3 ms·

Pretty interesting.. actually looks somewhat related to my thesis: http://www.cs.utexas.edu/~jderick/thesis.pdf http://www.cs.utexas.edu/~jderick/thesis.pdf I

by jderick 18y ago

Pretty interesting.. actually looks somewhat related to my thesis: http://www.cs.utexas.edu/~jderick/thesis.pdf http://www.cs.utexas.edu/~jderick/thesis.pdf I'll have to look at it more closely when I get a chance. I was working on learning proofs, rather than programs, but I think there are a lot of similarities.

mnemonicsloth 18y ago
I was working on learning proofs, rather than programs, but I think there are a lot of similarities. I'm just a mathematician, and my mind has obviously been warped by continua and the Axiom of Choice but aren't they the same thing? http://en.wikipedia.org/wiki/Curry-Howard_correspondence http://en.wikipedia.org/wiki/Curry-Howard_correspondence Actually, I've gotten interested in learning more about theoretical CS lately, but my everyone in my department is too old/applied/Russian to care, and my university's CS people only seem to care about operating systems and e-commerce. Can you recommend a logical starting place?
- sdp 18y ago
  The equivalence seems to hold only when the program is complete. Thus, while all proofs can be expressed as programs, not all programs can be expressed as proofs.
  - sdp 18y ago
    Programs are equivalent to proofs given their correctness, rather. Additionally, according to the wikipedia article, extracting a proof from a program requires a very richly typed programming language.
  - 13ren 18y ago
    Can't you restrict the domain, and so that the program is complete (for that restricted domain)?
    - sdp 18y ago
      I meant correct, not complete. I refer to the article on wikipedia which states: "A converse direction is to use a program to extract a proof, given its correctness. This is only feasible if the programming language the program is written for is very richly typed: the development of such type systems has been partly motivated by the wish to make the Curry-Howard correspondence practically relevant."
- ntoshev 18y ago
  The notation you'd use to represent the program/proof is very important.
- jderick 18y ago
  One interesting difference is that if you are learning a proof, then you know when you are successful. At least for my research, I generated a bunch of small cases of the theorem I was trying to prove, then tried to learn the general case. If I was successful, I would have a proof of the original conjecture. When learning programs from some data, you can always test your program against the data but never can be sure that you are generalizing the data properly.
moshe 18y ago
Thanks for the pointer. I was actually just looking at ACL the other day - I have a friend here at Google who is investigating integrating it with plop as a 20% project. I'm very excited that your work combines theorem-proving with learning, kudos! Do you do anything probabilistic to decide which generalizations to try?
- jderick 18y ago
  It is more just a set of techniques for generalizing theorems. At this point in time I would say there are a few roadblocks to doing some kind of data-driven probabilistic search for inductive proofs (like ACL2 does). The first is that there just aren't that many generalization techniques already implemented to choose from. The second is that most provers don't even have the basic backtracking mechanisms built in to support that kind of behavior. Although this may seem surprising, the search space for a typical theorem is so huge (technically infinite, but practically speaking just too complex to bother with) that usually if you don't find a proof right away with a high-confidence technique, then you are probably just going to go off into the weeds. Now that more processing power than ever is available, I'm sure this will gradually change. But a lot of theorem proving technology is very old (dates to the 70s) so it may take some time. The main technique I came up with looks at a particular finite case of a conjecture and generates a proof for that case using simple rewriting. You then look for patterns in that proof and use them to create a generalization. As a simple example, imagine your trying to prove a theorem about lists. First, I consider the case where a list is only 4 elements. Since this is a finite case, I can prove this easily with rewriting. Now I look at that proof and notice that rule X was applied 4 times. If I do the same thing when the list has 5 elements, that rule is applied 5 times. So in the general case, apply the rule n times. One novel thing about that I came across in this work is something I called "Hybrid proof terms". I found that it is very difficult to find patterns in sequences of terms (the typical way of representing a rewrite based proof). So I used this representation that factors out the things that do not change from one line of the proof to the next. Feel free to email me if you would like to discuss any more details. I'm not really working on this project anymore but I still find it a fascinating area of study. Also, for your friend who is thinking about working with ACL2, he should know that the acl2-help mailing list is quite friendly and helpful.

