Understanding growth function of closed intervals in $\mathbb{R}$

I as studying VCdimensions and growth functions and found the following example on Wikipedia:

The domain is the real like $ \mathbb{R}$ . The set H contains all the real intervals, i.e., all sets of form $ \{c \in [x_1, x_2] | x \in \mathbb{R}\}$ for some $ x_{0, 1} \in \mathbb{R}$ .

For any set C of m real numbers, the intersection $ H \cap C$ contains all runs of between 0 and m consecutive elements of C. The number of such runs of $ {m+1 \choose 2} + 1$ , so Growth(H, m) = $ {m+1 \choose 2} + 1$ .

Can anyone please explain to me what does the term "all runs of between 0 and m" refer to here and why the growth function is $ {m+1 \choose 2} + 1$ and not $ {m+1 \choose 2}$ ?

Thank you very much!

Are NP problems lower bounded by exponential order of growth?

My understanding of P. vs NP is quite limited. I can understand P refers to an algorithm with an upper bound (big O) with order of growth $ n^c$ for some constant c and variable n. My question is, do NP problems have a hypothesized lower bound order of growth (big Omega) of $ c^n$ for deterministic machines? I can’t find this stated anywhere and I’m trying to understand if this is something that is assumed or not.

Thanks.

Growth function for non-regular languages

For language $ L$ over an alphabet $ \Sigma$ denote $ \gamma_L(n)$ as the number of words of length $ n$ in the language $ L$ . It is known that for regular languages this function represents a sequence with rational generating function (which is equivalent to that $ \gamma_L(n)$ is linear-recurrent for sufficiently large coefficients).

However, I couldn’t find any information about non-regular languages. And it is not clear how to extend the result, stated above to some other types of languages.

Does anyone know the conditions for some non-regular classes of languages (for example, prefix-closed languages) to have rational geodesic growth function?

Does the area of the Spike Growth spell work around corners or through total cover?

The spike growth spell has a range of 150 feet, and its description states:

The ground in a 20-foot radius centered on a point within range twists and sprouts hard spikes and thorns. The area becomes difficult terrain for the duration. When a creature moves into or within the area, it takes 2d4 piercing damage for every 5 feet it travels.

The transformation of the ground is camouflaged to look natural. Any creature that can’t see the area at the time the spell is cast must make a Wisdom (Perception) check against your spell save DC to recognize the terrain as hazardous before entering it.

When a creature casts spike growth, do they need to be able to see the center of the circle which spawns the growth? Or can the center or entire growth be behind total cover? Does the area of effect continue around corners, or pass through total cover like a door or thin wall?

What happens if you cast Animal Growth in constrained environments?

Let´s say a wizard casts Animal Growth (PHB, p. 198) on a lion thus changing the lion´s size form large to huge. – What happens if there isn´t enough space for the lion to grow into?

In the spell description it says:

If insufficient room is available for the desired growth, the creature attains the maximum possible size and may make a Strength check (using its increased Strength) to burst any enclosures in the process. If it fails, it is constrained without harm by the materials enclosing it—the spell cannot be used to crush a creature by increasing its size.

Now I have a couple of questions:

  1. I understand it correctly that the spell takes effect no matter whether or not there is enough space for the enlarged lion to fit in, right?
  2. If there is some enclosure (like e.g. a wall) which the lion fails to break and is then constrained without harm – what exactly does this mean as a consequence? Is the lion squeezed? (Follow up question: If it is squeezed will it have to move to a legal space immediately when it is it´s turn to act? And what if there is no such space for a huge lion anywhere on a packed battlefield?)
  3. What if the enclosure isn´t an object but a creature? Maybe there is an orc standing in the corner of the 15 feet square the huge lion is supposed to fill up. Do the lion and the orc contest the space? Do they make strength checks?

(You get the same wording in the descriptive texts of the spells Enlarge Person and Righteous Might (PHB p. 226 / 273), so my question also adresses these spells.)

How to do an arbitrary expression growth in Mathematica?

I want to write a function that can be used to grow an expression from a seed or another existing expression based on part specification.

growExpression[expr_,growParts_,unspecified_:Null]:=... 

So if I start from a seed say "START" like follows:

expr = growExpression["START",{{1,2,3}->"B",{2,1}->"A"},Null] 

“START”[Null[Null,Null[Null,Null,”B”]],Null[“A”]]

Note I want it to accept a parameter which can be used to decide unspecified but necessary growth to support the rest of the structure. Next, if I now take this expression as a seed and do further growth as follows:

expr = growExpression[expr,{{1}->"C",{1,2,2,4}->"D",{1,2,1}->"A",{2,1}->"E"},f] 

“START”[“C”[Null,Null[“A”,Null[f,f,f,”D”],”B”]],Null[“E”]]

Note even if non-leaves are replaced the expression doesn’t shrink, i.e. {1}->"C" is basically equivalent to {1,0}->"C".

Is there a builtin function that can help to achieve this behavior without side effects?