#complexity-theory

[ohaithe-re]

Complexity of calculating, complexity of things

To follow up on this post: we had a complexity axiom that told us something about topology, apparently. I would like to elaborate on this and explain what it's "really" (in my opinion) doing. The topological statement is also, inherently, a complexity question. It's just that, instead of being couched in the usual langauge of minimum complexity of a Turing machine calculating something, it's a statement about the minimum complexity writing down a description of some knot. This is more natural than you might think!

Morally, it's very analogous to the "theorem" that the Hajos number of a graph can be exponentially big (under the complexity "axiom" that coNP != NP). The Hajos number is one way of measuring how hard to build a graph from simpler graphs. You start with complete graphs K_n. Of the steps you're allowed to use, none of them can ever decrease the chromatic number of the graph. It's also been shown you can always build a graph in this way where the chromatic number never changes. That is, the Hajos number sort of asks: how many steps do you need to build this graph, in a way where each step leaves the chromatic number unchanged?

So, if you have a graph that is "simple" by measure of the Hajos number (at most polynomially long), then it means that there's a short proof (a polynomial length sequence of steps to construct it) that it has a certain chromatic number k. This means there is no coloring of k, which is coNP-hard. So you would have a NP way to check coNP questions.

Generally you could imagine other versions of this... like, #3COL is the counting problem that asks how many 3-colorings there are of a graph. This is #P-hard, so if P^{#P} != NP, then this shouldn't be in NP. If I gave you a set of allowed "moves" to do on a graph that always change #3COL in simple ways (e.g. "I'm going to add a degree-one edge, which doubles the number" or "This graph has two distinct components, so I can reduce to counting on each component separately and then multiply"). Then I define the complexity of a graph com(G) as the number of steps it takes to reduce it to the empty graph. P^{#P} != NP would imply that com(G) must grow superpolynomially in the size of G.

I could generalize that somewhat. Instead of reducing to just trivial graphs, I could say, I want to reduce to graphs of small treewidth. This is 'enough' because on graphs of small treewidth, #3COL is easy to compute. Maybe the steps I'm allowed to use won't reduce treewidth, but I'm just sort of asking: can I take a graph and somehow canonicalize it to a graph of small twin-width that has the same number of colorings?

So if I start a graph G and can reach a graph H by applying these moves, then I say that G and H are "equivalent" within this space of moves. The distance(G,H) is how many moves you need to do this. I would be able to solve #3COL quickly on G if I can reduce it to H with only polynomially many moves, and then H has small treewidth. So then the theorem would be,

if there is a constant k such that (for all G, there is an equivalent H, with polynomial distance(G,H), and treewidth(H) <= k), then P^{#P} = NP.

so by flipping that around,

P^{#P} != NP --> for any k, there is some G, for which all equivalent H, either have treewidth(H) > k, or a superpolynomial distance(G,H).

This paper from Freedman is looking at a particularly nice, maybe natural example of this. Here instead of #3COL of a graph, the hard thing to calculate is the Jones polynomial of a knot. The allowed "steps" to change the knot are the standard Reidemeister moves together with Dehn surgery. Instead of talking about graphs of small treewidth, the measure of "complicated" knots is the girth of their plane diagram. The theorem there is

P^{#P} != NP --> for any b and b', there is some D, for which all equivalent D', either have girth(D') > b*log(size(D)) + b', or a superpolynomial distance(G,H).

[anomalys-bane]

Has anyone else realized that the Madonna/Whore dichotomy does not encompass all women, because there's pretty much always been a set of women who existed outside the dichotomy since the beginning? It's the women who were considered too undesirable to fit into the Whore side and too stuck up to fit into the Madonna side, and too man hating for either side. The old maid, the spinster aunt, the ugly women, the plain boring or bookish women, the old witches, the nagging old women, the LESBIANS. And recently, the harpy feminists, the lonely cat ladies, the weird girls. they've been around as long as this dichotomy has existed in the patriarchal conscious

[fins0up]

"unpopular opinion" (that should be popular): i think public schools should start teaching ab intersex people.

[shaylogic]

Charles Bag of Tricks = Go Bag from Life

Thinking about how Charles' cricket bat sat unused beside him during the abuse flashback, and how he now has a magic one in his possession as a ghost. In a couple cameos asking about the bat, Jayden says he figures that it was a normal bat that became what it is when Charles touched it.

I suspect it's a part of his soul as a ghost, something significant from his life, representing violence and protection of the self and others.

Edwin's endless notebook is probably part of his soul, with his family crest on it from his own home/school days. George said it was important to him personally to keep the notebook in Edwin's breast pocket, over his heart, despite his conflicted feelings about his parents.

