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It seems like the person you're responding to has decided a type checker that outputs 1 when the proof object is valid according to the rules & 0 otherwise shou

It seems like the person you're responding to has decided a type checker that outputs 1 when the proof object is valid according to the rules & 0 otherwise should be considered a scoring function. The objective function in this case is not differentiable so the usual techniques from deep learning will not work & genetic search algorithms like AlphaEvolve can in theory be used in these cases. Someone still has to come up w/ the type system & verify the soundness of the rules b/c there is no finite codification of all valid & sound type systems like there are for problems in the linked blog post.

omnicognate 1y ago
An evolutionary algorithm doesn't need a differentiable objective function, but it can't work with a simple "1 if you got it right, 0 otherwise". It is an iterative approach and at each step it needs to be able to distinguish "better" from "worse" candidates. You certainly can't just say to AlphaEvolve "off you go, I'll tell you when you got it right". Taking the Collatz Conjecture I used as an example just now, you can trivially write an objective function that outputs 1 for a correct Lean proof of it and 0 for an incorrect one, but AlphaEvolve won't be able to work on that. It needs to be able to assess a collection of incorrect proofs to identify the most promising ones for the next step. I don't know how you'd even start on that, and it's certainly not what they've been doing with AlphaEvolve.
- measurablefunc 1y ago
  It can and it does work w/ such objective functions. Lots of people have used evolutionary algorithms to evolve chess playing neural networks¹ & they have been successful w/ very sparse reward signals where the the final trajectory is scored w/ 0 or 1 according to a win condition. You can say this is not likely to work for proof search & I'd be inclined to agree but the strategy has proven to work in simpler settings so whether it can be used in more complex settings is yet to be determined. If Collatz is not independent of existing axiomatic foundations then a brute force search will find a solution so any heuristics added on top of it that cut out paths to unsuccessful attempts will increase the probability of finding the proof object. ¹https://arxiv.org/abs/1711.08337 https://arxiv.org/abs/1711.08337
  - omnicognate 1y ago
    From that chess paper: > Each position in the ECM test suite has a predetermined “best move”. Each chromosome processes all of the 879 positions, and for each position it attempts to find this predetermined best move as fast as possible. > Instead of counting the number of correctly “solved” positions (number of positions for which the organism found the best move), we used the number of nodes the organism had to process in order to find the best move. That isn't a 1-or-0 objective function, and the paper isn't an example of using an evolutionary loop in which the objective function doesn't give you any information on which candidates are better in a given iteration. Because that isn't possible. Re brute forcing by evaluating the countable set of correct proofs within a given formal system, people have been trying that since computers were invented and it hasn't resulted in the magic proof factory yet. People continue to work on better and better heuristics for trimming the search and I understand some of the stuff people have been doing in that direction with Lean is actually useful now, but there hasn't been a huge breakthrough in it and nobody expects a system like that to spit out a proof of the Collatz Conjecture any time soon. More to the point of this discussion, it's not what AlphaEvolve does. Anyway, I need to go to bed. It's been fun.
    - measurablefunc 1y ago
      The same applies to proof search. Once you fix a finite foundational set of axioms the game proceeds exactly as in chess.
      - wizzwizz4 1y ago
        You can't use minimax with alpha-beta pruning for proof search, but that's sufficient to play chess at a high level. I don't see what you're seeing. Chess and mathematics are completely different kinds of problem.
- 11101010001100 1y ago
  In other words, proofs of existence are measure zero in the space of evolution?
ants_everywhere 1y ago
> It seems like the person you're responding to has decided a type checker... should be considered a scoring function To clarify, I didn't decide this, this is a valid scoring function in AlphaEvolve. The scoring function is generic and can even be an LLM writing prose giving feedback on the solution followed by another LLM scoring that prose numerically. There needs to be a numeric score to rank solutions. Typically type checkers give more output than 1 or 0, though. For example, they'll often give you information about where the first error occurred. That doesn't mean it's a great scoring function or even a good one. But it is a scoring function and without any scoring function at all progress would be impossible. To the extent that math is about writing proofs, it's a valid and essential scoring function for any problem. In practice, to make progress you need more than just the ability to write a logical proof, you need to build on previous results, add extra conditions, compute examples, etc. But in the context of the discussion, the point is that there is always some way to measure progress, which is why AlphaEvolve includes this mechanism. > Someone still has to come up w/ the type system & verify the soundness of the rules b/c there is no finite codification of all valid & sound type systems like there are for problems in the linked blog post. This is true, but it's also true that typically mathematicians fix a logic or universe before getting down to work. So AlphaEvolve and human mathematicians are on equal footing in that respect.
- measurablefunc 1y ago
  > it's also true that typically mathematicians fix a logic or universe before getting down to work This is true for programmers, it's not true for mathematicians. You can say programming is a subset of mathematics but mathematics is more than programming so proof search does not exhaust all the activities of a mathematician but it does exhaust all the activities of a programmer.
  - ants_everywhere 1y ago
    > it's not true for mathematicians It actually is true for mathematicians. Most of us work in ZFC. Some people choose to work in constructive logic. Some people choose to only use countable choice. But we always know what logic we're working in and which theorems we can invoke. E.g. if you choose not to accept the axiom of choice you also have a sense of which results depend on Zorn's lemma and have to work around them. All of this is normal background for mathematicians. So there's no need to allow the foundations of mathematics to vary unless you're working in logic and need to quantify over foundational systems or compare them. You would certainly never start a problem and then switch the logic midway through. That sort of thing wouldn't even be coherent.

