Büyük açık karmaşıklık boşlukları ile ilgili sorunlar

Bu soru, bilinen alt sınır ve üst sınır arasında büyük bir açık karmaşıklık boşluğu olduğu, ancak karmaşıklık sınıflarının kendilerindeki açık sorunlar nedeniyle olmadığı sorunlarla ilgilidir.

Daha açık olmak gerekirse, edelim bir sorun vardır ki boşluk sınıfları ile ( bir ⊆ B ise benzersiz tanımlanmamış) bir bunun bir kanıtlayabileceğiniz maksimal sınıf bir -Sert ve B üst sınırı bilinen bir minimumdur yani B'de sorunu çözen bir algoritma var . Bu demektir ki, sorun olduğunu bulmaya sonunda eğer Cı -tamamlamak ile bir ⊆ Cı ⊆ B bulgusuyla aksine, genel olarak, bu olmaz etki karmaşıklığı teorisi P bir algoritması N P Komple sorun.$A,B$$A\subseteq B$$A$$A$$B$$B$$C$$A\subseteq C\subseteq B$$P$$NP$

ve B = N P ile ilgili sorunlara ilgi duymuyorum çünkü bu zaten bu sorunun amacı .$A\subseteq P$$B=NP$

Ben mümkün olduğunca boşluk sınıfları ile ilgili sorun örnekleri arıyorum. Kapsamı ve kesin bir soru sınırlamak için, bir sorun özellikle ilgi ben ve B ⊇ e X, P , T ı M E hem de üye anlamına p ve e X, P , T ı M E -completeness mevcut bilgi ile tutarlıdır , bilinen sınıfları çökertmeden ( bu listeden sınıfları söyleyin ).$A\subseteq P$$B\supseteq EXPTIME$$P$$EXPTIME$

The Knot Equivalence Problem.

Given two knots drawn in the plane, are they topologically the same? This problem is known to be decidable, and there do not seem to be any computational complexity obstructions to its being in P. The best upper bound currently known on its time complexity seems to be a tower of $2$s of height $c^n$, where $c = 10^{10^{6}}$, and $n$ is the number of crossings in the knot diagrams. This comes from a bound by Coward and Lackenby on the number of Reidemeister moves needed to take one knot to an equivalent one. See Lackenby's more recent paper for some more recent related results and for the explicit form of the bound I give above (page 16).

Here's a version of the minimum circuit size problem (MCSP): given the $2^n$ bit truth table of a Boolean function, does it have a circuit of size at most $2^{n/2}$?

Known to be not in $AC0$. Contained in $NP$. Generally believed to be $NP$-hard, but this is open. I believe it's not even known to be $AC0[2]$-hard. Indeed, recent work with Cody Murray (to appear in CCC'15) shows there's no uniform NC0 reduction from PARITY to MCSP.

The complexity of computing a bit (specified in binary) of an irrational algebraic number (such as $\sqrt{2}$) has the best known upper bound of $\mathsf{P^{{{PP}^{PP}}^{PP}}}$ via a reduction to the problem $\mathsf{BitSLP}$ which known to have this upper bound [ABD14]. On the other hand we do not even know if this problem is harder than computing the parity of $n$ bits - for all we know this problem could be in $\mathsf{AC^0}$. Notice however that we know that no finite automaton can compute the bits of an irrational algebraic number [AB07]

Another natural topological problem, similar in spirit to Peter Shor's answer, is embeddability of 2-dimensional abstract simplicial complexes in $\mathbb{R}^3$. In general it's natural to ask when can we effectively/efficiently decide that an abstract $k$-dimensional simplicial complex can be embedded in $\mathbb{R}^d$. For $k=1$ and $d=2$ this is the graph planarity problem and has a linear-time algorithm. For $k=2$ and $d=2$ there is also a linear time algorithm. The $k=2$, $d=3$ case was open until last year, when it was shown to be decidable by Matousek, Sedgwick, Tancer, and Wagner. They say that their algorithm has a primitive recursive time bound, but larger than a tower of exponentials. On the other hand they speculate that it might be possible to put the problem in NP, but going beyond that would be challenging. However, there doesn't seem to be any strong evidence that a polytime algorithm is impossible.

The latter paper has many references for further reading.

Multicounter automata (MCAs) are finite automata equipped with counters that can be incremented and decremented within one step but only take integers >=0 as numbers. Unlike Minsky machines (aka counter automata), MCAs are not allowed to test whether a counter is zero.

One of the algorithmic problems with a huge gap related to MSCs is the Reachability problem. E.g., whether the automaton can reach, from a configuration with the initial state and all counters zero, a configuration with an accepting state, and all counters zero again.

The problem is hard for EXPTIME (as shown by Richard Lipton in 1976), decidable (Ernst Mayr, 1981) and solvable in Fω3 (thanks, Sylvain, for pointing this out). A huge gap.

$\mathsf{QMA}(2)$ (Quantum Merlin-Arthur with two unentangled provers): certainly $\mathsf{QMA}$-hard, but only known to be in $\mathsf{NEXP}$.

The computational problem associated to Noether's Normalization Lemma for explicit varieties ("explicit" in the sense of this paper [freely available full version]). Best known upper bound is $\mathsf{EXPSPACE}$ (note, SPACE, not TIME!) but it is conjectured to be in $\mathsf{P}$ (and indeed, its being in $\mathsf{P}$ is essentially equivalent to derandomizing PIT).

The Skolem problem (given a linear recurrence with integer base cases and integer coefficients, does it ever reach the value 0) is known to be NP-hard and not known to be decidable. As far as I know anything in between would be consistent with our current knowledge without any collapses of standard complexity classes.