6 ms·

See also: https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...

by chills 4y ago

peter_d_sherman 4y ago
Wowzer! That is quite the page... There is definitely something there! That page is decidedly worth reading and re-reading many times in the future... I think it boils down to the following: >"Curry–Howard correspondence [...] is the direct relationship between computer programs and mathematical proofs." And: >"If one abstracts on the peculiarities of either formalism, the following generalization arises: a proof is a program, and the formula it proves is the type for the program." In fact, I'm going to go for "full crackpot" here... If all computer programs are algorithms, and all mathematical proofs are algorithms, and all types are algorithms -- then a "grand unifying theory" between Computer Programs and Mathematics -- looks like this: It's all Algorithms. Algorithms on top of algorithms. You know, like turtles on top of turtles... This makes sense too, if we think about it... Algorithms are just series of steps, with given inputs and given outputs. That is no different than Computer Programs. And that is no different than Math... You can call this series of steps an Algorithm, you can call it a Function, you can call it y=f(x), you can call it a Type, you can call it a Computer Program, you can call it a Math Equation, you can call it a logical proposition, and you can use whatever notation and nomenclature you like -- but in the end, in the end, it all boils down to an Algorithm... A series of steps... Now, perhaps that series of steps -- uses other series of steps -- perhaps that Algorithm uses other Algorithms, perhaps that function uses other functions, perhaps that Computer Program uses other computer programs, perhaps that Math Equation uses other math equations, etc., etc. -- But in the end, in the end... It's all a series of rigorously defined steps... It's all an Algorithm... Or Algorithm consisting of other Algorithms... (recursively...) Patterns of steps -- inside of patterns of other steps (again, recursively...) Anyway, great page, definitely worth reading and re-reading!
- schoen 4y ago
  If you're excited about that, you might enjoy https://softwarefoundations.cis.upenn.edu/ https://softwarefoundations.cis.upenn.edu/ Officially you're supposed to download the book and then edit the exercise solutions into it with Emacs (or VSCode, I think), as you can then run the exercises to see if they type-check (i.e., if they're correct!). However, there's also a not-necessarily-up-to-date interactive in-browser version: https://jscoq.github.io/ext/sf/ https://jscoq.github.io/ext/sf/ I haven't used Idris, so I'd say it's quite possible that working through the Idris book is also just as fun and relevant for understanding the implications of the Curry-Howard correspondence.
- rbonvall 4y ago
  I think you will enjoy this talk: https://www.youtube.com/watch?v=IOiZatlZtGU https://www.youtube.com/watch?v=IOiZatlZtGU
  - peter_d_sherman 4y ago
    I did, immensely! Thank you for the excellent link!
- consilient 4y ago
  > all mathematical proofs are algorithms They're not. Only constructive proofs have corresponding programs (namely, the program that actually carries out the relevant construction). You can embed nonconstructive proofs indirectly via double-negation translation (computational counterpart: continuation-passing style), but equivalence of classical formulas isn't preserved. > all types are algorithms Definitely not the case. Types correspond to propositions, not their proofs.
  - peter_d_sherman 4y ago
    Let me rephrase then... >all mathematical proofs are algorithms All mathematical proofs consist of a series of steps. >all types are algorithms All types can be expressed as a series of steps -- with a given input -- and an output of True or False following those steps. True if the given input is a member of that Type. False if a given input is not a member of that Type. If the definition of an 'Algorithm' -- is 'a series of steps', then both mathematical proofs and types -- must be Algorithms... If we have any debate -- then we are debating the semantics of "what constitutes a step" -- what rigorously defines it -- and what steps may be permitted when... It may very well be that the Lambda Calculus is the best definition of what constitutes these steps -- but it may very well be that there is a different/better paradigm for looking at them (I don't know myself -- I am trying to determine this)... Here, you might like the following video (graciously submitted by rbonvall!) for an overview of some of the different possible paradigms that these steps -- might be considered in: https://www.youtube.com/watch?v=IOiZatlZtGU https://www.youtube.com/watch?v=IOiZatlZtGU
    - consilient 4y ago
      > All types can be expressed as a series of steps -- with a given input -- and an output of True or False following those steps. No, this is a fundamental misunderstanding of what types are. They're not, in general, subsets of some larger universe of values, and you can't have terms without types attached. (Of course there are "gradually typed" programming languages, but these are really languages with a top type a la `object`, subtyping, and generally a healthy dose of unsoundness).
- still_grokking 4y ago
  I think you're missed the elephant in the room. "Programs as proves" is only a thing in the context of mathematically pure languages. Almost all programming languages aren't pure. That's on the other hand's side why prove assistants are very unusable as programming languages; you can't do anything with them usually besides proving stuff. Running actually "useful" code is mostly not possible. Things like Haskell or Idris try to bridge both worlds, but this isn't straight forward. How to actually do anything in a pure programing language is still an open question. Monads are some kludge but not very practicable for most people… So to summarize: "Normal" programs don't correspond to proves in any way!
  - peter_d_sherman 4y ago
    You might wish to have a look at "Propositions as Types" by Philip Wadler (thanks rbonvall, for the link!) -- at 21:30 (1290 seconds) into the video: https://youtu.be/IOiZatlZtGU?t=1290 https://youtu.be/IOiZatlZtGU?t=1290 "Evaluation corresponds exactly to simplification of proofs..." (The rest of the video, both before and after this statement, contain more context...) That being said, I agree with you that languages which are used primarily for theorem proving (AKA, "proof assistants") -- are usually not as applicable to as broad a range of programming paradigms and problems as most general purpose computer programming languages are...
    - still_grokking 4y ago
      You might wish to re-watch the linked talk as you just restated your confusion (which was already the cause of a lot of down-votes in this thread). Wadler says there: "Evaluation [of simply typed lambda calculus!] corresponds exactly to simplification of proofs…" If you didn't get that context you actually missed the whole talk. You really need to understand: "Programs as proves" is only a thing in languages with strongly normalizing type-systems. This implies that the language is pure and does not contain general recursion. In a language with mutation (which is obviously not pure) you can destroy any "prove" by just writing to a variable, e.g. switching a single bit in the case of a boolean value. Terms can be obviously only proves if it's not possible to change a term ("prove") form true to false (or the other way around) at will! Once some term is determined as having some type it may not change any more. This rules out obviously any language with mutation or I/O. That's why you can't write applications in Agda, or mathematical proves in Haskell. (In the general case; you could do both with "tricks"; but those are indeed tricks).

