All Questions
Tagged with computer-science lambda-calculus
19 questions
23
votes
1
answer
1k
views
What, mathematically speaking, does it mean to say that the continuation monad can simulate all monads?
In various places it is stated that the continuation monad can simulate all monads in some sense (see for example http://lambda1.jimpryor.net/manipulating_trees_with_monads/))
In particular, in http://...
- 639
20
votes
1
answer
2k
views
What is Chemlambda? In which ways could it be interesting for a mathematician?
I${}^{*}$ have randomly come across a couple of websites (Chemlambda project, chorasimilarity) that seem to be about a certain "thing" (a computer program, I think) called Chemlambda that does "stuff" ...
- 23.3k
17
votes
3
answers
3k
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? ...
- 8,803
16
votes
2
answers
3k
views
Why is there no product type in simply typed lambda-calculus?
$\DeclareMathOperator\Pair{Pair}\DeclareMathOperator\First{First}\DeclareMathOperator\Second{Second}\DeclareMathOperator\Left{Left}\DeclareMathOperator\Right{Right}\DeclareMathOperator\Choice{Choice}$...
- 271
7
votes
1
answer
531
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 x.xx)\...
- 461
7
votes
0
answers
181
views
CCCs, computational calculi and point-surjectivity
The models of some computational calculi are in a correspondence with Cartesian Closed Categories with an object $U$ that has some relationship to its exponential object $U^U$ e.g. a retraction ...
4
votes
1
answer
278
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 ...
- 476
4
votes
1
answer
967
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
0
answers
95
views
$\omega$ incompleteness of $\lambda$ calculus
In Plotkin's 'The $\lambda$-Calculus is $\omega$-Incomplete' (The Journal of Symbolic Logic Vol. 39, No. 2 (Jun., 1974), pp. 313-317), an example is given of two (untyped) $\lambda$-terms $M$ and $N$ ...
- 358
3
votes
1
answer
404
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 $\beta$-...
- 31
3
votes
2
answers
790
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 ...
3
votes
0
answers
264
views
Upward confluence in the interaction calculus
The lambda calculus is not upward confluent, counterexamples being known for a long time. Now, what about the interaction calculus? Specifically, I am looking for configurations $c_1$ and $c_2$ such ...
3
votes
0
answers
266
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 ...
2
votes
2
answers
181
views
Background for Kierstead terms
I was looking at some slides of John Longley's here, where he mentions "the Kierstead functional"
$$\lambda f.f(\lambda x.f(\lambda y.x)) \ ,$$
(where $f$ should be of type $2$, and $x,y$ of ground ...
- 269
2
votes
1
answer
286
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 ...
1
vote
0
answers
137
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 ...
1
vote
1
answer
169
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
0
answers
111
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
1
answer
223
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\ |\ ...