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Propositions as Types (2014) [pdf]

108 points by nill0 1y ago

deleted 1y ago
[deleted]
elikoga 1y ago
In my opinion it's a tragedy there are so few resources in using "Propositions as Types"/"Curry–Howard correspondence"[0] in didactics in tandem with teaching functional programming to teach structured proof-writing. Many students do not feel comfortable with proof-writing and cannot dispatch/discharge implications or quantifications correctly when writing proofs and I believe that a structured approach using the Curry-Howard correspondence could help. [0]: https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...
- mietek 1y ago
  One of the best modern resources is "Programming language foundations in Agda", co-authored by Wen Kokke and the same Philip Wadler, and used for teaching at multiple universities. https://plfa.github.io/ https://plfa.github.io/
- jerf 1y ago
  While I am first in line to say that programming is math whether you like it or not, it is certainly the case that few programmers are using that sort of math in their heads when they are programming. I suspect that if, oh, about 60-70 years of trying to change that has not had any headway that there is little reason to suspect that's going to change in the next 30. But you can sort of backdoor the idea with something like this: https://jerf.org/iri/post/2025/fp_lessons_types_as_assertions/ https://jerf.org/iri/post/2025/fp_lessons_types_as_assertion... I won't claim it is exactly the same, but it is certainly similar. Tell a programmer, "Hey, you can build a type that carries certain guarantees with it, so that the rest of the program doesn't need to be constantly validating it", a problem any 5+ year dev ought to be very familiar with, and I think you make more progress in the direction of this notion than a paper full of logic diagrams in a notation even most computer scientist undergrads will never encounter. (I'm not even certain if I directly encountered that in my Master's degree. It's been a while. I think I did, but I think it was only one class, and not a huge part of it. I'm not sure because I know I did more with it on my own, so I can't remember if I was formally taught it or if I picked it up entirely myself.)
  - skybrian 1y ago
    While I generally agree with defining new types to assert that validation has been done, I think your blog post could have explained more about what kinds of validation are practical to do. For example: > Address that represents a “street address” that has been validated by your street address to exist What does it even mean to verify that a street address exists? Verifying real-world relationships is complex and error-prone. I’m reminded of the genre of articles starting with “Falsehoods programmers believe about names” [1]. In practice, the rules that can be enforced are often imposed by the system itself. It simply doesn’t accept data that isn’t in correct format or isn’t consistent with other data it already has. And we need to be cautious about what harm might be caused by these rejections. Having the validation logic in one place will certainly help when fixing mistakes, but then what do you do about data that’s already been accepted, but is no longer “valid?” This sort of thing makes long-running systems like databases hard to maintain. [1] https://www.kalzumeus.com/2010/06/17/falsehoods-programmers-believe-about-names/ https://www.kalzumeus.com/2010/06/17/falsehoods-programmers-...
    - fmbb 1y ago
      Far too many programmers forget that time passes.
      - daxfohl 1y ago
        Yeah, the issue is that proofs are harder than people think, even for trivial things (try a few easy leetcode problems in Lean with a proof of correctness), and less useful than people think, especially when you get past low level algorithms and into domain logic (your point exactly). They also don't serialize well, so a database or API call with a "proof" field would be susceptible to fudging, or the definition could change between deployments. They're also easy to make incompatible: one library requires bounds proofs for signed ints, but your app uses unsigned ints, so you have to either rewrite your app or rewrite the library, or cast, in which your type system has to handle like a "checked exception" and propagate that possibility throughout your whole type domain and app logic. I'm pretty convinced there's a good reason that while "propositions as types" is cool in theory, it's unlikely we'll ever see it catch on in practice.
- mrkeen 1y ago
  This may not go over as well as you'd think. I took Haskell in the same uni course with FSAs, Turing machines, Lambda calculus, natural deduction, Hoare logic and structural induction. So I was exposed to the "base case & step case" of structural induction at the same time as learning about it in Haskell. And for whatever reason it didn't leave a good impression. Implementing it formally was harder and more error-prone than shooting from the hip in an IDE. What was the point!? Now I smash out Haskell code daily as the quickest way to end up with little binaries that just-work-if-they-compile. It took me a while to realise that the upside of all this formality/mathiness is that other people did the proofs so that you don't need to. I get to take for granted that a boolean is true or false (and not some brilliant third value!) and that (map f . map g) is (map (f . g)).
  - AnimalMuppet 1y ago