See also: https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...

by chills 4y ago

See also: https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...

peter_d_sherman 4y ago

Wowzer! That is quite the page... There is definitely something there! That page is decidedly worth reading and re-reading many times in the future... I think it boils down to the following: >"Curry–Howard correspondence [...] is the direct relationship between computer programs and mathematical proofs." And: >"If one abstracts on the peculiarities of either formalism, the following generalization arises: a proof is a program, and the formula it proves is the type for the program." In fact, I'm going to go for "full crackpot" here... If all computer programs are algorithms, and all mathematical proofs are algorithms, and all types are algorithms -- then a "grand unifying theory" between Computer Programs and Mathematics -- looks like this: It's all Algorithms. Algorithms on top of algorithms. You know, like turtles on top of turtles... This makes sense too, if we think about it... Algorithms are just series of steps, with given inputs and given outputs. That is no different than Computer Programs. And that is no different than Math... You can call this series of steps an Algorithm, you can call it a Function, you can call it y=f(x), you can call it a Type, you can call it a Computer Program, you can call it a Math Equation, you can call it a logical proposition, and you can use whatever notation and nomenclature you like -- but in the end, in the end, it all boils down to an Algorithm... A series of steps... Now, perhaps that series of steps -- uses other series of steps -- perhaps that Algorithm uses other Algorithms, perhaps that function uses other functions, perhaps that Computer Program uses other computer programs, perhaps that Math Equation uses other math equations, etc., etc. -- But in the end, in the end... It's all a series of rigorously defined steps... It's all an Algorithm... Or Algorithm consisting of other Algorithms... (recursively...) Patterns of steps -- inside of patterns of other steps (again, recursively...) Anyway, great page, definitely worth reading and re-reading!

