Questions tagged [lambda-calculus]

λ-calculus is a formal system for function definition, function application and recursion which forms the mathematical basis of functional programming.

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Equality of lambda terms which do not have normal form

In the context of lambda calculus, how should one prove $\beta$-equality of terms that do not have normal form? In particular, how to prove that these are different combinators: $$ Y = λf.(λx.f(xx))(...
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Is α-renaming necessary for STLC?

Consider the following in untyped lambda calculus ( \ x. x x ) ( \ g. \ f. g f ) Even though each variable is uniquely named reducing this will require an $\alpha$-...
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Raising to the T in machine learning

What does it mean when in a machine learning paper there is $(arg)^{T}$, what does the T does to an arg in this 3b1b video on neural networks he puts the: $(w^{l-1})^{T}$
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Proving program termination in the $\lambda$-calculus

Turing's Checking a large routine: Finally the checker has to verify that the process comes to an end. Here again he should be assisted by the programmer giving a further definite assertion to be ...
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Association rules during execution in lambda calculus

I learned that expressions are left associative while abstraction are right associative, however, while solving some examples i faced this problem. Succ 3 Succ 2 when i applied left associativety i ...
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Is there a standard way to do Debrujin indices for an uncurried language?

With Debrujin indices, variable names are replaced with numbers that indicate which function binds them. This works fine if each function only binds one variable, but what if one's language is not ...
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How is `y λx.x y` parsed using the standard pure untyped lambda calculus conventions?

How would the following term in the pure untyped lambda calculus be parsed: y λx.x y The relevant conventions listed on https://en.wikipedia.org/wiki/...
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Simplify the definition of substitution in Lamdba calculus

Substitution in untyped Lambda calculus is complicated by variable capture. Can this boring technical complication be entirely avoided by some restriction on the standard formation rules? Something ...
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What is congruence in lambda-calculus

I see a lot of lecture notes where they use the term "congruence" (ex: congruence relation) or deriving usages such as "the expression e is alpha-congruent to e2". Could someone ...
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Proving equivalence of two substitutions by induction

I'm trying to prove the following reduction: $$ t\{x:=u\}\{y:=v\} = t\{y:=v\}\{x:=u\{y:=v\}\} $$ under the following assumptions: $x \neq y$ $x$ is not a free variable of $v$ (in symbols, $x \...
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Can a lambda expression be beta-equal to beta-normal forms?

Given a Lambda Expression Term T can it be beta-equal to two different Lambda Terms T1 and T2, both T1 and T2 are in beta-normal form?
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Computation equivalence of functional and procedural programming

I'm really interested in the idea of functional programming, it seems like a very modular way of doings things. I've seen some suggestion that functional programming is just as powerful as procedural ...
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Church Numerals represented as a List?

Afternoon All, Perhaps an easy one for many, but am new to Lambda calculus: Is the below list, Ni, a representation of the Church numerals? I think it is but how do I prove it...if not, what is it...? ...
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λ -terms that correspond to Haskell functions

Evening All I already have a "grasp" of haskell (not terrible, about 6 month experience) and am trying to learn the fundamentals that sit behind it, thus am now turning my attention to ...
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Show that term cons works by showing all beta reductions

I'm new to functional programming. So the terms cons appends an element to the front of the list. Where cons ≜ λx:λl:λc:λn: c x (l c n). How should I go about proving that cons works correctly using ...
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Category Set language using simply typed lambda calculus

I am currently self learning Category Theory and Simply typed lambda calculus (STLC). For learning purposes, I have implemented an STLC interpreter as given in Types and Programming Languages book ...
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Is type unification a kind of search for alpha equivalence?

I was reading about type unification and it moves through of substitution of variables. To me it looks like a search for an alpha equivalence... I mean, two types are unifiable if they are alpha ...
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Why $((+ 1) ((+ 1) 1))$ is not a $\delta-$redex?