    “A man is rich in proportion to the number of things which he can afford to let alone.” ― Henry David Thoreau, Walden or, Life in the Woods If I have to do the proof, then as you say, it's probably harder and (in many cases) more error prone than if I didn't use a proof. If the language can do it for me, and I get the lack of errors without having to do the work (and without the chance of me making the mistakes)? Yeah, now we're talking. (Yes, I am aware that I still have to design the types to fit what the program is doing.) If a tool introduces complexity, it has to make up for it by eliminating at least that much complexity somewhere else. If it doesn't, the tool isn't worth using.
  - edef 1y ago
    > I get to take for granted that a boolean is true or false (and not some brilliant third value!) [True, False, undefined] :: [Bool]
    - immibis 1y ago
      The traditional third value is actually FileNotFound: https://thedailywtf.com/articles/what_is_truth_0x3f_ https://thedailywtf.com/articles/what_is_truth_0x3f_ but in Haskell, yes, it's undefined. Which isn't a real value! For example, infinite loops are undefined. Theorists like to call it a value of every type, but in practical terms, it's more like a computation that never produces a value. The builtin "undefined" can be written as an infinite loop ("undefined = undefined") and many other pure infinite loops can also act as undefined values. The runtime is able to catch some and crash instead of hanging, but not others.
- talkingtab 1y ago
  The connection to intuitionism (see https://en.wikipedia.org/wiki/Intuitionism https://en.wikipedia.org/wiki/Intuitionism) gives this kind of thinking much broader appeal and application. It seems to me we live in a time so dominated by analytical thinking that we completely ignore other useful and effective modes.
  - johnnyjeans 1y ago
    Intuitionism in this context just means the proofs have to be constructive. (no proof-by-contradiction, or other half-assed academic hacks)
    - AnimalMuppet 1y ago
      Why do you say that proof-by-contradiction is a "half-assed academic hack"? In particular, if someone isn't already an intuitionist or constructivist, what reason can you give that would be valid in their frame of reference?
      - johnnyjeans 1y ago
        Because they explain nothing and aren't useful for building new insights. It's not a direct verification of the existence or lack-of-existence of a thing with given properties. It relies on rules lawyering in a very specific logic to say something with basically no substance. Allowing it causes more problems for automated proofs than they solve, so long as your domain only deals with finites (which all real-world domains do exclusively.) They're also generally much easier to create than constructive proofs, which makes them a convenient go-to for lazy academics.
        ndriscoll 1y ago
        Proof by contradiction is fine for lack-of-existence (indeed, ¬P is defined in type theory as P -> false). I think also if I got my types right, you can do def eliminateTripleNot(f: ¬¬¬P): ¬P = {p: P => f({g: ¬P => g(p)})} For a constructive proof of ¬¬¬P => ¬P? So it's really just ¬¬P => P that causes trouble. It's not clear to me though whether LEM is actually okay in the "fast and loose reasoning is morally correct" sense (i.e. if it's okay as long as you don't cheat and do something like use a non-terminating loop or exception to make your ¬¬P) though? Are there cases where you didn't "obviously" cheat where it's "wrong" to use it? In some sense, I see ¬¬P => P as a "cast" from an opaque function type to specifically be a closure {f => f(p)} for some p: P.
- ndriscoll 1y ago
  I've been thinking the same thing and mentioned it the other day[0]. When I was learning proofs, I saw people struggle with the idea of needing to choose a "generic" element to prove a forall statement, for example. I suspect it might make more sense if you taught things in terms of needing to write a function x=>P(x). I think in some cases, thinking in terms of programming might also change how we think about structuring things. e.g. define a data structure for a "point with error tolerance" (x, epsilon), then continuity says given a toleranced-point y* at f(x), I can find a toleranced-point x* at x such that f(x*) is "compatible" (equal at the base point and within tolerance) with y*. This factoring lets you avoid shocking new students with quantifier soup. Likewise the chain rule is just straightforward composition when you define a "derivative at a point" data structure Df((x,m))=(f(x), m*f'(x)). This is not at all new, but I suppose it's currently a lot to ask students to learn proof mechanics while also unwinding multiple layers of definitions for abstractions. Computers can help do the unwinding (or lifting) automatically to make it easier to make small "quality of life" definitions that otherwise wouldn't be hugely useful when pen-and-paper-proofs would always be unwinding them anyway. Basically, math education could look a lot like software engineering education. The concerns and solution patterns are basically the same. e.g. typeclasses are pretty much how mathematics does polymorphism, and are probably usually the right way to do it in programming too. [0] https://news.ycombinator.com/item?id=43875101 https://news.ycombinator.com/item?id=43875101
  - immibis 1y ago
    I don't think the C-H correspondence is necessary for this. It would be a useful way to think even if the C-H correspondence were false.
- js8 1y ago
  I would recommend https://lean-lang.org/theorem_proving_in_lean4/ https://lean-lang.org/theorem_proving_in_lean4/, especially the first few chapters, if you're willing to use Lean.
- ngruhn 1y ago
  100% agree. I did not understand induction until I learned Coq. It really shows how mechanical proving can be.
- boxfire 1y ago
  There's a book that's explicitly about this, "Program = Proof", and though it's not beginner and needs maybe a light version for earlier learners, is an excellent example.
marcusb 1y ago
The author has given a few talks built around the same concept: https://www.youtube.com/watch?v=IOiZatlZtGU https://www.youtube.com/watch?v=IOiZatlZtGU