schoen 4y ago
If you're excited about that, you might enjoy https://softwarefoundations.cis.upenn.edu/ https://softwarefoundations.cis.upenn.edu/ Officially you're supposed to download the book and then edit the exercise solutions into it with Emacs (or VSCode, I think), as you can then run the exercises to see if they type-check (i.e., if they're correct!). However, there's also a not-necessarily-up-to-date interactive in-browser version: https://jscoq.github.io/ext/sf/ https://jscoq.github.io/ext/sf/ I haven't used Idris, so I'd say it's quite possible that working through the Idris book is also just as fun and relevant for understanding the implications of the Curry-Howard correspondence.
rbonvall 4y ago
I think you will enjoy this talk: https://www.youtube.com/watch?v=IOiZatlZtGU https://www.youtube.com/watch?v=IOiZatlZtGU
- peter_d_sherman 4y ago
  I did, immensely! Thank you for the excellent link!
consilient 4y ago
> all mathematical proofs are algorithms They're not. Only constructive proofs have corresponding programs (namely, the program that actually carries out the relevant construction). You can embed nonconstructive proofs indirectly via double-negation translation (computational counterpart: continuation-passing style), but equivalence of classical formulas isn't preserved. > all types are algorithms Definitely not the case. Types correspond to propositions, not their proofs.
- peter_d_sherman 4y ago
  Let me rephrase then... >all mathematical proofs are algorithms All mathematical proofs consist of a series of steps. >all types are algorithms All types can be expressed as a series of steps -- with a given input -- and an output of True or False following those steps. True if the given input is a member of that Type. False if a given input is not a member of that Type. If the definition of an 'Algorithm' -- is 'a series of steps', then both mathematical proofs and types -- must be Algorithms... If we have any debate -- then we are debating the semantics of "what constitutes a step" -- what rigorously defines it -- and what steps may be permitted when... It may very well be that the Lambda Calculus is the best definition of what constitutes these steps -- but it may very well be that there is a different/better paradigm for looking at them (I don't know myself -- I am trying to determine this)... Here, you might like the following video (graciously submitted by rbonvall!) for an overview of some of the different possible paradigms that these steps -- might be considered in: https://www.youtube.com/watch?v=IOiZatlZtGU https://www.youtube.com/watch?v=IOiZatlZtGU
  - consilient 4y ago
    > All types can be expressed as a series of steps -- with a given input -- and an output of True or False following those steps. No, this is a fundamental misunderstanding of what types are. They're not, in general, subsets of some larger universe of values, and you can't have terms without types attached. (Of course there are "gradually typed" programming languages, but these are really languages with a top type a la `object`, subtyping, and generally a healthy dose of unsoundness).
still_grokking 4y ago
I think you're missed the elephant in the room. "Programs as proves" is only a thing in the context of mathematically pure languages. Almost all programming languages aren't pure. That's on the other hand's side why prove assistants are very unusable as programming languages; you can't do anything with them usually besides proving stuff. Running actually "useful" code is mostly not possible. Things like Haskell or Idris try to bridge both worlds, but this isn't straight forward. How to actually do anything in a pure programing language is still an open question. Monads are some kludge but not very practicable for most people… So to summarize: "Normal" programs don't correspond to proves in any way!
- peter_d_sherman 4y ago
  You might wish to have a look at "Propositions as Types" by Philip Wadler (thanks rbonvall, for the link!) -- at 21:30 (1290 seconds) into the video: https://youtu.be/IOiZatlZtGU?t=1290 https://youtu.be/IOiZatlZtGU?t=1290 "Evaluation corresponds exactly to simplification of proofs..." (The rest of the video, both before and after this statement, contain more context...) That being said, I agree with you that languages which are used primarily for theorem proving (AKA, "proof assistants") -- are usually not as applicable to as broad a range of programming paradigms and problems as most general purpose computer programming languages are...
  - still_grokking 4y ago
    You might wish to re-watch the linked talk as you just restated your confusion (which was already the cause of a lot of down-votes in this thread). Wadler says there: "Evaluation [of simply typed lambda calculus!] corresponds exactly to simplification of proofs…" If you didn't get that context you actually missed the whole talk. You really need to understand: "Programs as proves" is only a thing in languages with strongly normalizing type-systems. This implies that the language is pure and does not contain general recursion. In a language with mutation (which is obviously not pure) you can destroy any "prove" by just writing to a variable, e.g. switching a single bit in the case of a boolean value. Terms can be obviously only proves if it's not possible to change a term ("prove") form true to false (or the other way around) at will! Once some term is determined as having some type it may not change any more. This rules out obviously any language with mutation or I/O. That's why you can't write applications in Agda, or mathematical proves in Haskell. (In the general case; you could do both with "tricks"; but those are indeed tricks).