Likewise, I've had a growing theory over the past few days that I've been chatting about with friends in the Dead Gay Detectives discord server about Charles' backpack.

It seems like a go bag to grab quick on the run from volatile situations, like bullies at school and an abusive home.

Both George & Jayden have stated more than once that they would like to have seen an episode exploring the inside of Charles' backpack, as there's more to be discovered there. Of course, this is a popular curiosity of the fans, as well, but it seems poignant to me that they've spoken on this, like important backstory and worldbuilding mechanics may be involved.

If the bag is also a part of Charles' ghost inventory, part of his soul, heart, mind, etc, that's massively symbolic!

I'll share screenshots of the notes I've been making for myself for my fic-writing on this:

What if Charles ran from home and boarding school and all of it but somehow still ended up with his "friends" at the lake, where he was killed?

@wordsinhaled @dearheartdont

[burningfunobject]

S2 arcane spoilers!!

Anyone else notice how dim viktors eyes look in the new season like

And it's like this for nearly every other shot

Compared to his eyes in s1

And he's rlly out of character too

And sure that could be the trauma of EVERYTHING that's happened

But I feel like it's more than that

The hexcore is controlling him obviously so maybe we'll see him try to fight against it??

I really hope he does bc I don't think the whole magic arcane Jesus thing is really working for me tbh

OR if he doesn't manage to get free of the hexcore maybe he decides to lean into it by baking himself even more robotic buy powered by hextech or whatever the new equivalent would be

I've also been thinking maybe he'll start upsetting the chem barrens?? But it seems kinda unlikely

[anthurak]

Here’s something that might be a bit of a hot take:

I think Summer is inadvertently responsible for basically the ENTIRE current/‘modern’ plot of RWBY.

As in, Summer’s whole super-secret-sapphic-suicide mission with Raven wasn’t just the defining and inciting incident of Ruby’s and Yang’s whole character arcs. She also unwittingly kicked off all of SALEM’s current plotting and scheming.

I think basically EVERYTHING we’ve been seeing Salem do over the course of the show, from creating Grimm Hybrids to using Cinder to steal the maiden powers to even recruiting ALL of her current followers to her current plots to destabilize the kingdoms, ALL STARTED with whatever happened when she met Summer.

For one, I think it’s pretty apparent that Salem ‘learning she could do something new…’ as Ruby put it, didn’t just lead to her Grimm Hybrid program, but ALSO the ‘Grimm-Implants’ that have allowed Cinder to steal the maiden powers. Thereby giving Salem a far more direct way of getting to the Vaults and acquiring the Relics.

And that’s not even going into what ELSE Salem might have learned from her encounter with Summer and Raven. Say for example, things about the Silver Eyes that nobody else has realized…

I think 15+ years before the show started, Salem probably didn’t have ANY schemes or plots in mind and may have even been close to giving up on whatever her true goals actually are*. But then SUMMER not only inadvertently emboldened Salem, she gave her KNOWLEDGE that provided Salem the proverbial missing piece(s) she needed to finally accomplish her goals.

In fact I wouldn’t be surprised if we find out that, through Salem, Summer’s actions butterflied out to affect/inform pretty much ALL the major events that have shaped the present day of RWBY. Shaping not just Ruby’s and Yang’s lives, but maybe even Weiss’s and Blake’s too. Like perhaps Salem’s machinations as part of her new schemes are what led to the death(s) of Ironwood’s predecessors and lead to him being installed as headmaster, general and the unexpected new member of the Ozluminati, greatly altering the future of Atlas. Or maybe Salem’s recruiting of Leonardo informed waves of anti-Faunus actions in Anima via deliberate mismanagement/non-action from Leo, which in turn lead to the further radicalization of the White Fang?

Basically, given just how central she is to the stories of two of our main characters, and RWBY’s theme of decrying ‘heroic sacrifices’ and the implications that Summer’s mission is going to end up being framed as a huge MISTAKE on her part, I think it would be rather appropriate if Summer also unwittingly kicked off the ENTIRE modern plot of RWBY.**

Also you may note that I’ve been stressing that Summer unintentionally did all of this, because as I’ve stated elsewhere I think the idea that Summer might have willingly joined Salem just doesn’t work with everything we’ve learned so far. Particularly with Summer being the ‘patient zero’ of Salem’s Grimm Hybrid Program.

*Considering that Oz’s GUESS about her goal being a suicidal-cosmic-temper-tantrum is such an OBVIOUS red-herring at this point that it is baffling how some people still think it’s actually legit

**Also it would make a very fun twist considering how so much of the fandom seems to think that Oz is somehow at the center of everything.

