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4225f277-bda7-4df6-a612-1886a27b1dde | StampyAI/alignment-research-dataset/eaforum | Effective Altruism Forum | Map of maps of interesting fields
I love seeing cool visualizations (maps) of interesting intellectual fields. I’m also a big fan of making [lists](https://forum.effectivealtruism.org/posts/6trt8mTsKfqJJbfJa/suggestion-eas-should-post-more-summaries-and-collections)!
Accordingly, I compiled a bunch of maps of fields that I thought were interesting. Please comment if you know of more such maps, and I’ll include them.
Scott Alexander’s map of Effective Altruism ([2020](https://slatestarcodex.com/blog_images/ea_imagemap8.html))
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[](https://substackcdn.com/image/fetch/f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2Fcb6489df-1353-4467-af61-89a3a2b58ad7_1600x1034.png)
mariekedev’s mindmap of EA organisations ([2023](https://forum.effectivealtruism.org/posts/Avi9XgSikH5BdHzKu/mindmap-with-overview-of-ea-organisations-via-tinyurl-com))
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[](https://substackcdn.com/image/fetch/f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F20913dd1-158e-48aa-957b-1c7fe5d522cd_621x797.png)
Hamish Doodles’ [aisafety.world](http://aisafety.world/) ([2023](https://forum.effectivealtruism.org/posts/mDG49CJyxzeN99ELz/aisafety-world-is-a-map-of-the-ais-ecosystem))
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[](https://substackcdn.com/image/fetch/f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2Fc929fc7f-8f4d-4287-9101-b6b7aa77d7f8_1600x1006.png)
On a side note, I would be very keen for someone to create a similar map of relevant organizations in the [biosecurity & pandemic preparedness](https://forum.effectivealtruism.org/topics/biosecurity-and-pandemics) space. I plan to post a minimum viable product (bullet point list) soon.
James Lin’s map of biosecurity interventions ([2022](https://forum.effectivealtruism.org/posts/npNt43QRnaRNRixXK/map-of-biosecurity-interventions))
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[](https://substackcdn.com/image/fetch/f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2Fa03dd670-dbd3-4633-977b-aed645c34095_1600x718.png)
Scott Alexander’s map of the rationality community ([2014](https://slatestarcodex.com/2014/09/05/mapmaker-mapmaker-make-me-a-map/))
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[](https://substackcdn.com/image/fetch/f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2Feef5f780-b90c-4192-b60b-5c925ad3b904_1600x875.png)
Dan Elton’s map of progress studies ([2021](http://www.moreisdifferent.com/2021/11/19/map-of-progress-studies))
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[](https://substackcdn.com/image/fetch/f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2Fbafe257a-e4d6-46d9-be64-7e52fc66d7ee_1600x958.png)
Ada Nguyen’s map of the Longevity Biotech Landscape ([2023](https://www.adanguyenx.com/longevity))
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[](https://substackcdn.com/image/fetch/f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F03317458-38b5-441b-80e7-8f3e0fce5684_2500x5972.png)
Nadia Asparouhova’s map of climate tribes ([2022](https://nadia.xyz/climate-tribes))
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[](https://substackcdn.com/image/fetch/f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F0b780f96-4f73-4536-8340-e35eedbc630c_844x1078.png)
Samuel Arbesman’s Catalog of New Types of Research Organizations ([2023](https://arbesman.net/overedge/))
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This one is more of a long list, but I thought it was very interesting nonetheless!
[](https://substackcdn.com/image/fetch/f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F065ffcbe-0dac-405a-b842-784a8502d46d_1600x1157.png)
Honorable mentions
------------------
These are other mapping efforts that met my "I'm curious about it" but not my "This is super interesting" bar. Some of them also seem outdated.
