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What's a concrete example of LIN ⊊ NLIN? (self.math)

submitted 4 days ago by CanadaPlus to c/math

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I've read the old papers proving that fact, but honestly it seems like some of the terminology and notation has changed since the 70's, and I roundly can't make heads or tails of it. The other sources I can find are in textbooks that I don't own.

Ideally, what I'm hoping for is a segment of pseudocode or some modern language that generates an n-character string from some kind of seed, which then cannot be recognised in linear time.

It's of interest to me just because, coming from other areas of math where inverting a bijective function is routine, it's highly unintuitive that you provably can't sometimes in complexity theory.

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[–] CanadaPlus 3 points 3 days ago* (last edited 3 days ago) (1 children)

Complexity classes, specifically the ones for algorithms taking asymptotically linear time in the size of their input. One is deterministic, the other nondeterministic.

Other famous classes like L, P or EXP are trivially in their nondeterministic equivalents, but it's unknown if they're strict subsets. In the case of P it's one of the Clay problems. LIN is the main (only?) case where the answer is known, and it's positive.

An example of a problem that's in NP, but we've kind of bet our civilisation on not being in P, is finding a string with a given hash value. It's a function, it gives the same answer every time, but it's preimages are (we think) completely intractable. What I've described as a possible concrete example is a kind of linear-time version of a hash algorithm. That's less useful, but apparently the (linear-time) irreversibility can be proven in that case.

[–] jacksilver@lemmy.world 1 points 3 days ago (1 children)

Thanks for the explanation!

I'm familiar with O() notation, but hadnt seen LIN before, which would be O(1). But that may be because I stick more to the papers written for computer scientists and don't go too deep into mathematic papers.

[–] CanadaPlus 2 points 3 days ago* (last edited 2 days ago) (1 children)

Ah sorry, I had no idea, you could have been a topologist who doesn't like computers or something.

LIN is unusual to hear about, probably because it's pretty well understood. Are you more of a coder, or an actual, academic computer scientist? If the latter, what do you know about pebbling games on nondeterministic machines?

[–] jacksilver@lemmy.world 1 points 2 days ago (1 children)

Oh no worries, I think I stumbled on this in a computer science crosspost.

While I do lean a bit in the academics, my area is mostly in ML / AI so not well read in pebbling games (although it sounds interesting).

[–] CanadaPlus 1 points 2 days ago* (last edited 2 days ago)

Yeah, the first paper I read was pretty heavily reliant on them. As far as I can tell they're laying the pebbles on the execution tree of a nondeterministic machine and then proving something with that.