[wellofdean]

I got this one hobby horse, and god knows I'm gonna saddle up and ride it: Supernatural was a legitimately good show and it was queer as hell in its very bones.

[hellisntreal]

With Happles, I'm about halfway on my paper Masters of the Bazaar project!! Here they are so far :D

[karebear923]

They made fun of me for thinking that Chris was just some dude who looked like Way (no one did)

But you’ll see! You’ll all see!

YOU 🫵 will be the real clowns 🤡 when it turns out that there’s no secret clone subplot and Chris really is Just Some Guy™️ who happens to look like Way

[lorynna]

this moid is the perfect example of someone who displays a flawless victim-complex and has to always be the center of attention. Here he complains with his fancy shrimp earrings about being kicked out of a restaurant when he was literally asked to move to a smaller table because someone else had already reserved the one he was sitting at for an event. He wasn't kicked out, he was asked to move.

and OF COURSE he also identifies as a lesbian! Because a male totally can be anything he wants if he just identifies as it! what else could it be! hope he dies.

when will they ever leave us alone. you're not a woman. never will be.

[chessboredom]

I keep seeing people depict Sage of Truth as Shamil's precorrupted self, and it makes me think that him being the holder of the Soul Jam of Knowledge as seen in his own gacha intro just didn't matter to any of you. 😭 The lore is just not loring.

And I pretty much prefer the Korean titles because they makes more sense.

Sage of Truth is literally the equal result of Pure Vanilla Cookie becoming The Hermit of Truth(Kr) being devoured by Deceit except he was dawned upon the truth.

Pioneer of Knowledge(Kr): Black, gold and blue (Choosing Deceit)-> Beast Shamil: Black and blue (Truth shined upon him)-> Sage of Truth: White, gold and blue

And Sage coming from a skin showcase means he came from an AU where he was radically changed by the Truth, but he still has his expressive mannerisms as a Cookie of Deceit. (Theatre kid lol)

Also, this is another fault of the English Translations. It's called "Diverging Paths" when it's actually called "A Possibility of Another World" in Korean. Both skins are NOT from the canon timeline, which could also be seen in their skin story, because most of the time that's what in-game skins are for. For Alternate Universes ot "possibilities" or cute outfits with stories.

I even have a couple of posts showing the differences. (Link, link)

[lune-moon-nuit]

Someone commented, "How can people say their relationship is toxic? She literally saved his life," under someone's TikTok that highlighted how Mike was willing to sacrifice his own life to spare Dustin's baby teeth (and thus proving the lack of values ​​he has towards himself and his own life) and he would be dead if El wasn't there.

“But she saved him.” Yeah… and that’s exactly why it’s complicated — and why their relationship might not be as healthy as it seems.

Let’s talk about the psychological weight behind this kind of dynamic.

When someone literally saves your life — especially during a traumatic, high-stakes moment — it creates a powerful emotional imprint. The person you associate with relief, safety, and survival can easily become idealized in your mind. But here’s the thing: that’s not the same as love.

The savior complex / emotional debt syndrome

When one person saves another — especially in an extreme (traumatic or life-threatening) moment — it can create a deep emotional debt, where the saved person feels they owe something profound to their savior. This can evolve into a distorted emotional attachment, which is often perceived as love.

Emotional debt ≠ romantic connection

Mike may feel like he owes El everything because she saved him. That sense of emotional debt can evolve into what looks like devotion, loyalty, even “love” — but at its core, it’s about guilt and obligation, not emotional compatibility or genuine desire.

When someone stays in a relationship because they feel they “should,” not because they want to, it creates an imbalance. It’s not real intimacy — it’s emotional submission masked as love.

Gratitude/savior love syndrome

Though not a clinical term, in psychological or romantic literature, people sometimes talk about “savior love,” where the attraction doesn’t stem from real romantic desire, but from a mix of gratitude, admiration, emotional dependence, and relief. This type of feeling can be very intense — but also fleeting — because it’s rooted in a dramatic event rather than a truly built relationship.

The “Savior Complex” and distorted attachment

This is a textbook example of what psychology often refers to as emotional debt attachment or “savior love.” When a person is rescued in a moment of trauma, they might unconsciously attach themselves to their savior, mistaking relief and admiration for romantic love.

It can feel very intense — even destiny-like — but it often fades or becomes damaging once the adrenaline of survival fades and there’s no emotional foundation strong enough to carry the relationship.

