Church's λ-Calculus (2023) [pdf]

http://www.cs.cmu.edu/~rwh/pfpl/supplements/ulc.pdf

by jstrieb • 2 years ago

155 46 2 years ago

cs.cmu.edu

ashton314• 2 years ago

I've got Harper's Practical Foundations for Programming Languages and it's a great book—he writes clearly and succinctly.

Knowing the lambda calculus helped me one time when I was working as a software engineer: I had just added functions to a little DSL interpreter that was going to make it easy to customize behavior of our product to different customers. (It wasn't ever going to be used by customers directly; it was primarily a tool for the internal team.) It was at this point that I realized we needed some kind of execution time-out: since we could encode functions, we could write down e.g. the Y combinator or the omega combinator and we could get non-terminating programs in this DSL.

Now I work as a programming languages researcher, so the lambda calculus has direct application to my day job.

Those curious might be interested in ISWIM, [1,2] which is an extension of the lambda calculus with an arbitrary set of operators. Like the lambda calculus, this is an abstract language. However, you can add numbers and other operators, and ISWIM-like languages are often used to illustrate new ideas in programming languages.

Syntax is incidental. Boil the syntax away from languages and reduce them to their distilled semantics to get out the essential differences—this is the kind of thing that the lambda calculus makes easy.

I highly recommend reading Landin's The Next 700 Programming Languages [2] as it is a great, short, clear read.

[1] https://en.wikipedia.org/wiki/ISWIM [2]: https://www.cs.cmu.edu/~crary/819-f09/Landin66.pdf

JonChesterfield• 2 years ago

It appears there are a few like the opening post, e.g. https://www.cs.cmu.edu/~rwh/courses/oplss/tlc-semeq.pdf. This suggests the book will be worth the time, thank you for mentioning it.

trueismywork• 2 years ago

Can you read Harpers book without knowing lambda calculus?

rkrzr• 2 years ago

You can learn the lambda calculus in a few hours.

If you read just the first page of the linked paper and work through a few examples, you will likely already know enough about it to read the book. It's really just like equational reasoning in mathematics.

an-allen• 2 years ago

When I took Fundamentals of Programming Languages 20 years ago - I nearly failed the class. Lambda calculus was simply too esoteric for me to appreciate and much less understand intuitively.

Fast forward 20 years, and I see the fundamentals of Alonzo Church’s system in every computation problem I encounter. It’s one of those concepts that age like wine. The only other concept I put on the same level is Shannons “informational entropy” and maybe Wolfram’s Ruliad.

_delirium• 2 years ago

It's kind of interesting that the history is in the order it is. I could completely imagine it being reversed: first, 1000 programming languages are invented, then later, in an attempt to put order to this madness, and understand whether some of them are in a fundamental sense equivalent to others or not, you invent minimalist languages like Turing machines or the lambda calculus, and start developing a theory of reductions.

Kind of odd that the Turing machines and lambda calculus predate almost all the others! I mean there are good reasons for it, especially if you try to put yourself in a 1930s mathematics mindset (which is why it actually happened that way), but it is, I'd submit, a bit surprising to learn from a 2020s perspective if you didn't already know it.

andoando6• 2 years ago

Interesting thought. Perhaps we'll discover a language that is in some way of higher order than Turing complete.

thesz• 2 years ago

Before LC there were combinators [1].

[1] https://en.wikipedia.org/wiki/Combinatory_logic

These are even more primitive.

dexwiz3• 2 years ago

There has to be some programmer rite of passage involving learning lambda calculus. I went through this a few years ago when something similar was posted. I learned the notation, marveled at its dual simplicity and completeness, and and did a few exercises. But I came out the other side none the wiser. I think I was looking for some great epiphany on the nature of computation. Alas, it eluded me in the end. It was fun but ultimately useless for me.

somat• 2 years ago

You will probably be wanting unlambda then. an implementation of the lambda calculus without the lambda forms.

http://www.madore.org/~david/programs/unlambda/

tombert6• 2 years ago

I love theory, and I also really like lambda calculus, but I feel like this sentiment applies to most theory, particularly stuff after undergrad.

I spent a not-insignificant amount of time learning how to do proofs with Isabelle. I learned a lot about inductive proofs, set theory, meta-logic, and challenged myself to prove a lot of the stuff I had previous taken for granted (e.g. proving that different sorts refine each other). I enjoyed it, and similarly was convinced that this was going to be some life-changing thing that changes my career trajectory and...

Nothing changed. No one in charge of companies gives a shit about theory. They all claim that they love theory, they claim that they are very research focused, they claim that they value all the time you spent learning this stuff, but in reality they really just want you to change the color of buttons, or change the format of dates, or add a field to a JSON. It sometimes feels like no software engineer but me actually wants to learn any math, and will refuse to touch anything even resembling it.

