## Can “partial” surprise happen? Would it be an acceptable rule?

I’m thinking the "surprise round" rule just isn’t fair. Say there’s a 2 vs 2 encounter (teams PC and Creatures (Cs)), then PC1 could notice the 2 Cs and PC2 just one of them. Also for the sake of discussion PC1 is mute for whatever reason, so he can’t quickly communicate with PC2 and he also plays last according to iniciative order. Then PC2 loosing an entire round seems unfair to me, because he could perfectly be taking actions against the C he found. Also if PC2 gets attacked by the Cs it would be doubly unfair, even if one has advantage and the other don’t, it’s 2 actions against none. So I figured you could ask PC2 to just turn around when it isn’t his turn during the surprise round, so as not to reveal the concealed C position to him and then, when he plays his turn remove the hidden C and let PC2 play as always…or if he gets attacked by hidden C before his turn he now sees it. What do you think?

## Mathematica Partial Fraction Decomposition full steps

I hope to get full steps instead of getting the result. Any ways to do so?

## Can archers bypass partial cover by arcing their shot?

There are two basic ways an archer fires at a target. In close quarters engagements, archers (and anyone using a projectile weapon) would likely use "direct fire", ie. fire at an angle nearly parallel to the ground. At longer distances and especially when targets are hiding behind terrain and walls, archers instead use "indirect fire", ie. firing at angle greater than 45 degrees, in order to lob arrows over and behind cover.

Imagine an archer firing at a range of 100 feet on a creature using 5-foot tall wall for cover.

The rules for cover on a grid state the following:

Choose a corner of the attacker’s space or the point of origin of an area of effect. Then trace imaginary lines from that corner to every corner of any one square the target occupies. If one or two of those lines are blocked by an obstacle (including another creature), the target has half cover. If three or four of those lines are blocked but the attack can still reach the target (such as when the target is behind an arrow slit), the target has three-quarters cover.

Using these rules, it’s easy to see how the wall could provide half or three-quarters cover against direct fire. The trajectory of the arrow will always intersect with the wall, and if enough lines from the archer intersect with the wall, then partial cover is granted. This is consistent with a physical understanding of the scenario, because the arrows will follow a nearly straight line from the archer to their target.

But what if the archer chooses to fire indirectly at their target? In the physical world a wall would provide no cover against an attack that falls from above. Drawing lines from the archer, however, results in the same result as direct fire, granting partial cover in a way which is inconsistent with reality.

Are there any rules that would allow the archer to use indirect fire to bypass partial cover?

## Sidebar widget not partial refreshing for first widget on add and remove action

When you add the first widget in the sidebar it’s a refreshing customizer completely and not partial refreshing.

• Go to Customizer
• Go to the Widgets panel.
• Remove all the widgets one by one.
• At the last widget when you remove it, it refreshes the customizer.
• When adding the first widget it again refreshes the customizer.
• First onward, it partially refreshing the customizer as expected.

Please find the attached video for further reference: https://a.cl.ly/WnulD9r6

Is it the right behavior?

## In Ghost Ops do NPCs get free attacks only on total Bullet Time failure or also on partial failure?

I have the original version of Ghost Ops (which uses Fudge dice), not the Savage Worlds version or the OSR version. This question is about that original version, but if you think the rules in one of the other versions can throw some light on this, please chip in.

On page 132 of the core rulebook there is an example of a failed Bullet Time action. The PC was attempting to shoot 3 NPCs in the head, and needed an 8 but only got a 6.

The book then has some more rules:

The Handler can decide that the Operator succeeded in some of the attempt. Maybe they barged the door and managed to get 2 of the attempted headshots off but missed the third. Failing a Bullet Time event places the Operator as prone for 1 round, allowing any Tangos free attacks. Deciding to attempt Bullet Time is risky but can be ultimately rewarding.