* xkcd’s map of online communities ([2010](https://xkcd.com/802/))
* Julia Galef’s map of the Bay Area memespace ([2013](https://www.lesswrong.com/posts/WzPJRNYWhMXQTEj69/a-map-of-bay-area-memespace))
* Joe Lightfoot’s The Liminal Web: Mapping An Emergent Subculture Of Sensemakers, Meta-Theorists & Systems Poets (2021 & updated in [2023](https://www.joelightfoot.org/post/the-liminal-web-mapping-an-emergent-subculture-of-sensemakers-meta-theorists-systems-poets))
Shoutout to Rival Voices, who created a similar-ish collection of maps ([2023](https://rivalvoices.substack.com/p/previous-maps)), and Nadia Asparouhova, who wrote a more meta-level post about Mapping digital worlds ([2023](https://nadia.xyz/mapping-digital-worlds)). |
b24806d1-559b-475e-bf96-34c5fd631b3f | trentmkelly/LessWrong-43k | LessWrong | On what research policymakers actually need
I saw this guest post on the Slow Boring substack, by a former senior US government official, and figured it might be of interest here. The post's original title is "The economic research policymakers actually need", but it seemed to me like the post could be applied just as well to other fields.
Excerpts (totaling ~750 words vs. the original's ~1500):
I was a senior administration official, here’s what was helpful
> [Most] academic research isn’t helpful for programmatic policymaking — and isn’t designed to be. I can, of course, only speak to the policy areas I worked on at Commerce, but I believe many policymakers would benefit enormously from research that addressed today’s most pressing policy problems.
> ... most academic papers presume familiarity with the relevant academic literature, making it difficult for anyone outside of academia to make the best possible use of them.
> The most useful research often came instead from regional Federal Reserve banks, non-partisan think-tanks, the corporate sector, and from academics who had the support, freedom, or job security to prioritize policy relevance. It generally fell into three categories:
>
> 1. New measures of the economy
> 2. Broad literature reviews
> 3. Analyses that directly quantify or simulate policy decisions.
>
> If you’re an economic researcher and you want to do work that is actually helpful for policymakers — and increases economists’ influence in government — aim for one of those three buckets.
New data and measures of the economy
> The pandemic and its aftermath brought an urgent need for data at higher frequency, with greater geographic and sectoral detail, and about ways the economy suddenly changed. Some of the most useful research contributions during that period were new data and measures of the economy: they were valuable as ingredients rather than as recipes or finished meals...
>
> These data and measures were especially useful because the authors made underlying numbers avail |
598970ee-20b4-4993-a0d0-d5c12eb2f835 | trentmkelly/LessWrong-43k | LessWrong | Rhetoric for the Good
The topics of rationality and existential risk reduction need their own Richard Dawkins. Their own Darwin. Their own Voltaire.
Rhetoric moves minds.
Students and masochists aside, people read only what is exciting. So: Want to make an impact? Be exciting. You must be heard before you can turn heads in the right direction.
Thus, I've decided to try harder and actually put effort into the quality of my writing instead of just cranking stuff out quickly so I can fill in inferential gaps and get to the cutting edge of the research subjects I care about.
That's why I asked LWers for their picks of best nonfiction writing on Less Wrong.
It's also why I've been reading lots of good science writing, focusing on those who manage to be exciting while covering fairly complex subjects: Dawkins, Sagan, Gleick, Zimmer, Shermer, Ramachandran, Roach, Sacks, Hawking, Greene, Hofstadter, Penrose, Wilson, Feynman, Kaku, Gould, Bryson, Pinker, Kurzban, and others.
I've also been re-reading lots of books and articles on how to write well: Keys to Great Writing, Style: Lessons in Clarity and Grace, Elements of Style, On Writing Well, The Classic Guide to Better Writing, The Book on Writing, Telling True Stories, Writing Tools, Ideas into Words, The Chicago Guide to Communicating Science, A Field Guide for Science Writers, Six Rules for Rewriting, Writing, Briefly, and Singularity Writing Advice. (Conversations with Eliezer also helped.)
