A reduction from $HP$ to $\{(\langle M \rangle, \langle k \rangle) : \text{M visits in at list $k$ states for any input}\}$

I tried to define the next reduction from $ HP$ to $ \{(\langle M \rangle, \langle k \rangle) : \text{M visits in at list $ k$ states for any input}\}$ .

Given a couple $ (\langle M\rangle , \langle x\rangle)$ we define $ M_x$ such that for any input $ y$ , $ M_x$ simulates $ M$ on the input $ x$ . We denote $ Q_M +c$ the number of states needed for the simulation of $ M$ on $ x$ , and define more special states for $ M_x$ $ q_1′,q_2′,…,q_{Q_M + c+ 1}’$ when $ q’_{Q_M +c+1}$ is defined as the only final state of $ M_x$ . Now, in case $ M_x$ simulation of $ M$ on $ x$ halts (i.e $ M$ reach one of its finite state) $ M_x$ move to $ q_1’$ and then continue to walk through all the special states till it reaches $ q_{Q_M + c + 1}$ .

We define the reduction $ (\langle M \rangle , \langle x \rangle) \longrightarrow (\langle M_x \rangle , \langle Q_M +c+1 \rangle)$

In case $ ((\langle M \rangle , \langle x \rangle) \in HP$ then for any input $ y$ , $ M_x$ walks through all the special states and thus visits in at least $ Q_m + c+ 1$ steps. Otherwise, $ M$ doesn’t stop on $ x$ so $ M_x$ doesn’t visit any special state, thus visits at most $ Q_M +c$ states (the states needed for the simulation).

It is ok? If you have other ideas or suggestions please let me know.

reduction of independence problem and cluster problem

independent problem is: there is a simple and undirected graph, we are looking for the maximum vertex in which there is no edge between any two of them.

cluster problem is: there is a simple and undirected graph, we are looking for the maximum number of the vertex in which there are proximity every two vertexes ( there is an edge between any two vertexes)

how can I reduct independent problem to cluster problem and vise versa?

Damage reduction and damage resistance: how to calculate?

Assume a character has both damage reduction and damage resistance vs an incoming attack.

One example of damage reduction is the Heavy Armor Master feat:

While you are wearing heavy armor, bludgeoning, piercing, and slashing damage that you take from non magical weapons is reduced by 3.

One example of damage resistance is the blade ward cantrip:

You extend your hand and trace a sigil of warding in the air. Until the end of your next turn, you have resistance against bludgeoning, piercing, and slashing damage dealt by weapon attacks.

Let’s assume a character with both effects above is hit by a nonmagical weapon attack dealing slashing damage. The attack deals 10 damage. What happens:

  1. Damage reduction applies reducing damage to 7, then damage resistance (rounds down). Character takes 3 damage.
  2. Damage resistance applies, halving to 5, then damage reduction takes it to 2 damage.

Disagreement with professor over NP reduction problem

I’m a slight disagreement with my professor over whether or not a certain reduction is possible. He asked us to reduce the problem of 3-Coloring to the problem of 3-Clique. The problem is that I’m fairly confident that 3-Coloring is NP-Complete, while 3-Clique is P. Correct me if I’m wrong (which is very likely), but for any k-clique where the k is fixed, is $ V^k$ , meaning the 3-clique is $ V^3$ , right? I asked my professor about this and his response was:

“3-clique is definitely not in P. You (apparently) have to examine all thrices of vertices to settle the matter.”

And I still don’t understand how this is not a $ V^3$ operation.

If I figured out a way to reduce 3-coloring to 3-clique wouldn’t I be millionaire?

How does Max-HP reduction affect wild-shaped/polymorphed creatures?

Certain creatures have abilities which can reduce a character’s maximum HP, and usually if it gets reduced to 0 the character dies outright.

Suppose a HP30 PC is wild-shaped/polymorphed to a creature with 50HP, they get into a fight with a Wraith and take a few hits dealing a total of 30HP. If they failed the con saves, that PC’s max-HP is reduced by 30, but it’s still at 20.

An interesting, perilous situation.