So, if the GM has said the failed roll can be partial success (hit 2 of the NPCs) and partial failure (miss the 3rd NPC), which of these applies?

1. It still counts as a normal fail – the PC is prone and subject to a free attack by all three NPCs (assuming the two he shot aren’t dead or disabled).
2. It still counts as a ‘reduced’ fail – the PC is prone but only the third NPC, who was not hit, gets a free attack.
3. It counts as a success – the PC is not prone and the NPC/s don’t get free attacks.
4. The GM decides on a case by case basis.

I’m hoping there is clarification for this question in one of the expansions, or in an updated version of the pdf (I only have a print copy). I’ve failed to find any errata on the internet.

## Termination of term rewiting using strict partial order on subterms

Are there any good books, research reports, surveys, theses, or papers that display proof techniques, with clear proofs of termination of term rewriting problems that have the following form…?

Terms are represented by directed acyclic graphs where the terms are vertices with arcs labelled $$arg_{1}…arg_{n}$$ pointing to the immediate sub-terms. There would be additional equality arcs between vertices. Thinking of the transitive closure of the equality arcs as representing equivalence classes of vertices that are "merged", the $$arg$$ arcs in the graph form a lattice (because or the strict order on sub-terms, and some sub-terms might be shared). A rewrite rule would add extra arcs, such that existing partial order would be preserved and added to, so the rewrite rules would be constructing a series of partial orders (represented in the graph state at each step) $$p_{0} \subset … \subset p_{m}$$ more and more "constraining" the partial order relation between vertices until either the re-write rules find nothing to re-write or a rewrite would introduce a cycle (immediately detectable by a depth first search). I think this kind of termination proof is correct because we can say every step was a reduction under the partial order $$p_{m}$$ but I’d like a formal justification because I have worries about my not knowing $$p_{m}$$ before hand, only when it is constructed. And if the rewrite finds a cycle then that cycle was implicit from the beginning. Again I think that’s OK because my re-write rules are prove-ably LHS iff RHS so they transform the problem to an equivalent problem. I call this "construct a partial order or die trying." Is there a more formal name for this kind of proof?

Ideally the proof examples would be constructive and mathematically thorough. I see some papers that assume a lot of prior knowledge, probably because of brevity requirement, and not wanting to bore an expert audience. And others with "wordy" explanations, which are great to give intuitive understanding, but proofs should not depend on them.

## Can partial Turing completeness be quantified as a subset of Turing-computable functions?

Can partial Turing completeness be coherently defined this way:
An an abstract machine or programming language can be construed as Turing complete on its computable subset of Turing-computable functions.

In computability theory, several closely related terms are used to describe the computational power of a computational system (such as an abstract machine or programming language):

Turing completeness A computational system that can compute every Turing-computable function is called Turing-complete (or Turing-powerful). https://en.wikipedia.org/wiki/Turing_completeness

## Partial correctness for array multiplication

Can someone help me to prove partial correctness and termination for array multiplication.

## annotations for partial correctness

I am reading about hoare login but I don’t really understand the annotations part. This is what I know about annotations:

The inserted annotation is expected to be true whenever the execution reaches the point of the annotation 

This function f(x,n) calculates x^n

For the given code, the answers for annotations are the following:

before while:

(K >0) ^ (Y x P^K = X^n) 

after while statement:

(K > 0) ^ (Y x P^K = X^n) 

I understand the K>0 part but I don’t get why we’re using the second part.

## How to show that a partial function is recursive?

I try to prove that this function is recursive: $$f(x_1,x_2)= \begin{cases} 2x_1-x_2 & \text{if x_1 \geqslant \sqrt{x_2} } \newline \bot & \text{otherwise} \end{cases}$$

I think that I need to use minimization operator but I don’t know how to do that. $$\qquad$$ $$\qquad$$ $$\qquad$$ Maybe i have to prove that $$\mu y(|x_1-\sqrt{x_2}| = 0)$$ ?