indyjo 1y ago
As much as the concept blew me away when I first heard of it, I can't shake the feeling that the Curry-Howard correspondence is somehow mis-marketed as something that would immediately cater to programmers. The idea to encode propositions into types sounds enticing for programmers, because there are indeed a lot of cases where something needs to be conveyed that can't be conveyed using type systems (or other features) of common programming languages. Examples include the famous "the caller of this function guarantees that the argument is a non-empty list" but also "the caller guarantees that the argument object has been properly locked against concurrent access before using the function". However, in my experience, the community is more interested in mathematics than in programming and I don't know if anybody is really working to provide propositions-as-types to mainstream programmers. This is certainly because it is hard enough to prove soundness of a strict functional programming language as Agda or Rocq, much more for anything imperative, stateful, "real-world", non-strict, non-pure, concurrent, ill-defined, you name it. So, for me the promise of "showing programmers that what they do is actually mathematics" is not really kept as long as the definition of "programmer" is so narrow that it excludes the vast majority of people who define themselves as programmers. What I'm trying to say: I wish there were more efforts to bring more powerful formal methods, especially as powerful as dependent types, to existing mainstream programming languages where they could be useful in an industrial context. Or at least try to come up with new programming languages that are more pragmatic and do not force the programmer into some narrow paradigm. Am I wrong? I hope so :)
- ndriscoll 1y ago
  Programmers regularly do this stuff under various names/phrases. "Make illegal states unrepresentable", "parse, don't validate", "resource acquisition is initialization", etc. It's all about encoding the fact that your structure isn't just data, but also represents some proof of provenance that guarantees some property. Scala is an industrial language that gives more tools than you might usually find for this while also letting you be pragmatic and integrate with Java's massive ecosystem (though you quickly learn you'd rather not because Java code tends to not do a good job of leveraging types in a sane way).
- AnimalMuppet 1y ago
  No, you're not wrong. I mean, in some circles, it's a battle to get programmers to use types at all. And, while not every proposition can currently be usefully encoded in a type, every type currently encodes a proposition. So if we can get the people who don't use types to start using them, that's probably the lowest-hanging fruit. From there, every step to improve the expressiveness of types allows you to encode more within the type system. For example, one could think about encoding that non-empty requirement in C++ or Java collection types. It would be nontrivial - a number of things would need adjusting - but you could think about doing it. (Or rather, it would be nontrivial to do with static types. You could more easily make it throw if the condition was not satisfied.) Your "properly locked" example is much harder. It would require telling the type system what the definition of "properly locked" is. Even at that, I can't see any way to do it as a static type. And that's a bummer, because I far prefer static types. I prefer my proofs to fail at compile time. But my main point is, "proving" is not binary. Every incremental advance in what types can check proves more things, and therefore leaves the programmers having to prove less in their heads.
aatd86 1y ago
Could have written further into set theory and set theoretic interpretation of types. A proposition being a set intension.
js8 1y ago
I really recommend [56] M. H. Sørensen and P. Urzyczyn. Lectures on the Curry-Howard isomorphism. Elsevier, 2006 on this topic. I found it online. It is very self-contained and explains some interesting topics such as relation of intuitionist and classical logic.
immibis 1y ago
The Curry-Howard correspondence seems like one of those things that's weird and unexpected but not actually very useful? Most of the types that correspond to propositions, and programs that correspond to proofs, aren't of much utility in actual programming. And most of the useful programs and their types don't correspond to interesting proofs and propositions. The paper relates it to other connections between fields, such as Cartesian coordinates linking geometry and algebra. This allows you to bring the power of algebra into geometric problems and geometric intuition into algebra problems. But does the Curry-Howard correspondence bring similar powers? -- This kind of rearrangement without really changing anything reminds me of the equivalence between first-order logic terms and sets. Consider [P] to be the set of models where statement P is true - then [A∧B] = [A]∩[B], [A∨B] = [A]∪[B] and so on. But this doesn't lead to any new insights. You basically just wrote the same thing with different symbols. In some contexts (proving soundness/completeness) it can allow you to step down one turtle, but you still have aleph-null turtles below you so that doesn't seem that useful either.
- colbyn 1y ago