omnicognate 1y ago

An evolutionary algorithm doesn't need a differentiable objective function, but it can't work with a simple "1 if you got it right, 0 otherwise". It is an iterative approach and at each step it needs to be able to distinguish "better" from "worse" candidates. You certainly can't just say to AlphaEvolve "off you go, I'll tell you when you got it right". Taking the Collatz Conjecture I used as an example just now, you can trivially write an objective function that outputs 1 for a correct Lean proof of it and 0 for an incorrect one, but AlphaEvolve won't be able to work on that. It needs to be able to assess a collection of incorrect proofs to identify the most promising ones for the next step. I don't know how you'd even start on that, and it's certainly not what they've been doing with AlphaEvolve.

measurablefunc 1y ago
It can and it does work w/ such objective functions. Lots of people have used evolutionary algorithms to evolve chess playing neural networks¹ & they have been successful w/ very sparse reward signals where the the final trajectory is scored w/ 0 or 1 according to a win condition. You can say this is not likely to work for proof search & I'd be inclined to agree but the strategy has proven to work in simpler settings so whether it can be used in more complex settings is yet to be determined. If Collatz is not independent of existing axiomatic foundations then a brute force search will find a solution so any heuristics added on top of it that cut out paths to unsuccessful attempts will increase the probability of finding the proof object. ¹https://arxiv.org/abs/1711.08337 https://arxiv.org/abs/1711.08337
- omnicognate 1y ago
  From that chess paper: > Each position in the ECM test suite has a predetermined “best move”. Each chromosome processes all of the 879 positions, and for each position it attempts to find this predetermined best move as fast as possible. > Instead of counting the number of correctly “solved” positions (number of positions for which the organism found the best move), we used the number of nodes the organism had to process in order to find the best move. That isn't a 1-or-0 objective function, and the paper isn't an example of using an evolutionary loop in which the objective function doesn't give you any information on which candidates are better in a given iteration. Because that isn't possible. Re brute forcing by evaluating the countable set of correct proofs within a given formal system, people have been trying that since computers were invented and it hasn't resulted in the magic proof factory yet. People continue to work on better and better heuristics for trimming the search and I understand some of the stuff people have been doing in that direction with Lean is actually useful now, but there hasn't been a huge breakthrough in it and nobody expects a system like that to spit out a proof of the Collatz Conjecture any time soon. More to the point of this discussion, it's not what AlphaEvolve does. Anyway, I need to go to bed. It's been fun.
  - measurablefunc 1y ago
    The same applies to proof search. Once you fix a finite foundational set of axioms the game proceeds exactly as in chess.
    - wizzwizz4 1y ago
      You can't use minimax with alpha-beta pruning for proof search, but that's sufficient to play chess at a high level. I don't see what you're seeing. Chess and mathematics are completely different kinds of problem.
11101010001100 1y ago
In other words, proofs of existence are measure zero in the space of evolution?

ants_everywhere 1y ago

> It seems like the person you're responding to has decided a type checker... should be considered a scoring function To clarify, I didn't decide this, this is a valid scoring function in AlphaEvolve. The scoring function is generic and can even be an LLM writing prose giving feedback on the solution followed by another LLM scoring that prose numerically. There needs to be a numeric score to rank solutions. Typically type checkers give more output than 1 or 0, though. For example, they'll often give you information about where the first error occurred. That doesn't mean it's a great scoring function or even a good one. But it is a scoring function and without any scoring function at all progress would be impossible. To the extent that math is about writing proofs, it's a valid and essential scoring function for any problem. In practice, to make progress you need more than just the ability to write a logical proof, you need to build on previous results, add extra conditions, compute examples, etc. But in the context of the discussion, the point is that there is always some way to measure progress, which is why AlphaEvolve includes this mechanism. > Someone still has to come up w/ the type system & verify the soundness of the rules b/c there is no finite codification of all valid & sound type systems like there are for problems in the linked blog post. This is true, but it's also true that typically mathematicians fix a logic or universe before getting down to work. So AlphaEvolve and human mathematicians are on equal footing in that respect.

measurablefunc 1y ago
> it's also true that typically mathematicians fix a logic or universe before getting down to work This is true for programmers, it's not true for mathematicians. You can say programming is a subset of mathematics but mathematics is more than programming so proof search does not exhaust all the activities of a mathematician but it does exhaust all the activities of a programmer.
- ants_everywhere 1y ago
  > it's not true for mathematicians It actually is true for mathematicians. Most of us work in ZFC. Some people choose to work in constructive logic. Some people choose to only use countable choice. But we always know what logic we're working in and which theorems we can invoke. E.g. if you choose not to accept the axiom of choice you also have a sense of which results depend on Zorn's lemma and have to work around them. All of this is normal background for mathematicians. So there's no need to allow the foundations of mathematics to vary unless you're working in logic and need to quantify over foundational systems or compare them. You would certainly never start a problem and then switch the logic midway through. That sort of thing wouldn't even be coherent.