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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Is `sort` typeable on elementary afine logic?

The following λ-term, here in normal form: ...
7
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1answer
196 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
46 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
166 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: ...
8
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1answer
85 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
138 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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68 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
42 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
361 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
70 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
88 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
52 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 ...
4
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40 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 ...
8
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2answers
321 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
49 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 ...
11
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3answers
425 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 ...
7
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1answer
202 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
60 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 ...
5
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1answer
296 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 ...
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3answers
366 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 ...
5
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3answers
174 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 ...
20
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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
77 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 ...
8
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2answers
570 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
31 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 ...
10
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3answers
338 views

Is there any known CCC closed under a probabilistic powerdomain operation?

Equivalently, is there a known denotational semantics for probabilistic higher-order functional programming languages? Specifically, is there a domain model of pure untyped $\lambda$-calculus extended ...
14
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3answers
319 views

Can typed lambda calculi express *all* algorithms below a given complexity?

I know that the complexity of most varieties of typed lambda calculi without the Y combinator primitive is bounded, i.e. only functions of bounded complexity can be expressed, with the bound becoming ...
3
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1answer
93 views

Constructing terms of function types out of the empty type

If a function $f$ is understood as its graph, i.e. a set of pairs $\langle x,y\rangle$ where $x$ is input and $y$ is output, then the empty set $\emptyset$ is a valid function, and for any set $A$, we ...
5
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1answer
85 views

Commutativity of addition in polymorphic lambda calculus

In the article "Extensional models of polymorphism" by Breazu-Tannen and Coquand, natural numbers are presented using a Church-like encoding: $polyint = \forall t . (t \to t) \to t \to t$ Addition ...
7
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1answer
142 views

Church-Rosser equivalent for concatenative languages?

Looking at the striking parallels between combinatory logic and concatenative languages makes me wonder how many theorems of the former hold in the latter. The Church-Rosser theorem is particularly ...
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112 views

Is there an algorithm to find whether 2 combinators form a Turing-complete system?

It is known that K = (λx.(λy.x)) and S = (λx.(λy.(λz.((x z) (y z))))) define a turing complete system, and we know procedures to ...
13
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2answers
483 views

Fixed points in computability and logic

This question has also been posted on Math.SE, http://math.stackexchange.com/questions/1002540/fixed-points-in-computability-nd-logic I hope it is ok to also post it here. If not, or if it is too ...
4
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1answer
149 views

What is a term of the type $\bot\rightarrow A$?

The sentence $\bot\rightarrow A$ is provable in intuitionistic logic for any type $A$. The proof is trivial: \begin{align} \bot&\vdash\bot \\ \hline \bot&\vdash A \\ \hline ...
6
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1answer
199 views

Are there presentations of set theory in terms of lambda-calculus?

I am planning to implement in software a set theory language, based on a binary function, which in set theory is the so called adjunction operation: $f(x, y) = x \cup$ {y}. Therefore, a presentation ...
13
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1answer
358 views

Scott's stochastic lambda calculi

Recently, Dana Scott proposed stochastic lambda calculus, an attempt to introduce probabilistic elements into (untyped) lambda calculus based on a semantics called graph model. You can find his ...
10
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1answer
348 views

Can affine lambda calculus solve every problem in P?

In Advanced Topics in Types and Programming Languages it is mentioned, in the chapter on sub-structural type systems, that a "carefully crafted" affine lambda calculus with a recursion combinator for ...
5
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1answer
438 views

Is this behavior in a programming language inconsistent?

I'm developing a tiny programming language to try to wrap my head around type inference, and I'm trying to figure out if its behavior makes sense or not. Here's the problem: The identity function ...
6
votes
1answer
110 views

A function is lambda-2-definable iff it is HG computable and provably type correct in lambda-PRED2

I'm having a problem regarding Theorem 5.4.40.3 of Barendregt's Lambda calculi with types (1992), a chapter in Handbook in logic in computer science. (I'm referring to the PostScript version available ...
6
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1answer
696 views

Historic Relationship between Typed Lambda Calculus and Lisp?

I was having a discussion with a friend recently (who is an advocate of strongly typed languages). He made the comment: The inventors of Lambda Calculus always intended it to be typed. Now we ...
7
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1answer
140 views

Are there stronger notions of equivalence over lambda terms than beta equivalence?

I should add the context that I am concerned with strongly normalizing systems like System-F. I have what I consider a very strong notion of equivalence for lambda terms that goes something like the ...
22
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3answers
3k views

P and NP classes explanation through lambda-calculus

In the introduction and explanation P and NP complexity classes often given through Turing machine. One of the model of computation is the lambda-calculus. I understand, that all of models of ...
5
votes
1answer
74 views

Upper bound on Chaitin's constant for lambda calculus and SKI combinatory logic

I'd like to have proof that Chaitin's constant for lambda calculus and/or SKI combinatory logic is pretty small. I've found some approximations (accurate to about 63 binary digits) for truing machines ...
2
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0answers
82 views

How to translate general recursion into a set of $\mu$-recursive operator applications?

I'm trying to find a scheme to translate a functional language with let rec into a set of primitives called "generalized arrows", i.e. $\kappa$-calculus with ...
5
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1answer
165 views

Can factorial be encoded in the Kappa-calculus with fixed point operator?

Suppose we have a $\kappa$-calculus with operator $fix$, that could be used to transform function with type $(1 \rightarrow a) \rightarrow a$ to a value of type $1 \rightarrow a$. We use a normal ...
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238 views

A bijection between ordered lambda terms and rooted planar maps?

Consider the following recurrence in two parameters $n$ and $k$: \begin{aligned} NF(0,k) &= 0 \\ NF(n,k) &= Neu(n,k) + NF(n-1,k+1) \\ Neu(n,k) &= [n=1 \wedge k=1] + ...
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198 views

Complete combinator basis for System F-omega

The S and K combinators form a complete (and Turing complete) basis when untyped. Within the Hindley-Milner type-system, and I believe within system $F$ as well, S and K can encode any well-typed ...
64
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7answers
22k views

What is the contribution of lambda calculus to the field of theory of computation?

I'm just reading up on lambda calculus to "get to know it". I see it as an alternate form of computation as opposed to the Turing Machine. It's an interesting way of doing things with ...
7
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146 views

Equivalence of categories of directed complete posets

I asked this question there: http://math.stackexchange.com/questions/700975/equivalence-of-categories-of-directed-complete-posets. Since I had no answer, I try here. In the book ``Domains and ...
4
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99 views

Defining a calculation-reversing function in the lambda calculus

Obviously, lambda calculus functions are not in general invertible. That is, there is no lambda function $V$ (for inVerse) such that $$ (V\; A)\; (A\; B) \to B $$ for every $A$ and $B$ such that $(A\; ...
17
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1answer
201 views

A Lambda calculus for invertible (r-Turing computable) functions

I'm interested in the concept of "r-Turing completeness", as defined by Axelsen and Glück (2011). A system is r-Turing complete if it can compute the same set of functions as a reversible Turing ...