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

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Incomplete basis of combinators

This is inspired by this question. Let $\mathcal{C}$ be the collection of all combinators which only have two bound variables. Is $\mathcal{C}$ combinatorially complete? I believe the answer is ...
5
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4answers
172 views

Do I have to give up the Law of the Excluded Middle in order to Learn $\lambda$-Calculus?

I know very little about what I am talking about in what follows, so I appreciate any all help in pointing out all of my mistakes -- otherwise I won't be able to learn more and advance in my knowledge ...
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59 views

Understanding between lambda-calculus and other abstract machines (like Turing machine and Markov algorithm)

If we look on abstract machines we could noticed analogue with modern computers (of course). What I mean? I mean this points: 1. Model of implementer (In Turing machine it is description of head, ...
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1answer
125 views

Can all mathematical operations be encoded with a Turing Complete language? [closed]

In High School Computing I was taught the Structured Program Theorem - that you could implement any mathematical operation using: Sequence Selection Iteration After completing a Computer Science ...
2
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1answer
63 views

How is Lambda Calculus a specific type of Term Writing system?

Now we can see that Church was associated with the Simply Typed Lambda Calculus. Indeed, it seems he explained the Simply Typed Lambda Calculus in order to reduce misunderstanding about the Lambda ...
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61 views

Denotational semantics of System $F_\omega$ with recursive types and general recursion

Is there a denotational semantics for System $F_\omega$ in literature that supports both recursive types and general recursion? I'm looking for a model of Ralf Hinze's variant of System $F_\omega$ [4]...
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2answers
356 views

How do you get the Calculus of Constructions from the other points in the Lambda Cube?

The CoC is said to be the culmination of all three dimensions of the Lambda Cube. This isn't apparent to me at all. I think I understand the individual dimensions, and the combination of any two seems ...
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1answer
51 views

Algorithm for extensional equality in combinator calculus

I'm dealing with combinator calculus, using the $S$ and $K$ combinators as a basis. Sometimes my code generates expressions that define equivalent functions, such as $$ (S\, K\, K) \qquad\text{and}\...
5
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177 views

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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1answer
59 views

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 ...
5
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71 views

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 ...
4
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1answer
85 views

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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1answer
213 views

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 ...
9
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1answer
302 views

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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3answers
436 views

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 ...
11
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2answers
134 views

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 ...
9
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0answers
74 views

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 ...
10
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1answer
863 views

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))))))))) ...
5
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1answer
48 views

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-...
4
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0answers
45 views

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, ...
5
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0answers
70 views

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 ...
3
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0answers
33 views

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-...
4
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1answer
70 views

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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61 views

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 ...
8
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2answers
186 views

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 ...
8
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1answer
208 views

Is `sort` typeable on elementary affine logic?

The following λ-term, here in normal form: ...
7
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1answer
226 views

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 ...
4
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1answer
63 views

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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1answer
185 views

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: ...
10
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1answer
190 views

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 ...
2
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1answer
152 views

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 ...
6
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90 views

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 ...
4
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1answer
48 views

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 ...
13
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2answers
444 views

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 ...
2
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1answer
82 views

Is there a pair of different lambda terms in the normal form that behave identically when applied to any input?

Let f and g be lambda terms in the normal form, such that f is intensionally different from <...
5
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2answers
128 views

Weakly normalizing + confluent = strongly normalizing?

I was reading this abstract and saw that they prove weak normalization and confluence. My limited understanding suggests that those two properties should provide strong normalization, which then ...
0
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1answer
54 views

Why is the polymorphic weight 1

I am reading through through a paper called HMF: Simple Type Inference for First-Class Polymorphism by Daan Leijen of Microsoft Research. In the paper it describes how to calculate the polymorphic ...
6
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0answers
53 views

Decidability of equality between higher-kinded equirecursive types (or: between nonregular Böhm trees)

In §3 of Polytypic values possess polykinded types, Ralf Hinze described a calculus of types with higher-kinded recursive types. There is a fixed-point combinator $$\mu_\kappa : (\kappa \rightarrow \...
10
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2answers
447 views

Is there a typed lambda calculus which is consistent and Turing complete?

Is there a typed lambda calculus where the corresponding logic under the Curry-Howard correspondence is consistent, and where there are typeable lambda expressions for every computable function? This ...
3
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1answer
64 views

What is contextual equivalence ignoring non-termination called?

Contextual equivalence ($M_1 \cong_{ctx} M_2$) is often defined as: $C[M_1] \Downarrow V \iff C[M_2] \Downarrow V$ Which is to say for any context $C$, $C[M_1]$ terminates with value $V$ iff $C[M_2]$...
13
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3answers
496 views

Why do constructivists not seem to care too much about call/cc

So a little while back I first had someone tell me that call/cc could allow proof objects for classical proofs by implementing Peirce's law. I did some thinking about the topic recently and I can't ...
9
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1answer
354 views

Why it's impossible to declare an induction principle for Church numerals

Imagine, we defined natural numbers in dependently typed lambda calculus as Church numerals. They might be defined in the following way: ...
2
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0answers
72 views

Resources for Church's paper “An Unsolvable Problem of Elementary Number Theory”?

I'm trying to understand and breakdown into simple English Church's paper for "An Unsolvable Problem of Elementary Number Theory" but I'm not finding anything useful online, only citations and links ...
6
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2answers
408 views

How would a theory of computation course that culminated in lambda-calculus as “the” model of computation, instead of Turing machines, look like?

Currently, our ToC (Theory of Computation) courses are designed with the following progression of topics: Finite automata and regular languages Pushdown automata and context-free languages Turing ...
2
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3answers
393 views

Why is lambda calculus so “function” oriented?

I've always had this question nagging at me subconsciously but have never been able to intuitively grasp it. Why does $\lambda$-calculus have a functional notation? Why is everything a function? It ...
6
votes
3answers
310 views

What is the relationship between intuitionistic logic, combinatory logic and lambda calculus?

I've been reading Lectures on the Curry-Howard Isomorphism and it talks about intuitionistic/constructive logic (IL) , combinatory logic (CL) and lambda calculus ($\lambda$c) before moving on to the ...
21
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2answers
2k views

What was the original intent for the creation of Lambda calculus?

I've read that initially Church proposed the $\lambda$-calculus as part of his Postulates of Logic paper (which is a dense read). But Kleene proved his "system" inconsistent after which, Church ...
4
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1answer
88 views

In which posets is the set of compact elements downwards closed?

In a poset $(D, \sqsubseteq)$, a compact element is an element $d \in D$ such that for every directed set $A$ which happens to have a supremum $\bigsqcup A \in D$ with $d \sqsubseteq \bigsqcup A$, it ...
10
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2answers
626 views

How exactly does lambda calculus capture the intuitive notion of computability?

I've been trying to wrap my head around the what, why and how of $\lambda$-calculus but I'm unable to come to grips with "why does it work"? "Intuitively" I get the computability model of Turing ...
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1answer
52 views

How is does a scheme lambda function relate to lambda calculus?

For example: (define fact (lambda (n) (if (< n 2) 1 (* n (fact (- n 1))))) How is this an example of applied lambda calculus? I tried to read the ...