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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14
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3answers
181 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 ...
-1
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0answers
37 views

Decidability of sets of free variables in Lambda Calculus

This question is also posed in the Mathematics SE: http://math.stackexchange.com/questions/1055621/decidability-of-sets-of-free-variables-in-lambda-calculus Let $\Lambda$ be the set of lambda-terms ...
3
votes
1answer
72 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
70 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
votes
1answer
84 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 ...
0
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0answers
97 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 ...
12
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2answers
401 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
123 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
116 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
250 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
283 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
votes
1answer
415 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 ...
5
votes
1answer
96 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 ...
4
votes
1answer
422 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
votes
1answer
130 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 ...
16
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3answers
2k 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 ...
4
votes
1answer
62 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
66 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
votes
1answer
135 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 ...
10
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0answers
222 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] + ...
5
votes
0answers
149 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 ...
57
votes
6answers
21k 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 ...
6
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0answers
139 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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0answers
77 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\; ...
15
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1answer
157 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 ...
-3
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1answer
92 views

Are two unbound variables alpha-equivalent? [closed]

Let's say we have the following cases: $a =_\alpha b$ $a =_\alpha a$ Which of the above cases are $\alpha$-equivalent? Or does $\alpha$-equivalence has no meaning in the context of just variables? ...
4
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3answers
326 views

How to make the Lambda Calculus strong normalizing without a type system?

Is there any system similar to the lambda calculus that is strong normalizing, without the need to add a type system on top of it?
2
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0answers
72 views

Evaluation contexts: outside-in vs inside-out

I heard that there exist two styles to define an evaluation context: outside-in and inside-out. Can someone give the definitions? Why are they so named (inside-out and outside-in)? What is the ...
2
votes
2answers
449 views

Implementation of alpha equivalence

I'm reading through "Type Theory & Functional Programming" by Simon Thompson and it says We shall not distinguish between expressions which are equivalent up to change of bound variable names ...
1
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2answers
280 views

Why is lambda calculus a “calculus”?

The only definition of "calculus" I'm aware of is the study of limits, derivatives, integrals, etc. in analysis. In what sense is lambda calculus (or things like mu calculus) a "calculus"? How does it ...
1
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0answers
53 views

Is combinatory strong reduction equivalent to lambda beta-eta reduction?

I'm familiar with the correspondence between combinatory logic and $\lambda$-calculus at equality level, ie $=_{\beta\eta}$ corresponds to $=_s$ (the equivalence relation generated by strong ...
5
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0answers
160 views

Regular languages in lambda calculus

With Turing machines, by imposing certain restrictions on the form of the transition function, one can get a machine that accepts only regular languages. I am wondering what is the counterpart in ...
2
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2answers
207 views

Is there a space efficient way to represent numbers on the lambda calculus?

This is something I've been thinking. While it is agreed that Lambda Calculus is equivalent to a Turing Machine in power, is it actually so? Church Numerals are not very space efficient and I'm not ...
1
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3answers
220 views

“lambda” term usage in programming

could any one please let me know what is the relation between "lambda" and anonymous functions in programming? in other words why we say lambda function to an anonymous function? I am here trying to ...
1
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1answer
123 views

A few questions about ISWIM

I recently read Landin's paper "The Next 700 Programming Languages". But I was a bit confused by ISWIM. In particular, are functions first-class objects in ISWIM? It seems not because every ...
-2
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8answers
551 views

What are the simplest turing-complete systems? [closed]

Lambda Calculus is very simple. Are there even simpler turing-complete systems? Which is the simplest of them all?
4
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2answers
199 views

Is there a systematic method for constructing lambda calculus terms that can distinguish between inputs?

For example, finding terms $\vec{a}$ such that: $\vec{a}(\lambda x.x) = T\\\vec{a}(\lambda xy.x) = F$ Is there a systematic method for finding terms with these types of constraint?
8
votes
1answer
280 views

η-conversion vs extensionality in extensions of lambda-calculus

I'm often confused by the relation between η-conversion and extensionality. Edit: According to comments, it seems I'm also confused about the relation between extensional equivalence and ...
16
votes
1answer
475 views

What is the difference between arrows and exponential objects in a cartesian closed category?

In a Cartesian Closed Category (CCC), there exist the so-called exponential objects, written $B^A$. When a CCC is considered as a model of the simply-typed $\lambda$-calculus, an exponential object ...
13
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3answers
536 views

funsplit and polarity of Pi-types

In a recent thread on the Agda mailing list, the question of $\eta$ laws popped up, in which Peter Hancock made thought-provoking remark. My understanding is that $\eta$ laws come with negative ...
8
votes
1answer
159 views

What is the benefit of Krivine's notation?

I saw some people uses Krivine's notation for function application when presenting the syntax for the $\lambda$-calculus. For example, the $\lambda$-term $\lambda f . \lambda x . \lambda y . f\ x\ y$ ...
7
votes
3answers
190 views

Are the $\lambda_I$-Calculus and the $\lambda_K$-Calculus equivalent?

I see here and there mention of the $\lambda_I$-Calculus (in which every variable must be used at least once) and the $\lambda_K$-Calculus (in which a variable can also be unused). Are they ...
11
votes
1answer
554 views

Can Scheme's call/cc implement all known control flow structures?

The page "Advanced Scheme: Some Naughty Bits" states: Continuations are a powerful control-flow construct from which nearly any other control-flow structure [...] may be derived. I thought ...
9
votes
1answer
194 views

A simple proof that decidability of typability in System F ($\lambda 2$) implies decidability of type checking?

Suppose we don't know Joe B. Wells's result from 1994 that both typability and type checking are undecidable in System F (AKA $\lambda 2$). In Barendregt's Lambda calculi with types (1992) I found a ...
16
votes
2answers
302 views

Is it possible to decide $\beta$-equivalence within System F (or another normalizing typed λ-calculus)?

I know that's impossible to decide $\beta$-equivalence for untyped lambda calculus. Quoting Barendregt, H. P. The Lambda Calculus: Its Syntax and Semantics. North Holland, Amsterdam (1984).: If A ...
5
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1answer
197 views

Characterizing closure under expansion/reduction for big-step semantics?

Two common ways of formulating operational semantics for programming languages based on lambda-calculus are big-step and small-step semantics. In a big step semantics, you give a relation $e ...
9
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0answers
480 views

What's the expressive power of Simply Typed Lambda calculus?

The standard approach to simply typed lambda calculus considers computations over Church numerals. If input and outputs are Church numerals always typed as $Int$, where $Int = (\tau \rightarrow \tau) ...
3
votes
1answer
181 views

Exponential blowup in Simple Proof of a theorem of Statman by Mairson

I'm studying "A simple proof of a theorem of Statman" by H.G. Mairson. At page 4, he encodes set/type theory in lambda calculus. In particular, note che "op" trick in the definition of $eq_{k+1}$. ...
10
votes
1answer
263 views

Extensionality of lambda calculus models

I'm translating a book on LISP and naturally it touches some elements of $\lambda$-calculus. So, a notion of extensionality is mentioned there alongside some models of $\lambda$-calculus, namely: ...
7
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1answer
269 views

Reading list on rewriting systems?

I am new to studying rewriting systems as a first year PhD student. I would like to propose a special topics course on rewriting theory, and I want to make sure I don't leave any of the original ...