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.

Filter by
Sorted by
Tagged with
2
votes
0answers
41 views

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
votes
1answer
38 views

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 ...
2
votes
1answer
74 views

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 ...
1
vote
2answers
65 views

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? $\forall X. X \rightarrow int \rightarrow X$ ? $...
1
vote
0answers
42 views

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 ...
0
votes
0answers
28 views

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
votes
1answer
58 views

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 ...
2
votes
2answers
59 views

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 ...
2
votes
0answers
15 views

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. ...
5
votes
0answers
105 views

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 ...
0
votes
0answers
15 views

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 ...
0
votes
1answer
54 views

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: ...
1
vote
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 ...
5
votes
2answers
573 views

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 ...
3
votes
0answers
35 views

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 ...
-2
votes
1answer
70 views

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 ...
0
votes
0answers
25 views

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-...
0
votes
1answer
39 views

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, ...
3
votes
0answers
21 views

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 ...
1
vote
0answers
16 views

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 ...
3
votes
0answers
70 views

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 ...
2
votes
0answers
48 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 ...
0
votes
1answer
52 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 \...
0
votes
0answers
41 views

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 ...
2
votes
1answer
46 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} ...
2
votes
0answers
22 views

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 ...
4
votes
1answer
41 views

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 ...
5
votes
0answers
47 views

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: ...
2
votes
0answers
61 views

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? ...
1
vote
0answers
43 views

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 = (\...
2
votes
1answer
20 views

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 \...
6
votes
1answer
72 views

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 ...
4
votes
2answers
134 views

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 ...
2
votes
2answers
49 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] ...
2
votes
1answer
43 views

How does the function to curry and uncurrying another function work?

The following is the code to curry or uncurry a function in Haskell: ...
2
votes
1answer
51 views

An operational semantics for lambda-calculus normal order evaluation strategy

TAPL book, page 56 reads: Under the normal order strategy, the leftmost, outermost redex is always reduced first. I understand this as a restriction of the full beta-reduction evaluation ...
2
votes
1answer
48 views

Mapping a free variable of type A to a cartesian closed category

In From Lambda Calculus to Cartesian Closed Categories, the author explains the interpretation of lambda calculus in cartesian closed category and at one point he explains how a term representing a ...
1
vote
1answer
75 views

Is monoid the category for untyped lambda calculus?

If cartesian closed categories are the model for simply typed lambda calculus, then can it be said that a monoid is a categorical model for untyped lambda calculus?
2
votes
1answer
26 views

lambda-calc program which halts on only one input

Does there exist a normal-form lambda calculus program $f$ such that $f (\lambda x . x)$ normalizes For all normal form $e \ne \lambda x . x$, $f e$ does not normalize
2
votes
1answer
77 views

How is β-reduction a 2-morphism in Category theory?

According to Categorifying CCCs: Computation as a Process, computation or β-reduction process in untyped-lambda calculus is in fact a 2-morphism in category theory. Can someone please describe me ...
2
votes
1answer
56 views

Is $\Gamma \vdash x x : T$ possible in the simply typed lambda calculus?

Is $\Gamma \vdash x x : T$ possible? This problem appears on page 104 of Benjamin Pierce's "Types and Programming Languages". My conclusion is that it is was the case then we would get $x: T_1 \to ...
0
votes
0answers
25 views

Is it possible to deduce type from the lambda form?

I was continuing the exploration of lambda world this summer. When I take a look at the simply typed lambda calculus, it looks like there is no use for usual chuch numerals and boolean forms anymore. ...
4
votes
0answers
69 views

Does type-1 lambda calculus exist?

I'm interested in the intersection of linguistics and computer science, I've been reading on Chomsky hierarchy, and would like to know if there exist lambda calculus types that are equivalent to the ...
3
votes
1answer
53 views

Is this lambda abstraction created as a generator of a recursive function?

In lambda calculus, a recursive function $f$ is obtained by $$ f = Y g $$ where $Y$ is the Y combinator and $g$ is the generator of $f$ i.e. $f$ is a fixed point of $g$ i.e. $f == g f$. In The ...
4
votes
0answers
144 views

How shall I understand the definitions of `let` expression?

let as used in programming languages is defined in lambda calculus as per https://en.wikipedia.org/wiki/Let_expression#Let_definition_defined_from_lambda_calculus ...
3
votes
0answers
61 views

Calculus of constructions, type-in-type and recursion

Does adding type-in-type to the calculus of constructions lead to (general) recursion? Such that one can write the Y combinator.
0
votes
1answer
48 views

Could following be a counter example to Church-Rosser (Confluence) theorem?

According to the "Type Theory and Formal Proof" book, Church-Rosser theorem (confluence) is as follow: Suppose that for a given term $M$, we have $M \twoheadrightarrow_\beta N_1$ and $M\...
2
votes
1answer
62 views

Are the definitions of constructs in terms of lambda terms issues in implementation/design or uses of functional languages?

In Lambda Calculus, natural numbers, boolean values, list processing functions, recursion, if function are defined in terms of lambda terms. For example, natural numbers are defined as Church numerals,...
3
votes
2answers
91 views

What is the name of the operator that translates from $X\rightarrow(Y\rightarrow Z)$ to $Y\rightarrow(X\rightarrow Z)$?

Is there a standard name for the operator that takes a function $f:X\rightarrow(Y\rightarrow Z)$ and returns the function $f':Y\rightarrow(X\rightarrow Z)$ that satisfies, for every $y \in Y$ and $x \...
5
votes
1answer
45 views

Explain auto continue passing style transformations

Recently I saw 3 cps transformation rules, but no explanations were given. expressions: $e :=x\left|e e^{\prime}\right| \lambda x \cdot e$ rules: $$ \begin{array}{l}{[[x]]=\lambda \kappa \cdot \...

1
2 3 4 5
9