What are the three points of view in Kolmogorov Complexity?

I was reviewing for my finals and find this question that I have totally no clue.

Compare the following to statements from three points of view:

  1. There exists a constant $ c > 0$ such that for all palindromes $ x \in \{0, 1\}^*$ we have $ K(x) \leq \lfloor x / 2 \rfloor + c$ .

  2. There exists a constant $ c > 0$ such that for all $ x \in \{0, 1\}^*$ we have $ K(\overline{x}) \leq K(x) + c$ where $ \overline{x}$ is the complement of $ x$ .

So what are the three points of view am I suppose to use and where should I start?

Scaling down a set of points into a smaller area

A visibility graph $ G(P) = (V,E)$ of a set $ P = \{p_1, \dots, p_n\}$ of points is defined as follows.

  • Each vertex $ u \in V$ corresponds to a point $ p_u \in P$ .
  • There exists an edge $ uv \in E$ if, and only if the line segment $ \overline{p_up_v}$ does not contain any other points in $ P$ .

Assume that the coordinates of every $ p_i \in P$ has $ O(n^c)$ digits, where $ c$ is a constant.

My aim is to scale the point set, without changing the relative positions, into a smaller area. For the sake of simplicity, assume that the area is a circle of raduis 1.

Consider three points: $ p_1(100, 340)$ , $ p_2(500,150)$ , $ p_3(240, -600)$ .

The new coordinates after scaling would be $ p’_1(0.1, 0.34)$ , $ p’_2(0.5, 0.15)$ , $ p’_3(0.24, -0.6)$ , and the relative positions stay the same, as well as the visibility relations.

My question is two folded:

  1. In the very simple example above, the coordiantes are divided by $ 100$ , and it is pretty easy to find that number. But what is the algorithm to find this number for any given point set?
  2. Assuming that the coordinates are of $ O(n^c)$ length, can there be a coordinate of $ O(d^n)$ (exponential) length?

Do negative Hit Points exist in D&D 5e?

Say that a character has 5 HP remaining and is dealt 10 damage from an attack. Of course, 5 – 10 = -5, so the character has dropped below 0 Hit Points and follows the rules for making death saving throws (provided they didn’t get dealt enough damage for instant death). However, it’s not clear to us if the character remains at -5 Hit Points or if they bounce back up to 0 Hit Points (like many rules in 5e, knowledge of past editions may be a hindrance to interpreting them).

If negative Hit Points exist, then characters will take longer to recover naturally (since stable characters recover at a rate of 1 HP per 1d4 hours). Also, a natural 20 on a death saving throw, which recovers one Hit Point, would not make them instantly conscious. Finally, it would mean that instant death is a greater possibility, as you need less to reach the threshold if you are attacked again.

However if negative Hit Points do not exist and characters bounce up to 0 after crossing the 0 HP threshold, then characters will always regain 1 HP and become conscious after waiting 1d4 hours or rolling a natural 20 on a death saving throw. Also, this would mean that instant death is far less likely because someone who attacks an unconscious character would always need to deal maximum HP damage or they don’t kill you (and if they don’t, then I guess their damage means nothing, which seems rather odd).

Unfortunately, the example provided with the basic rules isn’t helpful because it describes someone taking enough damage for instant death, but not someone who got less than that. Furthermore, the rules describe “Dropping to 0 Hit Points”, but not “Dropping Below 0 Hit Points” and seems to omit what happens when you take more damage than the HP you have remaining, but less than enough for Instant Death. Our group spent a while debating this when we played from the Starter Edition and we weren’t sure given that some previous versions of D&D had them while others didn’t. So do negative hit points exist or do characters “bounce up to 0 HP”?

Fitting an integral function given a set of data points

I have a set of measures of the resistivity of a given material at different thicknesses and I’m trying to fit them using the Fuchs-Sondheimer model. My code is:

data = {{8.1, 60.166323}, {8.5, 47.01784}, {14, 52.534961}, {15,     50.4681111501753}, {20, 39.0704975714401}, {30,     29.7737879177201}, {45, 22.4406}, {50, 15.2659673601299}, {54,     18.189933218482}, {73, 14.8377093467966}, {100,     15.249523361101}, {137, 15.249523361101}, {170,     10.7190970441753}, {202, 15.249523361101}, {230, 10.9744085456615}}  G[d_, l_, p_] := NIntegrate[(y^(-3) - y^(-5)) (1 - Exp[-yd/l])/(1 - pExp[-yd/l]), {y,0.01, 1000}];  nlm  = NonlinearModelFit[data, 1/(1 - (3 l/(2 d)) G [d, l, p]) , {{l, 200}, {p, 4}}, d, Method -> NMinimize] 