I was reading http://barrywatson.se/lsi/lsi_delta_reduction.html, where ((+ 1) 2) →δ 3 is a δ-reduction. ((+ 1) ((+ 1) 1)) is a not a δ-redex. Wouldn't $((+ 1) ((...
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Beta reduction of S combinator in pure lambda calculus

S is defined as S x y z = x z (y z) This suggest that (y z) should be evaluated just after ...
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What kind of automaton recognizes closed terms of the lambda calculus?

There seems to be an interesting model of computation involved in determining whether a term from some programming language has any free variables. It's a tree traversal that seems almost like the ...
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Fixed Point Combinator Turing proof

I have to proof that Turing's combinator is a fixed point operator, but I can't get it. I tried this: \begin{align*} Vg &= (UU)g = ((\lambda f.\lambda x.x(ffx)) (\lambda f.\lambda x.x(ffx)))g =...
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What is the diference between $\lambda x.1$ and $1$?

I know that I can $(\lambda x.1) 0 \rhd_\beta 1$. This is the constant 1, but can I contract it automatically? I mean, is $1$ the normal form of $(\lambda x.1)$? It seems reasonable to do it but when ...
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Is (λx.FV(A)) and (λx.FV(B)) β-equivalent?

Does free variables have some meaning in lambda calculus? If I follow this β-reduction ...
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Other encodings of the natural numbers that are not the Church encoding or the Scott encoding

Are there other encodings of the natural numbers in the untyped lambda calculus which are neither the Church encoding nor the Scott encoding?
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Imposing sortal restrictions on functions in the untyped lambda calculus

In the simply typed lambda calculus if we have an addition operator, $+ : n \to n \to n$, its typing imposes that we can only combine it with something of a given type. So if $n$ is the type of ...
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The difference between $\beta$-reduction and $let$

These are the reduction rules associated with $\beta$ reduction and $let$: $$(\lambda x. e_2) e_1 \to_{\beta} e_2 [e_1 / x]$$ $$let \,x = e_1 \textit{ in }e_2 \to e_2 [e_1/x]$$ These reduction rules ...
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Why should I care about lambda calculus

I am a programmer by hobby. I stumbled upon lambda calculus from Kevlin Henney's talk lambdas to the slaughter and I was sold! It was an interesting new way of thinking that's entirely different than ...
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How would you model Rust procedural macros?

In Rust programming language one can write a compiler extension function that works on abstract syntax tree, effectively modifying source code before it gets converted into machine instructions. In ...
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Normalization-by-evaluation for untyped lambda calculus which results in 𝛽𝜂-normal form

Usual NbE algorithms for untyped lambda calculus, which use (P)HOAS to embed terms to a host language constructs, results in a beta-normal form of a terms. Is there algorithms to (efficiently) exploit ...
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Sequencing or continuation-passing in pure lambda-calculus

I am trying to solve the following exercise given here. Consider the following number representation. How to define the addition? ...
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Lambda calculus simplification excercise

Below is the lambda expression which I am finding difficult to reduce i.e. I am not able to understand how to go about this problem. (λx.λy.yx)z (λw.w) I am lost with this. if anyone could lead me in ...
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Is the term $(\lambda x.x)(y y)$ a normal form in call-by-value reduction strategy?

I am learning λ-calculus and I have some confusion about it. Is the term $(\lambda x.x)(y y)$ a normal form in call by value reduction strategy? (where $y$ is a free variable) From the wikipedia, it ...
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What's the internal language of the opposite of a Cartesian closed category?

I have heard the simply typed lambda calculus is the internal language of Cartesian closed categories. What's the internal language of the opposite type of category? The rules dual to currying and ...
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How to write “∀x.F(x)” for “F(x)=λx.Φ(x)” in one expression (sequel from question about “∀(λφ. (φ x m→ φ y))”?

