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# Lecture 19 - Chapter 2: Chemistry and Scheduling

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51.

And as John's book explains on p.251, the complexity of reachability is at least doubly exponential. Doesn't this mean that reachability cannot be in NP?

Which book exactly?

Also, I don't see how you've managed to prove $$\textsf{NP} \subsetneq \textsf{2-EXP}$$.

Maybe you want to use $$\textsf{EXP} \subsetneq \textsf{2-EXP}$$? What's the proof of that?

Comment Source:> And as John's book explains on p.251, the complexity of reachability is at least doubly exponential. Doesn't this mean that reachability cannot be in NP? Which book exactly? Also, I don't see how you've managed to prove \$$\textsf{NP} \subsetneq \textsf{2-EXP}\$$. Maybe you want to use \$$\textsf{EXP} \subsetneq \textsf{2-EXP}\$$? What's the proof of that?
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52.

I was referring to p.251 of Quantum Techniques for Stochastic Mechanics, which states that any decision algorithm has a worst-case runtime that is at least doubly exponential. But now that I'm looking at it again, I see that that is a statement about Presburger arithmetic, not about the reachability problem for Petri nets. So I have to retract my claim, which was based on a too cursory reading---sorry!

Comment Source:I was referring to p.251 of [Quantum Techniques for Stochastic Mechanics](https://arxiv.org/abs/1209.3632), which states that any decision algorithm has a worst-case runtime that is at least doubly exponential. But now that I'm looking at it again, I see that that is a statement about Presburger arithmetic, not about the reachability problem for Petri nets. So I have to retract my claim, which was based on a too cursory reading---sorry!
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53.
edited May 2018

I see other comments have appeared while I was writing mine. Let me still post it.

Tobias wrote:

And as John's book explains on p.251, the complexity of reachability is at least doubly exponential.

No, I said that the complexity of deciding the validity of statements in Presburger arithmetic is at least doubly exponential. This is an axiom system for arithmetic that mentions addition and multiplication only. Unlike Peano arithmetic, it's decidable.

The situation for Petri net reachability is much less well understood. In 1976, Roger Lipton showed that its complexity is at least exponential. More precisely, he showed the for any $$c > 0$$ the worst-case run-time for deciding Petri net reachability exceeds $$2^{cn}$$ where $$n$$ is the size of the problem.

However, the best known upper bound for the complexity is much worse. In 1981 Ernst Meyr found an algorithm that decides Petri net reachability. But its runtime grows faster than every primitive recursive function! For example, faster than this series

$$1, 2^2, 2^{2^2}, 2^{2^{2^2}}, \dots$$ but in fact much faster than that.

There's a tantalizing connection between Presburger arithmetic and Petri net reachability, discussed in my book, but it hasn't (yet) sufficed to get a doubly exponential algorithm for Petri net reachability.

Comment Source:I see other comments have appeared while I was writing mine. Let me still post it. Tobias wrote: > And as John's book explains on p.251, the complexity of reachability is at least doubly exponential. No, I said that the complexity of deciding the validity of statements in Presburger arithmetic is at least doubly exponential. This is an axiom system for arithmetic that mentions addition and multiplication only. Unlike Peano arithmetic, it's decidable. The situation for Petri net reachability is much less well understood. In 1976, [Roger Lipton showed that its complexity is at least exponential](http://www.cs.yale.edu/publications/techreports/tr63.pdf). More precisely, he showed the for any \$$c > 0\$$ the worst-case run-time for deciding Petri net reachability exceeds \$$2^{cn}\$$ where \$$n\$$ is the size of the problem. However, the best known _upper_ bound for the complexity is much worse. In 1981 Ernst Meyr found an algorithm that decides Petri net reachability. But its runtime grows faster than every primitive recursive function! For example, faster than this series $$1, 2^2, 2^{2^2}, 2^{2^{2^2}}, \dots$$ but in fact much faster than that. There's a tantalizing connection between Presburger arithmetic and Petri net reachability, discussed in my book, but it hasn't (yet) sufficed to get a doubly exponential algorithm for Petri net reachability.
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54.

@John

The situation for Petri net reachability is much less well understood. In 1976, Roger Lipton showed that its complexity is at least exponential. More precisely, he showed the for any $$c > 0$$ the worst-case run-time for deciding Petri net reachability exceeds $$2^{cn}$$ where $$n$$ is the size of the problem.

It's worse than that - the proof regards space, not time. That paper establishes you need at least $$\mathcal{O}(2^{n})$$ memory to solve the reachability problem. Since you can't writing to all that memory demands the algorithm take the time to do it, it also entails the algorithm is in $$\textsf{EXP-TIME}$$.

@Tobias

The same Roger Lipton cited above cowrote another paper in 1976 establishing that any $$\textsf{EXP-SPACE}$$ algorithm can be expressed as a Petri net reachability problem.

I used this to argue $$\textsf{NP} \subsetneq \textsf{PETRI-REACH}$$ in comment #41 in this thread. I had a few false starts with this - do you think this argument is adequate?

Comment Source:@John > The situation for Petri net reachability is much less well understood. In 1976, [Roger Lipton showed that its complexity is at least exponential](http://www.cs.yale.edu/publications/techreports/tr63.pdf). More precisely, he showed the for any \$$c > 0\$$ the worst-case run-time for deciding Petri net reachability exceeds \$$2^{cn}\$$ where \$$n\$$ is the size of the problem. It's worse than that - the proof regards *space*, not *time*. That paper establishes you need at least \$$\mathcal{O}(2^{n})\$$ memory to solve the reachability problem. Since you can't writing to all that memory demands the algorithm take the time to do it, it *also* entails the algorithm is in \$$\textsf{EXP-TIME}\$$. @Tobias The same Roger Lipton cited above cowrote another paper in 1976 establishing that any \$$\textsf{EXP-SPACE}\$$ algorithm can be expressed as a Petri net reachability problem. I used this to argue \$$\textsf{NP} \subsetneq \textsf{PETRI-REACH}\$$ in [comment #41](https://forum.azimuthproject.org/discussion/comment/17886/#Comment_17886) in this thread. I had a few false starts with this - do you think this argument is adequate?
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55.

@Matthew: I haven't been able to access the Cardoza et al paper that you've been referring to, but otherwise I agree with your earlier argument. Very nice!

Comment Source:@Matthew: I haven't been able to access the Cardoza et al paper that you've been referring to, but otherwise I agree with your earlier argument. Very nice!
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56.

Feel free to shoot me a line on the gitter I set up and I can send the paper if you want.

Comment Source:Feel free to shoot me a line on the [gitter](https://gitter.im/Applied-Category-Theory-Course/Lobby) I set up and I can send the paper if you want.