To Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus (1996)

http://dkeenan.com/Lambda/

by dpatru • 7 years ago

102 29 7 years ago

dkeenan.com

genezeta• 7 years ago

Link has been updated

dang• 7 years ago

Whoops - changed from http://www.cs.virginia.edu/~evans/cs655-S00/readings/mocking....

This was an invited repost of https://news.ycombinator.com/item?id=890715 and I forgot to update the link from long ago.

See https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que... for more about invited reposts. There is a list of them at https://news.ycombinator.com/invited.

asplake• 7 years ago

Oh wow, a reference to Laws of Form (George Spencer-Brown), takes me back!

bordercases2• 7 years ago

Definitely an underrated piece of work.

carapace2• 7 years ago

"The Markable Mark" site is a great exposition of the Laws of Form: http://www.markability.net/

joe_the_user• 7 years ago

Is there are relationship between the lambda calculus and Brainfuck and other few instruction set languages?

Edit: well, an easy find, https://esolangs.org/wiki/Lambda_Calculus_to_Brainfuck

c1ccccc1• 7 years ago

Also: https://esolangs.org/wiki/Iota

posterboy2• 7 years ago

there's an isomorphism, because bf is np complete, as is simply typed lc.

simply speaking, bf can implement lc, and vice versa, which would proof the claim.

Edit: the ugly bit is that IO is always an ugly hack and potentially makes the program indetermined and thus impossible to proof a priory. but one can probably prove that they are equivalently unprovable.

anchpop• 7 years ago

because bf is np complete, as is simply typed lc.

Do you mean turing complete?

tromp2• 7 years ago

I wrote a Brainfuck interpreter in (binary) Lambda Calculus [1], which was included in my 2012 IOCC submission [2]. Writing a lambda calculus interpreter in BF would be a fun challenge.

[1] https://tromp.github.io/cl/Binary_lambda_calculus.html#Brain...

[2] http://www.ioccc.org/2012/tromp/hint.html

motohagiography3• 7 years ago

I have this naive intuition that graphs are a universal encoding scheme, all theorems of category theory can be expressed as graphs, with the implication there is a massive unifying leap forward in maths that will result from expressing existing problems in terms that are consistent with such a representation. Maybe it's literary sci fi handwaving, but there must be a rules based level of abstraction that contains all we can conceive of and express. These diagrams appear to be an example of it.

posterboy2• 7 years ago

Maybe it's literary sci fi handwaving, but there must be rules based level of abstraction that contains all we can conceive of and express.
rules based level of abstraction

that's not literature, that's bad grammar.

Sorry.

motohagiography3• 7 years ago

Please.If only egregious crimes against the language were the bar.

zozbot234• 7 years ago

What kinds of graphs? There are a number of graphical representations that turn out to be incredibly useful in category theory, but they're far from equivalent. IIRC, https://arxiv.org/abs/1803.05316 introduces some of them.

motohagiography3• 7 years ago

Just saying objects and morphisms. The types of each are found in CT, which abstract-up to these, but are bounded/encompassed by the limits we know of regarding elementary ideas like cliques, cycles, paths, colouring, various morphisms.

Compression, folding, encoding, and other information theory ideas have analogues represented as these, and the bottom up/depth first search of the problem space seems to come at it from the wrong direction.

We're well into indulging Internet crank territory here but I'm naively speculating that for every known theorem, there is a consistent object/morphism representation of it, and the rules we're getting good at finding for these relationships (graphs) will yield insights we were missing for lack of an encompassing consistent abstraction.

carapace2• 7 years ago

Are you channeling Grothendieck? :-)

https://en.wikipedia.org/wiki/Alexander_Grothendieck#Mathema...

nixpulvis• 7 years ago

What language expresses itself in the most intuitive way is a somewhat undefined problem as far as I understand it. Humans are fuzzy messy objects, our languages too.

