interchangeably of the calculus are defined inductively as follows:.

The latter two rules of term formation bear most of the meat. The second bullet corresponds to function application. As you can see from this language definition, everything is a function. For this reason, they are often referred to as higher-order functions or first-class functions in programming language theory. Though I referred to the second term above as accepting two arguments, this is actually not the case.

We will see later, however, that this does not in any way restrict us and that we can often think of nested abstractions as accepting multiple arguments. Before we proceed, a few notational conventions should be mentioned:. This depends on the notion of free and bound variables, which we will define only informally. Px is a bound variable, as are similarly bound variables in the subterm P ; variables which are not bound are free.

A term containing no free variables is called a combinator. Intuitively, these substitution rules allow us to replace all the free occurrences of a variable x with any term N. It captures the intuitive notion of function application.

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A term which is not normalizing is called divergent. For instance, Y as defined earlier does not normalize. Letting FX denote arbitrary terms, we can evaluate:. This result is exactly what we expected. The process of treating a function of multiple arguments as iterated single-argument functions is generally referred to as Currying. Although a proof of this statement requires machinery that lies beyond the scope of the present exposition, it is both an important property and one that is weaker than either normalization or strong normalization.

The notion of external and internal properties used here is meant to be intuitive only. In other words, two functions are equivalent when they yield the same output on the same input.

This assumption is an extensionality one because it ignores any differences in how the two functions compute that output. In its original formulation, Church allowed abstraction to occur only over free variables in the body of the function.