Transference

This is an unconscious mechanism where someone projects intense emotions — often linked to important figures from their past (like parents) — onto another person. In this case, the girl who saves becomes an idealized figure, almost a symbolic “savior,” and the love felt isn’t always based on who she truly is, but on what she represents emotionally: safety, life, salvation.

Transference and idealization

Mike might be projecting deep emotional needs onto El — needs for protection, unconditional presence, and safety — especially if his space to explore those needs safely disappeared one day before meeting El. In psychology, this is called transference: when we assign symbolic meaning to someone based on what they represent to us emotionally, not who they really are.

In this case, El isn’t just a person. She’s the girl who saved his life. She’s “the one who pulled him out.” She's a "superhero". And when someone becomes a symbol, not a person, the relationship loses balance. She becomes untouchable, unquestionable — and that is not mutual love. That’s idolization wrapped in trauma.

Post-traumatic emotional confusion

When someone goes through a traumatic experience, the brain seeks symbols of safety to hold onto. The person who was present in the moment of greatest fear — and helped overcome it — can become unconsciously associated with feelings of love, simply because they were the source of relief.

Post-traumatic emotional confusion

Trauma warps emotional perception. When we’re vulnerable, the brain clings to anything that feels like safety. It makes sense that Mike might associate El with peace and survival — but that association doesn’t always translate into a sustainable, reciprocal relationship. It can create a bond rooted in fear, not in freedom.

So yes, she saved him. And that matters. But it’s also part of the problem.

Because "You saved me, so I have to love you" is not romantic — it’s tragic. It’s a trap. It’s the kind of belief that keeps people in relationships that look loyal on the outside but are emotionally repressive on the inside.

Mike deserves to choose love freely — not stay in a relationship because he feels indebted to someone who once saved him. And El, too, deserves someone who loves her for her, not for what she did for him.

Gratitude is not love. Debt is not devotion. And saving someone doesn’t mean they owe you their heart.

In summary: This is most likely a post-traumatic attachment, combined with emotional transference and a sense of emotional debt, all being misinterpreted as love. It’s not necessarily fake or illegitimate — but it’s often an idealized kind of love, born from survival rather than a deep, mutual emotional connection.

If you’re looking for simpler terms to describe it: “Rescue-based emotional attachment” or “post-traumatic gratitude love” can work — even if they’re not official clinical expressions, they’re still meaningful and accurate.

And yes, it can become a toxic relationship, or at the very least unhealthy, if one of them stays out of guilt, debt, or gratitude instead of sincere love or mutual desire.

Here’s why:

Emotional imbalance If he stays out of obligation (because she saved him), and she believes he truly loves her, then there’s a fundamental emotional lie at the core. There’s no balance: one gives out of love, the other out of duty. And even if the intention is good (not wanting to hurt her, wanting to repay what she did), it builds a relationship based on a false premise.

The fear of hurting or “betraying” her He may feel like he owes her his life, and therefore has no right to leave, even if he’s not happy or in love. This chronic guilt can lead to a form of emotional submission, which will make both of them miserable over time.

The myth of “I owe them everything” This is a common mental trap: believing that because someone saved you, you must stay loyal to them for life — even at the cost of your own freedom or inner truth. It becomes a form of perpetual emotional debt that prevents you from listening to what you really want.

Possible consequences:

He may repress his true feelings, or even fall into depression.

She may feel that something is wrong, even if he never says it.

The relationship can become suffocating, built on illusion, and eventually lead to resentment, frustration, or even repressed anger.

The factual and logical consequences I describe here include all the relationships based on this dynamic. But the fact is that every consequence cited here and highlighted is evident and present in season 4 (especially in their argument scenes) speak volumes.

It’s not necessarily toxic at the beginning, but it becomes toxic if it continues without honesty.

Being grateful is one thing. Sacrificing your truth in the name of that gratitude is another.

Conclusion :

Yes, El did save Mike’s life — and that’s incredibly important. But love born out of a life-saving moment doesn’t automatically make a relationship healthy. In fact, staying with someone because you feel emotionally indebted to them, rather than truly in love, can actually be a sign of an unhealthy or even toxic dynamic.

There’s a psychological phenomenon where intense gratitude, admiration, or even trauma-bonding gets confused with love. When someone saves your life — literally or emotionally — they can become symbolically larger than life to you. You feel like you owe them everything. But that doesn’t necessarily mean you're romantically compatible, or even emotionally fulfilled in the long run. It just means they became a figure of safety during a moment of terror or pain.

If Mike is with El partly because he feels like he has to be, because she saved him, then that’s not a free, mutual love. That’s emotional debt. That kind of imbalance can lead to deep internal conflict — guilt, suppression of real feelings, fear of hurting the other person — and that’s where toxicity begins to seep in, even if no one means harm.