  To play devils advocate, I’m not a mathematician but I’ve heard that the topics that lead to modern cryptography was once considered absolutely useless. They say for centuries, number theory (especially areas like prime numbers, modular arithmetic, and whatnot) was seen as the peak of “pure” math with no real world utility. Personally, im all for static analysis and formal verification in software, particularly the kind where properties can be automatically verified by a computer and to my understanding this field is the on bleeding edge of what’s possible. From a big picture perspective, our world is dependent on software, lives can be at risk when software fails, so for this reason I think its worthwhile to explore ideas that may one day lead to inherently more robust software even if it’s comercial utility isn’t clear.
  - immibis 1y ago
    Perhaps it leads to something down the line, but for now it's more useful to proving (still not very) than to programming.
emorning3 1y ago
I dont want to prove that my code is correct. I would prefer to create specifications in logic and have the code derived for me. Other than something like Prolog, are there tools for working this way?
- Tainnor 1y ago
  This is impossible to solve in the general case due to Gödel's first incompleteness theorem.
  - YeGoblynQueenne 1y ago
    A compiler is a program synthesizer that takes in a specification in a high-level language and produces an equivalent program in machine code. Given that observation, can you explain what you mean that this is impossible to solve?
    - Tainnor 1y ago
      OP specifically asked for a program that would derive code from logical specifications, not a run-of-the-mill compiler where you have to write the code yourself (albeit on a higher level). If your specification logic is strong enough to enclude basic addition and multiplication (and if it's not, it's probably not very useful), it will be affected by incompleteness. So you can write down the Gödel sentence corresponding to your program synthesiser's inference rule and it won't be able to find either a proof (i.e. an implementation) or a disproof (i.e. failing explicitly), so it will have to loop forever. Another way to see it is that the set of valid theorems in FOL is uncomputable. To take a concrete example, if you write down the specification for "find the smallest number that is a counterexample to the Goldbach conjecture", finding an implementation for it would be equivalent to answering the conjecture in the negative while getting an error message would mean that the conjecture is true, but obviously nobody has solved it yet.
      - YeGoblynQueenne 1y ago
        I'm sorry but I think that when I replied to your comment it didn't have the "in the general case". I might have missed that, but I thought you meant it's impossible to do it at all. Btw: >> OP specifically asked for a program that would derive code from logical specifications, not a run-of-the-mill compiler where you have to write the code yourself (albeit on a higher level). There's nothing "run-of-the-mill" about compilers and if your high-level language is e.g. FOL, then a compiler that transforms it to low-level machine code is, still, a program synthesizer, and v.v.
gitroom 1y ago
I feel like every time I try to do super formal proofs for code, it ends up way more brutal than I expect and slower than just fixing stuff as I go. Still love types catching my mess-ups for me though.
- cbdumas 1y ago
  This is the eternal tradeoff of testing, brought to its logical (no pun intended) conclusion. Testing (and formal verification) don't get you anything for free: writing detailed specs or tests is at least as difficult as writing a correct implementation. In fact given a sufficiently detailed spec, a program conforming to the spec can be derived automatically.
devlovstad 1y ago
Slightly OT: I'm a master's student in computer science who focuses mostly on machine learning. Still, the best course I've ever taken was one on semantics and types, presenting many of the ideas in this article. Learning proof-writing using natural deduction with a ruthlessly rigorous teacher has made me much more precise when I write proofs in general, and learning about theory of computation and logic has given me a good mental model of how my programs execute. While the course is an elective mostly focused on students interested in programming languages, I think all computer scientists can benefit from taking such a course. In a time where everyone wants to do AI, the course only had around 12 students out of a class of maybe 200 students. Even more OT: Phil Wadler gave a talk at the programming language section of my university not too long ago, which I was much excited to see. Sadly, he chose a vague pop-sciency talk on AI which felt quite a bit outside his expertise.
- somethingsome 1y ago
  Hey, do you have interesting slides/homeworks to share? I would be interested in taking a look
  - devlovstad 1y ago
    The course did not use slides but instead wrote everything on a whiteboard. The lecture notes are not public afaik. The lecturer did suggest the following supplementary material: - Michael Huth and Mark Ryan. Logic in Computer Science: Modelling and Reasoning about systems (2nd ed.). Cambridge University Press 2004. (Mainly chapters 1, 2) - Glynn Winskel. The Formal Semantics of Programming Languages: An Introduction, MIT Press 1993. (Mainly chapters 1, 2, 3, 6, 11) - Benjamin C. Pierce. Types and Programming Languages. MIT Press 2002. (Mainly chapters 5, 8, 9, 11, 12)
dang 1y ago
Related (I think) - others? Propositions as Types: Explained (and Debunked) - https://news.ycombinator.com/item?id=38894706 https://news.ycombinator.com/item?id=38894706 - Jan 2024 (1 comment) Propositions as Types - https://news.ycombinator.com/item?id=17947379 https://news.ycombinator.com/item?id=17947379 - Sept 2018 (1 comment) Propositions as Types (2014) [pdf] - https://news.ycombinator.com/item?id=10553766 https://news.ycombinator.com/item?id=10553766 - Nov 2015 (6 comments) On Propositions as Types [video] - https://news.ycombinator.com/item?id=10154349 https://news.ycombinator.com/item?id=10154349 - Sept 2015 (3 comments)

