# Questions tagged [lambda-calculus]

Church's formal system used in computatability, programming languages and proof theory to represent effective functions, programs and their computation, and proofs.

202 questions
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### Can a fixed point combinator find the fixed point of a function that has no fixed point? [closed]

A fixed point combinator is supposed to find the fixed point of any function. Yet I am wondering what if a function happens to have no fixed point, such as the add1 ...
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### Why is the multi-step reduction of semantics reflexive?

I was reading Programming Languages and Lambda Calculi, which defines the multi-step reduction to be the reflexive-transitive closure of the one-step reduction. (Page 15, $\twoheadrightarrow_r$ is the ...
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### Subtyping rules for extension of System $F_\omega$ with subtyping and kind-level variance tracking

I need an extension of System $F_\omega$ with subtyping, and where the variance of type constructors is reflected in their kind. Unfortunately, System $F^\omega_{<:}$, as defined in chapter 31 of ...
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### Composition in explicit substitutions

In the classical λσ calculus of explicit substitutions, there is the following rewrite rule: (a[s])[t] ==> a[s ∘ t] where ...
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### Is it possible to implement tail recursion inside a lambda calculus built on top of functions?

Inside a lambda calculus implementation for ECMASCript 6, we are trying to implement new constructs such as type tags for strong typing, and conditionals such as the ...
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### Is there a “lambda cube” for interaction nets?

The lambda calculus is an untyped language that is often extended with logical frameworks such as the vertices of the λ-cube. Is there something similar to it, but for interaction nets? What about ...
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### What type system fits the subclass of λ-terms that can be reduced optimally?

There is a subset of λ-calculus terms that can be reduced by Lamping's Abstract Algorithm without using the Oracle. That is an interesting subset, because only for those terms it is proven that ...
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### Contradiction between Gödel's Second Incompleteness Theorem and the Church-Rosser's property of CIC?

On one hand, Gödel's Second Incompleteness Theorem states that any consistent formal theory that is strong enough to express any basic arithmetical statements can't prove its own consistency. On the ...
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### What are the negative consequences of extending CIC with axioms?

Is it true that adding axioms to the CIC might have negative influences in the computational content of definitions and theorems? I understand that, in the theory's normal behavior, any closed term ...
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### Church-Rosser property for dependently typed lambda calculus?

It is well-known that the Church-Rosser property holds for $\beta \eta$-reduction in simply-typed lambda calculus. This implies that the calculus is consistent, in the sense that not all equations ...
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### historical question: earliest description of beta-normal terms together with “neutral” terms in lambda calculus?

A bit of "folklore" in lambda calculus is the idea of characterizing the class of $\beta$-normal terms inductively as a syntactic category ($R$) defined in mutual induction with an auxiliary syntactic ...
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### Are optimal evaluators actually optimal?

The following term (using bruijn-indexes): BADTERM = λ((0 λλλλ((((3 λλ(((0 3) 4) (1 λλ0))) λλ(((0 4) 3) (1 0))) λ1) λλ1)) λλλ(2 (2 (2 (2 (2 (2 (2 (2 0))))))))) ...
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### Is it possible to evaluate interaction combinators efficiently using a path-traveling strategy?

Interaction combinators can be evaluated using a path traversing strategy. That is, instead of applying annihilation/commutation rules to active pairs, one simply walks through the graph using a 2-...
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### Is the question about closed reduction using token-passing nets still open?

The question about possible implementation of closed reduction using token-passing nets is asked on page 17 in Token-passing Nets for Functional Languages by Jose Bacelar Almeida, Jorge Sousa Pinto, ...
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### Is higher-order unification decidable for terms without abstractions within applications?

Consider the problem of higher order unification - that is, finding a substitution for the equation a = b, where a and ...
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### How to state the adequacy of an encoding of lambda calculus in itself?

In the paper Discriminating coded lambda terms - Henk Barendregt a coding $\ulcorner M \urcorner$ of a lambda term $M$ is a term such that $M$ (and its parts) can be reconstructed from it in a lambda-...
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### How can you encode natural numbers operations on interaction combinators?

The church-encoding for natural numbers is a natural mean of implementing addition, multiplication and so on on the lambda calculus. Interaction nets are said to be an alternative universal ...
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### Optimal reduction using token-passing nets

I am looking for implementation of optimal reduction for λ-calculus based on interaction nets (McCarthy's amb allowed) in the spirit of "Token-Passing Nets: Call-by-Need for Free" by François-Régis ...
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### What's the difference between reduction strategies and evaluation strategies?

From the evaluation strategy article on Wikipedia: The notion of reduction strategy in lambda calculus is similar but distinct. From the reduction strategy article on Wikipedia: It is similar ...
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### Is sort typeable on elementary affine logic?

The following λ-term, here in normal form: ...
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### An example where smallest normal lambda term is not fastest

Let the $size$ of $\lambda$-terms be defined as follows: $size(x) = 1$, $size(λx.t) = size(t) + 1$, $size(t s) = size(t) + size(s) + 1$. Let the complexity of a $\lambda$-term $t$ be defined as the ...
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### What is the formal definitions of the reduction related to the “call/cc” (call with the current continuation) operator?

In lambda calculus or in combinatory logic we formally define reduction/expansion rules for terms (and in their typed variants reductions must preserve the type). Then we can talk about properties of ...
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### What is the simplest known solver for a np-complete problem?

Lets define the simpler of two terms as the one with shortest description length on the untyped λ-calculus. Trying to find the simplest solver for a np-complete problem, I've got this: ...
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### How do you encode Lamping's abstract algorithm using interaction combinators?

Interaction combinators have been proposed as a compile target for the λ-calculus before. That paper implements the full λ-calculus. It is also known that it is possible to optimize interaction-net ...
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### Would a proof that the traveling salesman algorithm can't be encoded on LAL also prove P!=NP?

An answer to the traveling salesman (and similar) problems can be easily verified on light lambda-calculi. Also, if I understand correctly, the light lambda-calculi can compute every polinomial-time ...
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### Is it possible to unambiguously read back λ terms from interaction nets without node types?

A class of lambda terms can be evaluated using Lamping's abstract algorithm - that is, converting them to interaction nets and applying a set of rules. In order to get the result, you have to read ...
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### Is infinitary Böhm-reduction wrt. root-active terms for $\lambda$-calculus transitive?

I expect the answer to be "obviously yes", but to my inexperienced eye, that's not directly obvious, because the definition of infinite Böhm-reduction does not include a transitivity rule (it wouldn't ...
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### Smallest possible universal combinator

I am looking for the smallest possible universal combinator, measured by the number of abstractions and applications required to specify such a combinator in the lambda calculus. Examples of universal ...