However it returns me these errors:

NIntegrate::inumr: The integrand ((1-E^(-(yd/l))) (-(1/y^5)+1/y^3))/(1-pExp[-(yd/l)]) has evaluated to non-numerical values for all sampling points in the region with boundaries {{0.01,1000}}. 
NonLinearModelFit: the function value is not a real number at {l,p} = {200.,4.} 

I think that the problem is in the way in which I defined the integral function G[d, l, p], because I had to fit a different set of data points with a different function of only one variable which I defined through the NIntegrate function and it gave me no error. Could anyone please help me?

Do sorcery points made from converted spell slots vanish after a long rest?

Sorcerers are able to turn sorcery points into spell slots, but the same conversion rate applies to turn spell slots into sorcery points. If, by the end of the day, you have unused spell slots and you wanted to risk the overnight ambush, could you turn them into sorcery points to be used the next day? Or would they disappear anyway? Sorcery points are replenished after a long rest, but that only infers to ones that have been expended, and ones created by spell slots are theoretically one-use, so do they disappear daily or can they stack?

Joining points of polygon in correct order

I have a point’s array of some 2d shape(polygon). The polygon could be either crossed or convex, I don’t know it. And I want to join those points in the correct order.

My first thought was to take some point as an origin and start looking for the closest one and so on. But this approach doesn’t work for some crossed polygons, for example: on Image1, it would go from x3 to x5 because it is closer than to x4, but what I really want is to join x1-x2-x3-x4-x5-x6.

After some thinking, I’ve realized that my requirement correct order is very unclear because on Image2 red lines are in the correct order too. So let’s add the requirement that polygon lines shouldn’t at least cross.

I really confused, can someone point me in what direction I should move? Maybe it’s some famous algorithm but I just don’t know how to google it properly. Thanks.

Image1: enter image description here

Image2: enter image description here

Why would I ever choose rolling hit points?

In DnD 5th edition, all classes seemingly have the option of rolling hit points -or- just increasing their hit points by a set value.

That set value is defined as the dice roll’s average, rounded up.

Given that it is rounded up (and thus will statistically offer better HP than rolling), why would anyone ever roll for hit points? Rolling seems to provide the worst of both worlds, giving you lower HP than defaulting as well as possibly screwing your barbarian over with a roll of 1, and just risking plain bad rolls in general.

It makes me wonder if I’ve overlooked any rules that otherwise balances the options. Am I?

Update: From some of the answers given below, I sense an answer along the lines of “it’s a roleplaying game, so there”, which to me is an unsatisfactory answer due to all other roleplaying games I know of (outside the D&D/Pathfinder line, of course) relying on deterministic methods.

Does reduction of maximum hit points stick to the form it is applied to?

Following up on How does Max-HP reduction affect wild-shaped/polymorphed creatures?, which states:

Damage taken in animal form doesn’t affect your original form’s HP unless you’re dropped to 0 HP in animal form and there’s excess damage. Nowhere is it suggested that max-HP reduction would work any differently. Because Wild Shape/Polymorph gives you a new pool of HP, only that pool is affected by the reduction.

Context

A druid gets seduced by a succubus. They kiss while the druid is in bear form – this is not hypothetical as yesterday exactly this had happened. The druid gets lowered Maximum Hit Points because of this forced romance. So according to the linked Q&A, the reduction would only apply to the bear form.

Question

If the druid reverts back to normal, the HP reduction is not active anymore. What if the druid wild shapes another time, back into a bear: does it get a fresh “pool of HP”, or does the Reduced Max HP stay with its bear form until it gets “cured”?

In other words: are shapeshifters actually really resilient against abilities that reduce maximum hit points?

Example

In case it helps to clarify, let’s use these numbers:

  1. Druid: 45 HP
  2. Wild Shapes into Brown Bear: 34 HP, but reduced to 10 Maximum HP (after two kisses).
  3. Druid reverts back to normal: 45 HP
  4. Wild Shapes back into Brown Bear: 34 HP, or still at 10 HP?