108 points by nill0 1y ago

deleted 1y ago

[deleted]

In my opinion it's a tragedy there are so few resources in using "Propositions as Types"/"Curry–Howard correspondence"[0] in didactics in tandem with teaching functional programming to teach structured proof-writing. Many students do not feel comfortable with proof-writing and cannot dispatch/discharge implications or quantifications correctly when writing proofs and I believe that a structured approach using the Curry-Howard correspondence could help. [0]: https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...

mietek 1y ago
One of the best modern resources is "Programming language foundations in Agda", co-authored by Wen Kokke and the same Philip Wadler, and used for teaching at multiple universities. https://plfa.github.io/ https://plfa.github.io/
jerf 1y ago
While I am first in line to say that programming is math whether you like it or not, it is certainly the case that few programmers are using that sort of math in their heads when they are programming. I suspect that if, oh, about 60-70 years of trying to change that has not had any headway that there is little reason to suspect that's going to change in the next 30. But you can sort of backdoor the idea with something like this: https://jerf.org/iri/post/2025/fp_lessons_types_as_assertions/ https://jerf.org/iri/post/2025/fp_lessons_types_as_assertion... I won't claim it is exactly the same, but it is certainly similar. Tell a programmer, "Hey, you can build a type that carries certain guarantees with it, so that the rest of the program doesn't need to be constantly validating it", a problem any 5+ year dev ought to be very familiar with, and I think you make more progress in the direction of this notion than a paper full of logic diagrams in a notation even most computer scientist undergrads will never encounter. (I'm not even certain if I directly encountered that in my Master's degree. It's been a while. I think I did, but I think it was only one class, and not a huge part of it. I'm not sure because I know I did more with it on my own, so I can't remember if I was formally taught it or if I picked it up entirely myself.)
- skybrian 1y ago
  While I generally agree with defining new types to assert that validation has been done, I think your blog post could have explained more about what kinds of validation are practical to do. For example: > Address that represents a “street address” that has been validated by your street address to exist What does it even mean to verify that a street address exists? Verifying real-world relationships is complex and error-prone. I’m reminded of the genre of articles starting with “Falsehoods programmers believe about names” [1]. In practice, the rules that can be enforced are often imposed by the system itself. It simply doesn’t accept data that isn’t in correct format or isn’t consistent with other data it already has. And we need to be cautious about what harm might be caused by these rejections. Having the validation logic in one place will certainly help when fixing mistakes, but then what do you do about data that’s already been accepted, but is no longer “valid?” This sort of thing makes long-running systems like databases hard to maintain. [1] https://www.kalzumeus.com/2010/06/17/falsehoods-programmers-believe-about-names/ https://www.kalzumeus.com/2010/06/17/falsehoods-programmers-...
  - fmbb 1y ago
    Far too many programmers forget that time passes.
    - daxfohl 1y ago
      Yeah, the issue is that proofs are harder than people think, even for trivial things (try a few easy leetcode problems in Lean with a proof of correctness), and less useful than people think, especially when you get past low level algorithms and into domain logic (your point exactly). They also don't serialize well, so a database or API call with a "proof" field would be susceptible to fudging, or the definition could change between deployments. They're also easy to make incompatible: one library requires bounds proofs for signed ints, but your app uses unsigned ints, so you have to either rewrite your app or rewrite the library, or cast, in which your type system has to handle like a "checked exception" and propagate that possibility throughout your whole type domain and app logic. I'm pretty convinced there's a good reason that while "propositions as types" is cool in theory, it's unlikely we'll ever see it catch on in practice.
mrkeen 1y ago
This may not go over as well as you'd think. I took Haskell in the same uni course with FSAs, Turing machines, Lambda calculus, natural deduction, Hoare logic and structural induction. So I was exposed to the "base case & step case" of structural induction at the same time as learning about it in Haskell. And for whatever reason it didn't leave a good impression. Implementing it formally was harder and more error-prone than shooting from the hip in an IDE. What was the point!? Now I smash out Haskell code daily as the quickest way to end up with little binaries that just-work-if-they-compile. It took me a while to realise that the upside of all this formality/mathiness is that other people did the proofs so that you don't need to. I get to take for granted that a boolean is true or false (and not some brilliant third value!) and that (map f . map g) is (map (f . g)).
- AnimalMuppet 1y ago