And I'm not picking on Isabelle here; I've had similar results trying to pitch TLA+ and Coq and Agda for some of the more error-prone parts of the codebase, with different sales-pitches, and without fail the managers will always say that they "will look into it", and promptly do absolutely nothing. The first two times a manager said that, I believed them, but after that I realized that they're just trying to shut me up and tell me "no" politely.

It was enough to depress me, and it still kind of does.

I still think learning stuff for fun is worth it, but I'd be lying if I told you if I knew why.

hnoyrnndd• 2 years ago

A friend of mine worked at Rockwell Collins and wrote proofs, and theorem provers, all day.

If you work on webshit like me you won’t get to use these skills much.

(On the other hand, recursing over the structure of JSON-like data often feels like 80% of the job, so I think the skills come into play at least a tiny bit).

andoando6• 2 years ago

But now you can write your own json parser

jvanderbot• 2 years ago

I use the deep theory experience a ton when designing software or systems. But I don't do an actual proof hardly ever. Being able to reason about a problem and intuit the extent of what's technologically possible to solve is insanely useful. But in most places that I work everyone around me can do that too, so we're constantly battling program managers and C suite guys / gals who just don't see why everything is so slow, difficult, un-scalable, unpredictable, etc. They want a stupid thing that works half the time, and they'll just turn the camera off the other half of the time.

eddd-ddde2• 2 years ago

I did a lot of math before I began learning programming. This was my experience:

Recursion? Oh you mean proof by induction?

Cryptography? Oh you mean number theory?

Neural networks? Oh you mean calculus?

Almost everything I learn while programming can be associated with some theory I learnt before for math.

mjh2539• 2 years ago

I used it in a graduate course on formal semantics (https://www.wiley.com/en-us/Semantics+in+Generative+Grammar-... after I graduated I wanted to learn how to program so I googled "lambda calculus programming language" and found Haskell. That was ten years ago. Though I rarely use Haskell anymore, the lambda calculus still holds a special place.

DonaldPShimoda• 2 years ago

The "utility" of the lambda calculus, in my opinion, comes down to Church's original choice of terminology. He did not call the things denoted by lambda "functions"; rather, they were abstractions. What makes the lambda calculus so cool is that it lets us see that abstraction is the root of all computation, at least through one perspective (as opposed to the Turing point of view). Learning the lambda calculus directly will allow you to appreciate functions ("abstractions") and use them more effectively; you can reason about their computation and composition.

andoando6• 2 years ago

I dont quite think of functions and abstractions as the same. Abstractions seem to do more with having a self referential property. Though I suppose that can be seen as the identity function f->f.

IIRC Church was heavily influenced by Turing and vice versa.

082349872349872• 2 years ago

Might the terminology have been due to early awareness of the cardinality problem associated with the abstractions (D->D) being a subset of the values (D)?

NotCamelCase• 2 years ago

Funny that you bring this up -- I feel exactly the same way and I even had more math background than average SW person.

Being on the engineering side of things without much knowledge/focus on programming languages theory and the mathematical nature of computation, one feels as if all the niceties we have in popular PLs are just laws of nature that we just had to discover, whereas in reality they too had to be derived, tried, improved first. But then you study something like this and expect to be able to finally connect the dots and yet, it's still not immediately obvious how we arrived here!

agumonkey• 2 years ago

The Y combinator (npi) was also a very mind opening exercise. Even recreating pairs was quite fun.

renonce2• 2 years ago

One invention that drew me to this topic was Binary Lambda Calculus, invented by John Tromp (“Tromp” as in Tromp-Taylor Rules). It’s just a direct binary encoding of a lambda calculus term, but it gives you a concrete and concise representation which is useful for evaluating the complexity of an expression. I encountered it when viewing https://codegolf.stackexchange.com/questions/6430/shortest-t... and found that this language was the one that could write the most precise representation of a very large number with the fewest bits possible. I later learned that Graham’s number could be encoded in 120 bits (maybe 3~4 less), much more concise than equivalent mathematical language, and I’ve since been drawn into the field of googology. It was fascinating.

tromp3• 2 years ago

That Stack Exchange thread shows you can exceed Graham's number with the 49 bit lambda term (λ 1 1) (λ 1 (1 (λ λ 1 2 (λ λ 2 (2 1))))), or graphically

    ┬─┬ ┬─┬──────────
    └─┤ │ │ ──┬──────
      │ │ │ ┬─┼──────
      │ │ │ └─┤ ┬─┬──
      │ │ │   │ ┼─┼─┬
      │ │ │   │ │ ├─┘
      │ │ │   │ ├─┘
      │ │ │   ├─┘
      │ │ ├───┘
      │ ├─┘      
      └─┘

Related: https://oeis.org/A333479

renonce2• 2 years ago

Yeah but it’s not representing a known number - I would rather say it is a proof that the busy beaver function for BLC at 49 bits is higher than Graham’s number. The fact that it represents known numbers concisely is more interesting to me.

Church's λ-Calculus (2023) [pdf]

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