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3
votes
1answer
161 views

internal language for the 2-category of small categories

What is the internal language of the category Cat of small categories? I found an article by Glynn Winskel and his student Mario Jose Cáccamo about such calculus! However it is limited to a fragment ...
18
votes
3answers
980 views

Example of a space for which $V \cong Hom(V,V)$

Let $V$ be a topological linear space, and let $\operatorname{Hom}(V,V)$ be the space of continuous linear maps from $V$ back to $V$, equipped with a suitable topology. Is there a non-trivial ...
4
votes
1answer
99 views

Models of intuitionistic linear logic that reflect the resource interpretation

I am interested in models of intuitionistic linear logic, that is, the logic that you get if you take classical linear logic and restrict the set of operators to $\otimes$, $1$, $\multimap$, $\times$, ...
8
votes
1answer
278 views

Why did Alonzo Church choose the letter $\lambda$ as the “binding operator”?

Is there any known reason why Alonzo Church chose Greek $\lambda$ as the "binding operator" for the Lambda Calculus?
5
votes
2answers
460 views

Why is there no product type in simply typed lambda-calculus?

Consider simply typed $\lambda$-calculus that has only the unit type as primitive. We would like to encode the product and the sum types. An encoding of the product type in the untyped ...
1
vote
0answers
109 views

Is there an easy decision algorithm for the inhabitation problem for simple types?

Consider the basic system of simple types usually known as $TA_\lambda$. One can prove that (as a consequence of the Subject Reduction Property and the fact that any typable term is strongly ...
0
votes
1answer
145 views

Interaction-based approximation for HP-complete λ-theory?

We are looking for a proof or counter-examples for the following hypothesis. Two combinators $M$ and $N$ are solvable and equivalent in the HP-complete sensible $\lambda$-theory iff either $$ \exists ...
1
vote
1answer
117 views

combinator SSS(SS)SS is not strongly normalizing. Why?

I highly speculate that combinator SSS(SS)SS is not strongly normalizing. What is the argument for the non strong normalization?
1
vote
0answers
92 views

Schönhage's SMM with only one instruction

It is possible to implement $\lambda$-calculus in Schönhage's storage modification machine using an infinite set of nodes and one single program consisted exclusively of (about hundred) instructions ...
1
vote
1answer
192 views

Hypothesis: interaction-based model for λKβη

We are looking for a proof or counter-examples to the following Hypothesis. In interaction calculus $\langle \varnothing\ |\ \Gamma(M, x) \cup \Gamma(N, x)\rangle \downarrow \langle \varnothing\ |\ ...
2
votes
0answers
149 views

Is it possible to implement η-reduction in interaction nets?

There are several ways to encode λ-terms in interaction nets; for instance, using the original optimal algorithm by Lamping, or compiling λ-calculus into interaction combinators. However, all the ...
1
vote
0answers
61 views

Optimal Reduction in Interaction Calculus

We work in interaction calculus. Let $\Sigma = \{\lambda, \psi, \delta, \epsilon\}$, $\text{Ar}(\lambda) = \text{Ar}(\psi) = \text{Ar}(\delta) = 2$, and $\text{Ar}(\epsilon) = 0$. For any $\alpha ...
3
votes
2answers
535 views

Turing-complete primitive blind automata

Let $N$ be the set of natural numbers, $S$ be the set of finite binary sequences, and $Q = [N \rightarrow N] \times [N \rightarrow N],$ where $[N \rightarrow N]$ is the set of all computable ...
0
votes
0answers
100 views

Recursive relation using successor function

What is the recursive relation for H(m)=2^(m^2) using successor function recursive relation for multiplication: mult(x,0)=0; mult(x,S(y))=add(x,mult(x,y)) recursive relation for addition: add(x,0)=x; ...
0
votes
1answer
244 views

