4 ms·

It may well be that for some worst case instances of SAT/SMT that there is no parallel algorithm to solve them faster than a sequential algorithm unless P==NP.

by nullc 2y ago

It may well be that for some worst case instances of SAT/SMT that there is no parallel algorithm to solve them faster than a sequential algorithm unless P==NP. But most people who are not complexity theorists (or perhaps cryptographers) are not generally interested in worst case instances, but are instead interested in solving practical problems which are often not (or even-- are practically never) worst case instances. My own experience w/ real SMT problems is that I can usually home-brew a fair amount of parallelism by fixing some variables and then solving the derived problems w/ Z3 on each core. But the result often has fairly poor load distribution (e.g. some threads finish long before others), so it could obviously be done better by software the changes the variable its parallel on and how it distributes those variables adaptively as it goes.

almostgotcaught 2y ago
> But the result often has fairly poor load distribution (e.g. some threads finish long before others), so it could obviously be done better by software the changes the variable its parallel on and how it distributes those variables adaptively as it goes. It's astonishing to me that you're staring my point straight in the face and still not getting it. Let me restate in terms of portfolios: If I distribute my stock picks across many portfolios, without an oracle to tell me how to distribute them, some of the portfolios will perform poorly and some will perform well, but on average they will perform as if I just had a single portfolio.
- nullc 2y ago
  I do wonder how we're talking past each other. Usually when I use an SMT solver I'm interested in finding a satisfying solution. Breaking up the problem on some variable the solver would search across multiple processors can improve the wall-time until finding one. I doubt I'm following your metaphors/allusions, but the parallel may be that you're only hoping for any portfolio to have excess performance, and not the sum total.
  - deleted 2y ago
    [deleted]
  - almostgotcaught 2y ago
    > I doubt I'm following your metaphors/allusions, but the parallel may be that you're only hoping for any portfolio to have excess performance, and not the sum total. I'm sorry but I don't know how else to explain it to you when you're ignoring such an obvious fact: if you're hoping for any portfolio to hit big, but they're all picked without any intelligence (i.e., without an oracle) then you will always be only as good as the worst portfolio. It's just such a patently obvious logical argument I don't know how else to state it so that it becomes clearer. I mean I could easily write out the formula for expected value of the max of N uniform iid random variables but you've literally been witness to it by watching the wall-clock time on your threads! Let me put it without any metaphors/allusions: you simply don't understand search and/or SAT if you think that dividing a 2^N search space into N pieces will give you a speedup. It will not. It's basically the "geometric" definition of NP.
    - nullc 2y ago
      I'm afraid I'm still mystified and suspect we must be talking past each other, ... But, in any case, I'm grateful you took your time to try teaching me something.
    - madars 2y ago
      Of course, dividing search space into K pieces is not guaranteed to give you a speed-up _in the worst case_: you might pick an unlucky division where K-1 pieces are easily UNSAT but the last remaining piece is as hard as the original (so the wall clock time is unchanged). However, in practice variable-fixing can and often does give an _expected time_ speed-up, especially if the SAT instance has some inherent parallel search structure (e.g., key or midstate bits for ciphers) and such heuristic tactics are still useful.
      - almostgotcaught 2y ago
        > However, in practice variable-fixing can and often does give an _expected time_ speed-up Proof? Because as far as I know literally none of RP, BPP, ZPP relationships to NP are known.

by nullc 2y ago

almostgotcaught 2y ago

> But the result often has fairly poor load distribution (e.g. some threads finish long before others), so it could obviously be done better by software the changes the variable its parallel on and how it distributes those variables adaptively as it goes. It's astonishing to me that you're staring my point straight in the face and still not getting it. Let me restate in terms of portfolios: If I distribute my stock picks across many portfolios, without an oracle to tell me how to distribute them, some of the portfolios will perform poorly and some will perform well, but on average they will perform as if I just had a single portfolio.

nullc 2y ago
I do wonder how we're talking past each other. Usually when I use an SMT solver I'm interested in finding a satisfying solution. Breaking up the problem on some variable the solver would search across multiple processors can improve the wall-time until finding one. I doubt I'm following your metaphors/allusions, but the parallel may be that you're only hoping for any portfolio to have excess performance, and not the sum total.
- deleted 2y ago
  [deleted]
- almostgotcaught 2y ago
  > I doubt I'm following your metaphors/allusions, but the parallel may be that you're only hoping for any portfolio to have excess performance, and not the sum total. I'm sorry but I don't know how else to explain it to you when you're ignoring such an obvious fact: if you're hoping for any portfolio to hit big, but they're all picked without any intelligence (i.e., without an oracle) then you will always be only as good as the worst portfolio. It's just such a patently obvious logical argument I don't know how else to state it so that it becomes clearer. I mean I could easily write out the formula for expected value of the max of N uniform iid random variables but you've literally been witness to it by watching the wall-clock time on your threads! Let me put it without any metaphors/allusions: you simply don't understand search and/or SAT if you think that dividing a 2^N search space into N pieces will give you a speedup. It will not. It's basically the "geometric" definition of NP.
  - nullc 2y ago
    I'm afraid I'm still mystified and suspect we must be talking past each other, ... But, in any case, I'm grateful you took your time to try teaching me something.
  - madars 2y ago
    Of course, dividing search space into K pieces is not guaranteed to give you a speed-up _in the worst case_: you might pick an unlucky division where K-1 pieces are easily UNSAT but the last remaining piece is as hard as the original (so the wall clock time is unchanged). However, in practice variable-fixing can and often does give an _expected time_ speed-up, especially if the SAT instance has some inherent parallel search structure (e.g., key or midstate bits for ciphers) and such heuristic tactics are still useful.
    - almostgotcaught 2y ago
      > However, in practice variable-fixing can and often does give an _expected time_ speed-up Proof? Because as far as I know literally none of RP, BPP, ZPP relationships to NP are known.