I don't know if I can become the Voltaire of rationality and existential risk reduction, but it seems worth a shot. Every improvement in writing style is beneficial even if my starry goal is never met. Also, it appears I produce better writing without really trying than most people produce with trying. (If you've ever had to grade essays by honors English seniors, you'll know what I mean.) I expect to gain more by striving where I already excel than by pushing where I have little natural talent.
(I won't try to write everything well. Sometimes I s |
d6269362-67c5-436f-8cf6-33e174a9ca52 | trentmkelly/LessWrong-43k | LessWrong | Meetup : Melbourne Rationality Dojo, January - Lightning Talks
Discussion article for the meetup : Melbourne Rationality Dojo, January - Lightning Talks
WHEN: 15 January 2017 04:00:00PM (+1100)
WHERE: Janet Lady Clarke Memorial Rotunda, Queen Victoria Garden, Melbourne
The next meetup for Less Wrong Melbourne will be outdoors (weather permitting). on Sunday 15 January 2017.
THIS MONTH: Lightning talks. A number of speakers (including you?) at 5 to 10 min each (including questions) on the art of rationality, or something to improve our understanding of the world.
BRING:
• An idea or question to share.
• Perhaps a snack to share. (Healthy snack choice is optional but welcome.)
At about 6:30pm some of us generally head for dinner in the CBD, and you are welcome to join us.
WHEN: 4pm to 6:30pm (and then optional dinner). The meeting proper will begin at 4:15 sharp.
WHERE: Janet Lady Clarke Memorial Rotunda, Queen Victoria Gardens. https://goo.gl/maps/5dni87guhcU2 We'll meet a stone's throw from the rotunda - same side of the road, just over the walking path. If you have any trouble finding the venue on the day, please text or call Chris on 0439 471 632.
WHO ARE THESE PEOPLE AND WHY DO THEY MEET?
The Less Wrong Melbourne Rationality Dojos are self-improvement sessions for those committed to the art of rationality and personal growth. We welcome new members who are interested in exploring rationality. We're "aspiring rationalists" and always open to learn.
Discussion article for the meetup : Melbourne Rationality Dojo, January - Lightning Talks |
34246193-79f8-4c2e-b9da-7fbfb7862212 | trentmkelly/LessWrong-43k | LessWrong | Something to Protect
In the gestalt of (ahem) Japanese fiction, one finds this oft-repeated motif: Power comes from having something to protect.
I'm not just talking about superheroes that power up when a friend is threatened, the way it works in Western fiction. In the Japanese version it runs deeper than that.
In the X saga it's explicitly stated that each of the good guys draw their power from having someone—one person—who they want to protect. Who? That question is part of X's plot—the "most precious person" isn't always who we think. But if that person is killed, or hurt in the wrong way, the protector loses their power—not so much from magical backlash, as from simple despair. This isn't something that happens once per week per good guy, the way it would work in a Western comic. It's equivalent to being Killed Off For Real—taken off the game board.
The way it works in Western superhero comics is that the good guy gets bitten by a radioactive spider; and then he needs something to do with his powers, to keep him busy, so he decides to fight crime. And then Western superheroes are always whining about how much time their superhero duties take up, and how they'd rather be ordinary mortals so they could go fishing or something.
Similarly, in Western real life, unhappy people are told that they need a "purpose in life", so they should pick out an altruistic cause that goes well with their personality, like picking out nice living-room drapes, and this will brighten up their days by adding some color, like nice living-room drapes. You should be careful not to pick something too expensive, though.
In Western comics, the magic comes first, then the purpose: Acquire amazing powers, decide to protect the innocent. In Japanese fiction, often, it works the other way around.
Of course I'm not saying all this to generalize from fictional evidence. But I want to convey a concept whose deceptively close Western analogue is not what I mean.