Universality of blind graph rewriting

Let us consider $S(M) = \{(f_0, f_1) | f_0, f_1: M \rightarrow M\}$, where $M$ is a finite set. Each element of $S(M)$ is equivalent to a finite directed graph with the set of nodes $M$, which has ...
3
votes
1answer
631 views

Algebraic structure generated by primitive graph operations

Let $M$ be a finite set, and $S(M) = \{(f_0, f_1) | f_0, f_1: M → M\}$. Each element of $S(M)$ can be considered as a finite directed graph with the set of nodes $M$, which has exactly two arrows ...
4
votes
0answers
230 views

Difference between lambda-calculus with well-formed formulas vs properly-formed formulas

In S.C. Kleene's 1935 paper "$\lambda$-definability and recursiveness," he proves that all $\lambda$-definable functions are general recursive in the Herbrand-Godel sense and vice-versa. However, the ...
4
votes
7answers
4k views

What is some good introduction to lambda calculus?

I have some background in set theory and automata and I am looking for a good place to start with lambda calculus.
5
votes
4answers
638 views

Why is alpha-reduction in untyped $\lambda$-calculus substitutive?

This is something all introductory texts seem to avoid proving, and many even avoid stating. We consider untyped $\lambda$-terms on some countably infinite alphabet. If $x$ is a variable and $p$ is ...
5
votes
1answer
352 views

Are innermost reductions perpetual in untyped $\lambda$-calculus?

Background In the untyped lambda calculus, a term may contain many redexes, and different choices about which one to reduce may produce wildly different results (e.g. $(\lambda x.y)((\lambda ...
0
votes
1answer
242 views

Is it correct to state that basic primitive recursive functions are in fact combinators?

Is it correct saying that the Zero, Successor and Projection functions can be seen as combinators?
7
votes
6answers
21k views

Difference between a 'calculus' and an 'algebra'

What is really the conceptual difference between a calculus and an algebra. Eg. Is SKI combinator calculus really a calculus? A friend claims that free variables are fundamental for a calculus, and ...
13
votes
3answers
1k views

What is the history of the Y-combinator?

Inspired by the comments to this question, I wonder if someone can explain the history of the fixed point combinator (often called the Y combinator) in lambda calculus. Where did it first appear? ...
6
votes
4answers
1k views

[solved] sequent calculus as programming language

intuitionistic logic ~ programming natural deduction ~ lambda-calculus Hilbert system ~ combinatory logic {S, K} Gentzen system=sequent calculus ~ ? What would you write in place of the question ...
6
votes
4answers
658 views

What is the intuitive meaning of star and box in a pure type system?

The systems of the λ-cube have the axiom $\star:\square$. I've listed a few meanings that the Curry-Howard isomorphism gives to $t : T$ below. What are the intuitive meanings of $\star$ and ...
6
votes
1answer
880 views

Scott on the consistency of the lambda calculus

I have twice heard it attributed to Dana Scott that he said something to the effect that the consistency of the lambda-calculus was an accident. Does anyone have a reasonable-sounding source for ...
6
votes
4answers
727 views

Can dependent sums be encoded as dependent products?

Please forgive any unorthodox notation or obvious errors here... I'm trying to get an intuition for dependently typed languages, so I'm starting out by seeing which analogies I can take from the ...
7
votes
3answers
3k views

Is functional programming a branch of mathematics?

In Theory mainly concerned with lambda-calculus?, F. G. Dorais wrote, of the idea that the lambda-calulus defines a domain of mathematics: That would never stick unless there's another good ...
3
votes
5answers
1k views

Theory mainly concerned with $\lambda$-calculus?

Automata theory is mainly concerned with Turing machines and all its relatives-in-spirit. $\lambda$-calculus is rather rarely mentioned in textbooks on automata theory. What's the common name of the ...
12
votes
7answers
2k views

What is lambda calculus related to?

So I'm not much of a math guy but I've really enjoyed programming in Lisp and have become interested in the ideas of lambda calculus which it is based. I was wondering if anyone had a suggestion ...