Do they die instantly? Would feel a bit unfair since they’re standing there with a bunch of HP. Is the damage just shrugged off like normal damage upon return? The Druid’s wild-shape section is quiet on status conditions, though it’s pretty blatant about HP:

When you transform, you assume the beast’s hit points and Hit Dice. When you revert to your normal form, you return to the number of hit points you had before you transformed.

That sounds like a free pass, but it would reduce the danger of these fights considerably. I’ve been assuming the PC becomes a sort of ‘dead man walking’ where if they revert the HP reduction will carry and they’ll die instantly. But I’m not sure.

If that’s the case, they’ve got a ‘Crank’ like situation where the PC has less than an hour (before the wild-shape/polymorph wears off) to find a Heal or Remove Curse.

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?

Reduction from HALT to MPCP via TM Configuration Transitions

I have book with a very simple reduction from HALT to MPCP and I suspect that it is wrong. Here is the reduction:

Add the following dominos $ (top, bottom$ ):

  • Start: $ (\langle \varepsilon \rangle ,\langle \varepsilon\rangle \langle\kappa_0\rangle)$ for initial configuration $ \kappa_0$ .

  • Step: $ (\langle\kappa_i\rangle, \langle\kappa_j\rangle)$ for all configuration transitions $ \kappa_i \rightarrow \kappa_j$

  • Finish: $ (\langle\kappa_e\rangle \langle\varepsilon\rangle, \langle\varepsilon\rangle)$ for all final configurations $ \kappa_e$

This is much easier than what I see e.g. in Sipser. Is there an error in this proof? I think it is impossible to add a domino for all possible configuration transitions.


You may need the definition of configuration transitions:

Let $ T = (S, \Sigma, \Pi, \delta, s_0, \_, E)$ be a TM. Every triple $ (\nu,s,\omega)$ with $ \nu, \omega \in \Pi^{+}$ and $ s \in S$ is called configuration of $ T$ . The configuration transitions $ \rightarrow_{T}$ is defined as

movement to the right: $ \delta(s,\sigma) = (s’, \sigma’, \rightarrow), \varrho, \sigma \in \Pi$

\begin{align*} (\nu \varrho, s, \sigma \omega) \rightarrow_{T} \left\{\begin{array}{ll}% (\nu \varrho \sigma’ ,s’, \omega) & \mbox{if $ \omega \neq \epsilon$ } \ (\nu \varrho \sigma’ ,s’, \_) & \mbox{if $ \omega = \epsilon$ } \end{array}\right. \end{align*} movement to the left: $ \delta(s, \sigma) = (s’, \sigma’, \leftarrow), \varrho, \sigma \in \Pi$ \begin{align*} (\nu \varrho, s, \sigma \omega) \rightarrow_{T} \left\{\begin{array}{ll}% (\nu, {s’}, \varrho {\sigma’} \omega) & \mbox{if $ \nu \neq \epsilon$ } \ (\_, {s’}, \varrho {\sigma’} \omega) & \mbox{if $ \nu = \epsilon$ } \end{array}\right. \end{align*}


Reduction from $HALT$ to $A_{TM}$

I know the reduction to from $ A_{TM}$ to $ HALT$ . But is the following reduction from $ HALT$ to $ A_{TM}$ correct?

We are looking for total computable function $ f$ mapping from $ HALT$ to $ A_{TM}$ . The following TM $ F$ calculates the reduction $ f$ .

F = on input <T, w>     create the following TM T':     T' = on input v:        start T on v        if T accepts or rejects, *accept*     return <T',w> 

I think the line if T accepts or rejects, *accept* is correct, but it would be great if someone could check this.

Edit: I found the following slides, but I don’t think the construction in there is correct: http://slideplayer.com/slide/13791105/

Damage reduction vs Dying/Bleeding Out

When a character has less than 0 but more than -10 HP, they are considred “dying”. According to the SRD:

A dying character loses 1 hit point every round. This continues until the character dies or becomes stable.

However, a character or creature with damage reduction takes an x-amount less from any source of physical damage, which arguably includes the above. Obviously (to me), having damage reduction doesn’t automatically mean you’re immune to bleeding out. However, is there any official source that explains this? If so, what does that source have to say about the matter?