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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4
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2answers
132 views

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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1answer
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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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1answer
292 views

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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1answer
95 views

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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1answer
249 views

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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2answers
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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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1answer
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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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2answers
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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 ...
2
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1answer
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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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1answer
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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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3answers
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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 ...
3
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1answer
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What are Contexts in Lambda Calculus?

What is a Context? Is it like a scope in C? Does it have a start and an end? Can contexts contain other contexts? I see Contexts being used in lambda calculi type system rules, but I don't understand ...
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2answers
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Tightening application rules for STLC

The syntax STLC is usually written: $e ::= x |\lambda x : \tau . e|(e \space e)|c$ However, the application rule appears to accept all expressions on the left hand side. Shouldn't the application ...
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Reducing Kleene's predecessor for Church numerals

I am trying to "reinvent" Kleene's predecessor myself. The following code snippet should be self-explanatory. The idea is to make a 2-tuple and count up from zero, i.e. ...
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Where is typed lambda calculus on the Chomsky hiererchy?

The functions definable in untyped lambda calculus are the computable functions, for which it is in turn possible to define equivalences to the concepts of Turing machines, recursive enumerability and ...
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Does there exist a way, within Lambda Calculus, to discover if two free variables are the same?

Using Church's $\lambda x.(\lambda y.y))$ as false and $\lambda x.(\lambda y.x))$ as true, and given two free variables $g$ and $h$: Could there exist a function $eq?$ such that $(eq?\ g\ h)$ is ...
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1answer
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lambda calculus beta reductions: ((((lambda f (lambda x ((f x) f))) (lambda y (lambda g (g (* y y))))) 2) (lambda a a))

My question is in continuation to lambda calculus reduction: (((lambda f (lambda x (f x))) (lambda y (* y y))) 12) given the input: ...
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1answer
61 views

lambda calculus reduction: (((lambda f (lambda x (f x))) (lambda y (* y y))) 12)

given the input (((lambda f (lambda x (f x))) (lambda y (* y y))) 12) what does this step evaluate to: lambda x (f x) I am ...
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2answers
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Does the underlying computational calculus in type theories affect decidability?

I'm looking for a high-level explanation although if that isn't possible or difficult, I'd prefer references to books/papers. I understand that modern type theory is inspired by Curry-Howard ...
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Is there an abstract architecture equivalent to Von Neumann's for Lambda expressions?

In other words, was a physical implementation modelling lambda calculus (so not built on top of a Von Neumann machine) ever devised? Even if just on paper? If there was, what was it? Did we make use ...
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1answer
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Lambda Expression Reduction

I am unable to solve the following lambda expression using both normal order (Call-by-name) and applicative order (Call-by-value) reduction. I keep getting different answers for both. This is the ...
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Lambda calculus self reducer with explicit redex selection

In "Efficient Self-Interpretation in Lambda Calculus", Mogensen presents a self-reducer in lambda calculus which leaves redex selection to the underlying reduction. Is there some example of a self-...
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1answer
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Termination of Z combinator with call-by-value

I am trying to build my own λ-calculus interpreter. So far it supports both call-by-value and normal order. I now want to try recursion via fixed points. The $Y$ combinator works with normal order, ...
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Standardisation Theorem versus Leftmost reduction Theorem

According to Chris Hankin in his book (Lambda Calculus a Guide for Computer Scientists). A reduction sequence $\sigma: M_0 \to^{\Delta_0} M_1 \to^{\Delta_1}M_2 \to^{\Delta_2}\ldots $ is a standard ...
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Self reduction for fully introspective lambda calculus representations

For some representation scheme $\ulcorner \cdot \urcorner$, a self interpreter $R$ is a lambda expression where $R \ulcorner A \urcorner \underset{\beta}{=} A$, while a self reducer $E$ is a lambda ...
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Is it true that if $M : \forall \alpha . \left( \alpha \rightarrow \alpha \right)$ is a closed term then $M = \Lambda \alpha. \lambda x^{\alpha} . x$?

In system F, is every closed term $M$, which is of $\forall \alpha . \left( \alpha \rightarrow \alpha \right)$, $\alpha \beta \eta$-equivalent to $\Lambda \alpha. \lambda x^{\alpha} . x$? I have ...
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56 views

Normal Order Reduction - is Leftmost Outermost order simply Leftmost?

This is a quick question. I've been reading up on Lambda Calculus, and I see Normal Order as "outermost leftmost first", and Applicative Order described as as "innermost leftmost first". I think ...
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1answer
53 views

How to find a lambda term to complete a function?

I tried to complete this exercise but i stopped... Defining a $ \lambda $-term M such that: $$(<M,u>)<M,v> \: \simeq_{\beta} \: <M,u>$$ I chose $M=\lambda m \lambda a \lambda b \...
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in the lambda calculus with products and sums is $f : [n] \to [n]$ $\beta\eta$ equivalent to $f^{n!}$?

$\eta$-reduction is often described as arising from the desire for functions which are point-wise equal to be syntactically equal. In a simply typed calculus with products it is sufficient, but when ...
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1answer
50 views

Hindley-Milner system with let expansion

I'm reading these slides that present Hindley-Milner type inference. In the system HM, we have the following let rule: $\dfrac{\Gamma \vdash t:S \;\; \Gamma,x:S \vdash t':T }{\Gamma \vdash \text{let} ...
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Blum complexity measure for lambda calculus

Is there a formal complexity measure for lambda expressions which satisfies the Blum axioms and measures the complexity of reducing the expression to its normal form? I assume that the complexity ...
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1answer
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Zero-knowledge proof of $\beta\eta$ equality

Is there some way to give a zero-knowledge proof that two $\lambda$-terms are convertible, i.e. equal modulo $\beta\eta$? A usual (and not zero-knowledge) proof that two terms are convertible is a ...
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Rules for consistency with mutual inductive families?

I'm trying to use a proof assistant to define a type and a relation that are mutually dependent on each other: ...
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Lambda terms forming non-abelian groups

I was wondering what kind of groups could be constructed with Lambda terms, where the group operation is application? For example, $a* b =c$ would mean $ab\to_{\beta}^{*} c$. Is this even possible? ...
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Equivalent terms in call-by-name but not in call-by-value

Working in the untyped lambda-calculus, I'm asked to give two terms that are equivalent in call-by-name semantics but not in call-by-value. Call $\text{fls} = \lambda x. \lambda y. y$ and $\Omega = (\...
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1answer
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Finding two store typings that make the same store valid (lambda-calculus with references)

Problem 13.5.2 of Pierce's TAPL's book (page 167) asks: Can you find a context $\Gamma$, a store $\mu$ and two different store typings $\Sigma_1,\Sigma_2$ such that both $\Gamma | \Sigma_1 \vdash \...
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1answer
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What is meant by a full abstract model of a lambda-calculus like language?

The simply typed lambda-calculus with numbers and fix has long been a favorite experimental subject for programming language researchers, since it is the simplest language in which a range of subtle ...
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2answers
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What is meant when we say that divergence is the only side-effect of the lambda-calculus?

In the simply typed lambda-calculus, I was told that behavioral equivalence is taken in terms of divergence because "divergence is the only side-effect of such language". How should I understand ...
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2answers
50 views

What is the adjunct of the evaluation morphism in a closed monodical category?

According to the nlab article about evaluation map if $X, Y \in C$, a closed monodical category then the adjunct to evaluation morphism $[X, Y]\otimes X \rightarrow Y$ is the identity morphism $[X, Y] ...
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1answer
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How does the function to curry and uncurrying another function work?

The following is the code to curry or uncurry a function in Haskell: ...

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