This question is sequel from How to understand quantifier without predication " ∀(λφ. (φ x m→ φ y))"? which further explains the notation and context. So - I have anonymous Boolean-valued ...
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Can lists be defined in a special way so that they contain things of different type?

In https://www.seas.harvard.edu/courses/cs152/2019sp/lectures/lec18-monads.pdf it is written that A type $\tau$ list is the type of lists with elements of type $\tau$ Why must a list contain ...
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How to understand quantifier without predication “ ∀(λφ. (φ x m→ φ y))”?

I am reading about embedding/automation of modal logics in classical higher order logic (http://page.mi.fu-berlin.de/cbenzmueller/papers/C46.pdf) and Goedels proof of God's existence is prominent ...
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Creating a large tuple from smaller tuples via a monad or applicative

Suppose I have a term $a :\alpha$ of the Simply-Typed Lambda Calculus (in the following, $\alpha, \beta, \gamma$ stand for arbitrary types) and I want to lift it to a term $\lambda x_{\beta}. \;(x, \, ...
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(Co)-monads and terminating implementations

The bounty above should read 'I would like to know whether the example I discuss is a com-monad and why (why not).' Suppose we set $\mathbb{M} \alpha := r \to \alpha$, where $r$ is some fixed type, ...
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The meaning and relevance of the locution ''no terminating implementation'' in type theory

In the context of a discussion of Haskell https://stackoverflow.com/questions/62509788/the-intuition-behind-the-definition-of-the-co-reader-monad, I was told that There is no terminating ...
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Constructing a monad via type synonyms of a particular kind

We can define a reader/environment monad on the simply-typed lambda calculus, using the following three equations, where $r$ is some fixed type, $\alpha$ is any type (I subscript some terms with their ...
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Lambda Calculus Conversion

How can I take a Haskell data type or function (eg fold, list, String, zip) and convert or translate it to a lambda calculus abstraction? Example: If sum computes a sum of all elements in a list, and :...
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Inhabitation of STLC is in PSPACE

Urzyczyn: Inhabitation in Typed Lambda-Calculi (A syntactic approach) gives a proof that STLC inhabitation problem is in PSPACE (section 2, lemma 1). I don't understand certain aspects of the proof: ...
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Tried to derive the Z combinator and instead derived another

I was working to derive the Z-Combinator by starting with the factorial function and ended up deriving a different fixed-point combinator. What did I derive? Did I make a subtle mistake? Here are the ...
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alpha equivalence of lambda calculus

I am pretty new in λ calculus. And I am now trying to understand Alpha equivalence. Basically can I think it in this way: as long as I make sure all the bound variables and their corresponding ...
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Understanding $\lambda \mu$-calculus in more programming way

I am learning $\lambda \mu$-calculus (self-study). I learned it because it seems very useful for understanding Curry-Howard correspondence (e.g understanding the connection between classical logic ...
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Lambda calculus without free variables is as strong as lambda calculus?

First question: How would one prove that by removing free (unbound) variables from lambda calculus, and allowing only bound variables, its power is not reduced (it is still Turing-complete) ? Second ...
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What does Lambda Calculus teach us about data?

In Lambda Calculus, the distinction between data and code doesn't seem to exist. Is there something fundamental about this, or purely Lambda Calculus's thing? Some context: as a software developer, I ...
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Uncurrying and Polymorphism

How do we uncurry functions when they are polymorphic? For example, is it possible to uncurry the following types? If so what is the uncurried type? $\forall X. X \rightarrow int \rightarrow X$ ? $...
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Lambda calculus and runtime inspection of the term

This is possibly related to reflection and quoting but I don't want to assume anything beforehand. Here is my requirement. My typed lambda calculus (Curry style) is a simpler variant of Calculus of ...
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How does this dependently-typed boolean elimination function work?

In the companion code to A Tutorial Implementation of a Dependently Typed Lambda Calculus - prelude.lp - there is a rather intimidating definition of a ...

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