Gratitude isn’t the same as love. And saving someone’s life doesn’t mean they owe you a relationship forever. If a relationship survives solely on the basis of a heroic act from the past — and not on genuine, reciprocal connection in the present — then it might not be as healthy as it looks from the outside.

[alicentflorent]

I think the reason why Alys Rivers is the character I have enjoyed most this season is because she brings a unique, less privileged perspective (much like mysaria) and most importantly, She’s the only character female character that they have allowed to be morally grey and maintain a sense of agency. Unlike the main female characters who sit on councils wringing their hands unable to make decisions or being stripped of any agency during the war efforts that they should be leading. Alys clearly has the upper hand in her dynamic with daemon, she’s drugging him and invading his mind which definitely not a moral thing to be doing even if it feels like daemon deserves it. She exploits the image that someone like daemon might have of her - a powerless, bastard woman - when we know she is far from powerless. She’s also shown to care about the crimes being inflicted on the riverlands and it’s people, her people, which shows she has motive and good reason for messing with daemon, she’s not just tormenting him for fun. Alys is allowed to be a complex character with questionable methods and unclear motives but at the same time she’s a character that we can root for, an underdog who understands the cost of war on the innocents.

[loudclan-clangen]

im sure Eklutna will be okay. she will be so okay she will immediately take her newborns to the Crude Oil Moonpool and baptise them like god intended.

better yet, she will insist on giving birth by the pool, since its the closest way of interacting with StarClan. and they will help her. surely it will end well.

(pls dont let your cats give birth and dip their infants in crude oil.)

You're so right. Eklutna CAN'T die. She has PLANS.

If she dies who's gonna put her kids in the world's most dangerous dunk tank??? Also, yes, she HAS visited the Black Water Pool for a blessing, and yes, the healers were pissed, but they just don't understand. They don't understand that she is Starclan's special little girl.

(P.S. thank you for including your own disclaimer. Wonderful.)

Yes. Eklutna. All the time. She was one of the cats who first discovered it, then after getting into a fight with the rest of Ghostclan over whether or not they should tell the other clans about the Black Water Pool she limped her bleeding ass there to give herself a blessing before she continued up the mountain to Loudclan. Now she's taken a dip while pregnant to get Starclan to bless her unborn kits, which is a BIG taboo. But what are the healers to do? How do you punish someone for participating too enthusiastically in your state sponsored religion? She's not the only one to ever do it, of course, she's just kinda the most egregious about it.

-- Eklutna, 2024

*:･ﾟ✧*:･ﾟ✧Drama✧･ﾟ: *✧･ﾟ:*

[sheiireen]

Izuku and his emotions

I keep seing posts about how Izuku's reactions are disappointing to some readers, because he does not cry or react deeply in specific scenarios - but it's more complex than you think.

First of all, let me remind you that Izuku is an unreliable narrator.

He lies when blackwhip was triggered for the first time:

2. During Katsuki's apology, he asks Izuku if he remembers what he said before he was attacked by Shigaraki, and he says no.

Really? The boy who 'frets over every small detail', who remembers details about quirks and character traits of all the people he meets? He would not forget such an important moment. He purposely avoids to answer.

Izuku is stubborn. He tries to convince himself that he is fine all the time. We have seen this throughout the whole story.

Then - that one time in ch425 when he does try to reach out to his friends, he is rejected. We see how sad he is but we don't get a glimpse into his mind. You can clearly see the implications of how frustrated and sad he is, even if he does not voice his emotions.

He saw Katsuki's dead body. There is no way he would be over this (he must feel responsible and most likely blames himself because he was late...).

He won the fight against Shigaraki and we saw glimpses of his feelings about the fight at the hospital - how disappointed he must feel that he couldn't save Shigaraki. He wanted to be a hero who saves people. In ch 425 right after Fuwa says Aizawa showed them why they became heroes, we get this image:

He looks so pensive and absent-minded...after all, he lost his purpose and his powers.

All of this does not mean that we will not know how Izuku truly feels. He needs to get his emotions out, and he will. Just because we don't see Izuku's reactions right after something happens, doesn't mean that he is indifferent or doesn't care.

Katsuki will play a big part in this. Even though he has reached out to Izuku, Katsuki didn't get through to Izuku's inner self. Izuku didn't "take his hand", yet. This is why we will get a Katsuki and Izuku conversation aka DvK3. Katsuki will be the one who initiates it, because he knows Izuku best and he will reach out his hand to Izuku again, but this time, Izuku will take it.

We will get an intense, heartfelt conversation and it will be cathartic for Izuku.