Lambda calculus is partially fun just because of the minimal extent of its description.

galaxyLogic3• 7 years ago

Graphs are a natural way to express structures, of things being composed of parts which are somehow related to each other connected to each other.

What are some alternatives to graphs? Well some kind of language, say a programming language. But underlying any program written in any programming language is a graph of the parts of the program and their connections to each other. For instance a variable may be written in one section of the code and read in another, and if it is the same variable then those locations in code are part of the graph showing the data-flow of the program.

So it is my intuition too that graphs are the fundamental representation of reality as we see it, the parts of reality and the connections and relations between those parts.

Knowledge is knowledge about containers and parts and how those related to each other. We can describe it in natural language or perhaps in algebra but really the picture we are expressing in any language is that of wholes and parts and their connections.

This is my intuition about it I don't know if such a general intuition is much good in actual theory formation. Theories are graphs I think.

galaxyLogic3• 7 years ago

I guess what I'm after here is Semantic Web, which was supposed to be able to represent all knowledge, and would be a graph.

fyp• 7 years ago

Since there's a correspondence between proofs and programs (Curry-Howard), maybe you're just looking for a graph-based visual programming language.

But given the failure (of adoption) of dataflow programming languages I don't think there will really be that much of a jump in understanding just by visualizing stuff as graphs.

visarga• 7 years ago

I have this naive intuition that graphs are a universal encoding scheme

I have thought of the same thing in the domain of neural nets. You want to understand an image? Build a graph of the objects and their relations. You want to understand a text? Turn it into a graph. Need to simulate the environment? You can represent it as a graph. Want to mine a huge database of common facts? Represent it as a graph (database of triplets subject-relation-object). Even the execution of a program could be better conceptualised as a graph.

Graphs seem like an universal format. Graph Neural Nets have been a blooming subfield lately, ever since Thomas Kipf's paper [1] which in three years has accumulated over 2300 citations.

The main difference between classical neural nets and GNNs is that GNNs have permutation invariance. You can rename the nodes of the graph and still have the same graph.

Previously neural nets only had translation invariance (CNN) and time invariance (RNNs). Graphs would solve the combinatorial explosion by virtue of their permutation invariance - a problem which limits previous models of neural nets. Instead of learning all possible combinations of input objects you learn the pairwise relations, then you can generalise those relations to new configurations.

On a side note, the insanely popular transformer architecture (based on soft attention) is a kind of implicit graph, where the connectivity is evaluated based on the dot product similarity of the key and query objects.

[1] https://scholar.google.ro/scholar?hl=en&as_sdt=0%2C5&q=kipf+...

xorand3• 7 years ago

Thanks for the link to Kipf's paper, I was not aware of it. I'm a mathematician working on graph rewrite systems, I would like to explore in the NN direction, in case you're interested contact me.

slifin• 7 years ago

I'm definitely coming around to graph first systems like Fulcro and Datomic, with Datahike being a great Datomic introduction

ratmice• 7 years ago

Similarly all propositions in modal logic can be expressed as kripke models, which have quite the correlation to graphs

As good a reference as I could find... https://rkirsling.github.io/modallogic/

xorand3• 7 years ago

I think, similarly, that there is an underground graphical rewriting formalism for all of mathematics, category theory included. Category theory is somehow a step towards it, but we don't keep categories in our heads, only the graph reducer.

galaxyLogic3• 7 years ago

Another approach for visualizing lambda calculus is presented in https://ycombinator.chibicode.com/functional-programming-emo... .

It looks quite different and I wonder which one is better? Or are they the same really?

tromp2• 7 years ago

Yet another is my Lambda Diagrams [1], an obsolete link for which appears at the bottom of the article.

[1] https://tromp.github.io/cl/diagrams.htm

xorand3• 7 years ago

A js lambda to graphs parser and reducer https://mbuliga.github.io/quinegraphs/lambda2mol.html

To Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus (1996)

Related Stories