  “A man is rich in proportion to the number of things which he can afford to let alone.” ― Henry David Thoreau, Walden or, Life in the Woods If I have to do the proof, then as you say, it's probably harder and (in many cases) more error prone than if I didn't use a proof. If the language can do it for me, and I get the lack of errors without having to do the work (and without the chance of me making the mistakes)? Yeah, now we're talking. (Yes, I am aware that I still have to design the types to fit what the program is doing.) If a tool introduces complexity, it has to make up for it by eliminating at least that much complexity somewhere else. If it doesn't, the tool isn't worth using.
- edef 1y ago
  > I get to take for granted that a boolean is true or false (and not some brilliant third value!) [True, False, undefined] :: [Bool]
  - immibis 1y ago
    The traditional third value is actually FileNotFound: https://thedailywtf.com/articles/what_is_truth_0x3f_ https://thedailywtf.com/articles/what_is_truth_0x3f_ but in Haskell, yes, it's undefined. Which isn't a real value! For example, infinite loops are undefined. Theorists like to call it a value of every type, but in practical terms, it's more like a computation that never produces a value. The builtin "undefined" can be written as an infinite loop ("undefined = undefined") and many other pure infinite loops can also act as undefined values. The runtime is able to catch some and crash instead of hanging, but not others.
talkingtab 1y ago
The connection to intuitionism (see https://en.wikipedia.org/wiki/Intuitionism https://en.wikipedia.org/wiki/Intuitionism) gives this kind of thinking much broader appeal and application. It seems to me we live in a time so dominated by analytical thinking that we completely ignore other useful and effective modes.
- johnnyjeans 1y ago
  Intuitionism in this context just means the proofs have to be constructive. (no proof-by-contradiction, or other half-assed academic hacks)
  - AnimalMuppet 1y ago
    Why do you say that proof-by-contradiction is a "half-assed academic hack"? In particular, if someone isn't already an intuitionist or constructivist, what reason can you give that would be valid in their frame of reference?
    - johnnyjeans 1y ago
      Because they explain nothing and aren't useful for building new insights. It's not a direct verification of the existence or lack-of-existence of a thing with given properties. It relies on rules lawyering in a very specific logic to say something with basically no substance. Allowing it causes more problems for automated proofs than they solve, so long as your domain only deals with finites (which all real-world domains do exclusively.) They're also generally much easier to create than constructive proofs, which makes them a convenient go-to for lazy academics.
      - ndriscoll 1y ago
        Proof by contradiction is fine for lack-of-existence (indeed, ¬P is defined in type theory as P -> false). I think also if I got my types right, you can do def eliminateTripleNot(f: ¬¬¬P): ¬P = {p: P => f({g: ¬P => g(p)})} For a constructive proof of ¬¬¬P => ¬P? So it's really just ¬¬P => P that causes trouble. It's not clear to me though whether LEM is actually okay in the "fast and loose reasoning is morally correct" sense (i.e. if it's okay as long as you don't cheat and do something like use a non-terminating loop or exception to make your ¬¬P) though? Are there cases where you didn't "obviously" cheat where it's "wrong" to use it? In some sense, I see ¬¬P => P as a "cast" from an opaque function type to specifically be a closure {f => f(p)} for some p: P.
ndriscoll 1y ago
I've been thinking the same thing and mentioned it the other day[0]. When I was learning proofs, I saw people struggle with the idea of needing to choose a "generic" element to prove a forall statement, for example. I suspect it might make more sense if you taught things in terms of needing to write a function x=>P(x). I think in some cases, thinking in terms of programming might also change how we think about structuring things. e.g. define a data structure for a "point with error tolerance" (x, epsilon), then continuity says given a toleranced-point y* at f(x), I can find a toleranced-point x* at x such that f(x*) is "compatible" (equal at the base point and within tolerance) with y*. This factoring lets you avoid shocking new students with quantifier soup. Likewise the chain rule is just straightforward composition when you define a "derivative at a point" data structure Df((x,m))=(f(x), m*f'(x)). This is not at all new, but I suppose it's currently a lot to ask students to learn proof mechanics while also unwinding multiple layers of definitions for abstractions. Computers can help do the unwinding (or lifting) automatically to make it easier to make small "quality of life" definitions that otherwise wouldn't be hugely useful when pen-and-paper-proofs would always be unwinding them anyway. Basically, math education could look a lot like software engineering education. The concerns and solution patterns are basically the same. e.g. typeclasses are pretty much how mathematics does polymorphism, and are probably usually the right way to do it in programming too. [0] https://news.ycombinator.com/item?id=43875101 https://news.ycombinator.com/item?id=43875101
- immibis 1y ago
  I don't think the C-H correspondence is necessary for this. It would be a useful way to think even if the C-H correspondence were false.
js8 1y ago
I would recommend https://lean-lang.org/theorem_proving_in_lean4/ https://lean-lang.org/theorem_proving_in_lean4/, especially the first few chapters, if you're willing to use Lean.
ngruhn 1y ago
100% agree. I did not understand induction until I learned Coq. It really shows how mechanical proving can be.
boxfire 1y ago
There's a book that's explicitly about this, "Program = Proof", and though it's not beginner and needs maybe a light version for earlier learners, is an excellent example.