by jderick 18y ago

mnemonicsloth 18y ago

I was working on learning proofs, rather than programs, but I think there are a lot of similarities. I'm just a mathematician, and my mind has obviously been warped by continua and the Axiom of Choice but aren't they the same thing? http://en.wikipedia.org/wiki/Curry-Howard_correspondence http://en.wikipedia.org/wiki/Curry-Howard_correspondence Actually, I've gotten interested in learning more about theoretical CS lately, but my everyone in my department is too old/applied/Russian to care, and my university's CS people only seem to care about operating systems and e-commerce. Can you recommend a logical starting place?

sdp 18y ago
The equivalence seems to hold only when the program is complete. Thus, while all proofs can be expressed as programs, not all programs can be expressed as proofs.
- sdp 18y ago
  Programs are equivalent to proofs given their correctness, rather. Additionally, according to the wikipedia article, extracting a proof from a program requires a very richly typed programming language.
- 13ren 18y ago
  Can't you restrict the domain, and so that the program is complete (for that restricted domain)?
  - sdp 18y ago
    I meant correct, not complete. I refer to the article on wikipedia which states: "A converse direction is to use a program to extract a proof, given its correctness. This is only feasible if the programming language the program is written for is very richly typed: the development of such type systems has been partly motivated by the wish to make the Curry-Howard correspondence practically relevant."
ntoshev 18y ago
The notation you'd use to represent the program/proof is very important.
jderick 18y ago
One interesting difference is that if you are learning a proof, then you know when you are successful. At least for my research, I generated a bunch of small cases of the theorem I was trying to prove, then tried to learn the general case. If I was successful, I would have a proof of the original conjecture. When learning programs from some data, you can always test your program against the data but never can be sure that you are generalizing the data properly.

moshe 18y ago

Thanks for the pointer. I was actually just looking at ACL the other day - I have a friend here at Google who is investigating integrating it with plop as a 20% project. I'm very excited that your work combines theorem-proving with learning, kudos! Do you do anything probabilistic to decide which generalizations to try?

jderick 18y ago
It is more just a set of techniques for generalizing theorems. At this point in time I would say there are a few roadblocks to doing some kind of data-driven probabilistic search for inductive proofs (like ACL2 does). The first is that there just aren't that many generalization techniques already implemented to choose from. The second is that most provers don't even have the basic backtracking mechanisms built in to support that kind of behavior. Although this may seem surprising, the search space for a typical theorem is so huge (technically infinite, but practically speaking just too complex to bother with) that usually if you don't find a proof right away with a high-confidence technique, then you are probably just going to go off into the weeds. Now that more processing power than ever is available, I'm sure this will gradually change. But a lot of theorem proving technology is very old (dates to the 70s) so it may take some time. The main technique I came up with looks at a particular finite case of a conjecture and generates a proof for that case using simple rewriting. You then look for patterns in that proof and use them to create a generalization. As a simple example, imagine your trying to prove a theorem about lists. First, I consider the case where a list is only 4 elements. Since this is a finite case, I can prove this easily with rewriting. Now I look at that proof and notice that rule X was applied 4 times. If I do the same thing when the list has 5 elements, that rule is applied 5 times. So in the general case, apply the rule n times. One novel thing about that I came across in this work is something I called "Hybrid proof terms". I found that it is very difficult to find patterns in sequences of terms (the typical way of representing a rewrite based proof). So I used this representation that factors out the things that do not change from one line of the proof to the next. Feel free to email me if you would like to discuss any more details. I'm not really working on this project anymore but I still find it a fascinating area of study. Also, for your friend who is thinking about working with ACL2, he should know that the acl2-help mailing list is quite friendly and helpful.