marcusb 1y ago

The author has given a few talks built around the same concept: https://www.youtube.com/watch?v=IOiZatlZtGU https://www.youtube.com/watch?v=IOiZatlZtGU

indyjo 1y ago

As much as the concept blew me away when I first heard of it, I can't shake the feeling that the Curry-Howard correspondence is somehow mis-marketed as something that would immediately cater to programmers. The idea to encode propositions into types sounds enticing for programmers, because there are indeed a lot of cases where something needs to be conveyed that can't be conveyed using type systems (or other features) of common programming languages. Examples include the famous "the caller of this function guarantees that the argument is a non-empty list" but also "the caller guarantees that the argument object has been properly locked against concurrent access before using the function". However, in my experience, the community is more interested in mathematics than in programming and I don't know if anybody is really working to provide propositions-as-types to mainstream programmers. This is certainly because it is hard enough to prove soundness of a strict functional programming language as Agda or Rocq, much more for anything imperative, stateful, "real-world", non-strict, non-pure, concurrent, ill-defined, you name it. So, for me the promise of "showing programmers that what they do is actually mathematics" is not really kept as long as the definition of "programmer" is so narrow that it excludes the vast majority of people who define themselves as programmers. What I'm trying to say: I wish there were more efforts to bring more powerful formal methods, especially as powerful as dependent types, to existing mainstream programming languages where they could be useful in an industrial context. Or at least try to come up with new programming languages that are more pragmatic and do not force the programmer into some narrow paradigm. Am I wrong? I hope so :)

ndriscoll 1y ago
Programmers regularly do this stuff under various names/phrases. "Make illegal states unrepresentable", "parse, don't validate", "resource acquisition is initialization", etc. It's all about encoding the fact that your structure isn't just data, but also represents some proof of provenance that guarantees some property. Scala is an industrial language that gives more tools than you might usually find for this while also letting you be pragmatic and integrate with Java's massive ecosystem (though you quickly learn you'd rather not because Java code tends to not do a good job of leveraging types in a sane way).
AnimalMuppet 1y ago
No, you're not wrong. I mean, in some circles, it's a battle to get programmers to use types at all. And, while not every proposition can currently be usefully encoded in a type, every type currently encodes a proposition. So if we can get the people who don't use types to start using them, that's probably the lowest-hanging fruit. From there, every step to improve the expressiveness of types allows you to encode more within the type system. For example, one could think about encoding that non-empty requirement in C++ or Java collection types. It would be nontrivial - a number of things would need adjusting - but you could think about doing it. (Or rather, it would be nontrivial to do with static types. You could more easily make it throw if the condition was not satisfied.) Your "properly locked" example is much harder. It would require telling the type system what the definition of "properly locked" is. Even at that, I can't see any way to do it as a static type. And that's a bummer, because I far prefer static types. I prefer my proofs to fail at compile time. But my main point is, "proving" is not binary. Every incremental advance in what types can check proves more things, and therefore leaves the programmers having to prove less in their heads.

aatd86 1y ago

Could have written further into set theory and set theoretic interpretation of types. A proposition being a set intension.

js8 1y ago

I really recommend [56] M. H. Sørensen and P. Urzyczyn. Lectures on the Curry-Howard isomorphism. Elsevier, 2006 on this topic. I found it online. It is very self-contained and explains some interesting topics such as relation of intuitionist and classical logic.

immibis 1y ago

The Curry-Howard correspondence seems like one of those things that's weird and unexpected but not actually very useful? Most of the types that correspond to propositions, and programs that correspond to proofs, aren't of much utility in actual programming. And most of the useful programs and their types don't correspond to interesting proofs and propositions. The paper relates it to other connections between fields, such as Cartesian coordinates linking geometry and algebra. This allows you to bring the power of algebra into geometric problems and geometric intuition into algebra problems. But does the Curry-Howard correspondence bring similar powers? -- This kind of rearrangement without really changing anything reminds me of the equivalence between first-order logic terms and sets. Consider [P] to be the set of models where statement P is true - then [A∧B] = [A]∩[B], [A∨B] = [A]∪[B] and so on. But this doesn't lead to any new insights. You basically just wrote the same thing with different symbols. In some contexts (proving soundness/completeness) it can allow you to step down one turtle, but you still have aleph-null turtles below you so that doesn't seem that useful either.

colbyn 1y ago
To play devils advocate, I’m not a mathematician but I’ve heard that the topics that lead to modern cryptography was once considered absolutely useless. They say for centuries, number theory (especially areas like prime numbers, modular arithmetic, and whatnot) was seen as the peak of “pure” math with no real world utility. Personally, im all for static analysis and formal verification in software, particularly the kind where properties can be automatically verified by a computer and to my understanding this field is the on bleeding edge of what’s possible. From a big picture perspective, our world is dependent on software, lives can be at risk when software fails, so for this reason I think its worthwhile to explore ideas that may one day lead to inherently more robust software even if it’s comercial utility isn’t clear.
- immibis 1y ago
  Perhaps it leads to something down the line, but for now it's more useful to proving (still not very) than to programming.

emorning3 1y ago

I dont want to prove that my code is correct. I would prefer to create specifications in logic and have the code derived for me. Other than something like Prolog, are there tools for working this way?

Tainnor 1y ago
This is impossible to solve in the general case due to Gödel's first incompleteness theorem.
- YeGoblynQueenne 1y ago
  A compiler is a program synthesizer that takes in a specification in a high-level language and produces an equivalent program in machine code. Given that observation, can you explain what you mean that this is impossible to solve?
  - Tainnor 1y ago
    OP specifically asked for a program that would derive code from logical specifications, not a run-of-the-mill compiler where you have to write the code yourself (albeit on a higher level). If your specification logic is strong enough to enclude basic addition and multiplication (and if it's not, it's probably not very useful), it will be affected by incompleteness. So you can write down the Gödel sentence corresponding to your program synthesiser's inference rule and it won't be able to find either a proof (i.e. an implementation) or a disproof (i.e. failing explicitly), so it will have to loop forever. Another way to see it is that the set of valid theorems in FOL is uncomputable. To take a concrete example, if you write down the specification for "find the smallest number that is a counterexample to the Goldbach conjecture", finding an implementation for it would be equivalent to answering the conjecture in the negative while getting an error message would mean that the conjecture is true, but obviously nobody has solved it yet.
    - YeGoblynQueenne 1y ago
      I'm sorry but I think that when I replied to your comment it didn't have the "in the general case". I might have missed that, but I thought you meant it's impossible to do it at all. Btw: >> OP specifically asked for a program that would derive code from logical specifications, not a run-of-the-mill compiler where you have to write the code yourself (albeit on a higher level). There's nothing "run-of-the-mill" about compilers and if your high-level language is e.g. FOL, then a compiler that transforms it to low-level machine code is, still, a program synthesizer, and v.v.

gitroom 1y ago

I feel like every time I try to do super formal proofs for code, it ends up way more brutal than I expect and slower than just fixing stuff as I go. Still love types catching my mess-ups for me though.

cbdumas 1y ago
This is the eternal tradeoff of testing, brought to its logical (no pun intended) conclusion. Testing (and formal verification) don't get you anything for free: writing detailed specs or tests is at least as difficult as writing a correct implementation. In fact given a sufficiently detailed spec, a program conforming to the spec can be derived automatically.

devlovstad 1y ago

Slightly OT: I'm a master's student in computer science who focuses mostly on machine learning. Still, the best course I've ever taken was one on semantics and types, presenting many of the ideas in this article. Learning proof-writing using natural deduction with a ruthlessly rigorous teacher has made me much more precise when I write proofs in general, and learning about theory of computation and logic has given me a good mental model of how my programs execute. While the course is an elective mostly focused on students interested in programming languages, I think all computer scientists can benefit from taking such a course. In a time where everyone wants to do AI, the course only had around 12 students out of a class of maybe 200 students. Even more OT: Phil Wadler gave a talk at the programming language section of my university not too long ago, which I was much excited to see. Sadly, he chose a vague pop-sciency talk on AI which felt quite a bit outside his expertise.

somethingsome 1y ago
Hey, do you have interesting slides/homeworks to share? I would be interested in taking a look
- devlovstad 1y ago
  The course did not use slides but instead wrote everything on a whiteboard. The lecture notes are not public afaik. The lecturer did suggest the following supplementary material: - Michael Huth and Mark Ryan. Logic in Computer Science: Modelling and Reasoning about systems (2nd ed.). Cambridge University Press 2004. (Mainly chapters 1, 2) - Glynn Winskel. The Formal Semantics of Programming Languages: An Introduction, MIT Press 1993. (Mainly chapters 1, 2, 3, 6, 11) - Benjamin C. Pierce. Types and Programming Languages. MIT Press 2002. (Mainly chapters 5, 8, 9, 11, 12)

dang 1y ago

Related (I think) - others? Propositions as Types: Explained (and Debunked) - https://news.ycombinator.com/item?id=38894706 https://news.ycombinator.com/item?id=38894706 - Jan 2024 (1 comment) Propositions as Types - https://news.ycombinator.com/item?id=17947379 https://news.ycombinator.com/item?id=17947379 - Sept 2018 (1 comment) Propositions as Types (2014) [pdf] - https://news.ycombinator.com/item?id=10553766 https://news.ycombinator.com/item?id=10553766 - Nov 2015 (6 comments) On Propositions as Types [video] - https://news.ycombinator.com/item?id=10154349 https://news.ycombinator.com/item?id=10154349 - Sept 2015 (3 comments)