I have touched before on the idea that |
e1d2bd7d-ec06-4b05-9da7-9ea4c6f2a201 | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | If interpretability research goes well, it may get dangerous
I've [historically](https://twitter.com/robbensinger/status/1602835145488113664) been pretty publicly [supportive](https://www.lesswrong.com/posts/3pinFH3jerMzAvmza/on-how-various-plans-miss-the-hard-bits-of-the-alignment) of interpretability research. I'm still supportive of interpretability research. However, I do not necessarily think that all of it should be done in the open indefinitely. Indeed, insofar as interpretability researchers gain understanding of AIs that could significantly advance the capabilities frontier, I encourage interpretability researchers to keep their research [closed](https://www.lesswrong.com/posts/tuwwLQT4wqk25ndxk/thoughts-on-agi-organizations-and-capabilities-work).
I acknowledge that spreading research insights less widely comes with real research [costs](https://intelligence.org/2018/11/22/2018-update-our-new-research-directions/#section3-7). I'd endorse building a cross-organization network of people who are committed to not using their understanding to push the capabilities frontier, and sharing freely within that.
I acknowledge that public sharing of research insights could, in principle, both shorten timelines and improve our odds of success. I suspect that [isn’t the case in real life](https://www.lesswrong.com/posts/vQNJrJqebXEWjJfnz/a-note-about-differential-technological-development).
It's much more important that blatant and direct capabilities research be made private. Anyone fighting for people to keep their AI insights close to the chest, should be focusing on the capabilities work that's happening out in the open, long before they focus on interpretability research.
Interpretability research is, I think, some of the best research that can be approached incrementally and by a large number of people, when it comes to improving our odds. (Which is not to say it doesn't require vision and genius; I expect it requires that too.) I simultaneously think it's entirely plausible that a better understanding of the workings of modern AI systems will help capabilities researchers significantly improve capabilities. I acknowledge that this sucks, and puts us in a bind. I don't have good solutions. [Reality doesn't have to provide you any outs.](https://www.lesswrong.com/posts/sYgv4eYH82JEsTD34/beyond-the-reach-of-god)
There's a tradeoff here. And it's not my tradeoff to make; researchers will have to figure out what they think of the costs and benefits. My guess is that the current field is not close to insights that would significantly improve capabilities, *and* that growing the field is important (and would be hindered by closure), *and also* that if the field succeeds to the degree required to move the strategic needle then it's going to start stumbling across serious capabilities improvements before it saves us, and will need to start doing research privately before then.
I reiterate that I'd feel ~pure enthusiasm about a cross-organization network of people trying to understand modern AI systems and committed not to letting their insights push the capabilities frontier.
My goal in writing this post, though, is mostly to keep the Overton window open around the claim that there is in fact a tradeoff here, that there are reasons to close even interpretability research. Maybe those reasons should win out, or maybe they shouldn't, but don't let my praise of interpretability research obscure the fact that there are tradeoffs here. |
b024d61a-ec9e-4d7c-8e22-2d9460044d4e | trentmkelly/LessWrong-43k | LessWrong | Which animals can suffer?
In this recent paper, the author study whether the octopi can suffer. Well, how are you going to answer that? Ask him?
It turns out there are two ways you can decide whether the animal can suffer or cannot. The first is simple. You just define that vertebrates can suffer (therefore experiments with them should follow ethical regulations) and that all other animals can not. Simple, but kind of arbitrary. And yes, obviously octopi can't suffer then by definition.
Another approach is functional. If an animal can learn to avoid pain (not just avoid it right now, but memorize what happened at given conditions and avoid these conditions), then the animal can suffer. According to this definition and experiments conducted in the abovementioned paper, octopi can suffer - so should be treated as vertebrates.
The second definition looks more logical to me. However, there is one problem with this one as well. If we make it purely functional, we should agree that an artificial neural network under the procedure of reinforcement learning suffers. Or instead of introducing an arbitrary threshold of "vertebrate - invertebrate", we introduce not less arbitrary threshold "biological neurons - artificial neurons".
Does it mean we should worry about the suffering of our models? And this is not the whole story. From the mathematical point of view, many things can be mapped onto the learning process. Does it all involve suffering?
So far, I see the following possible answers. Maybe there are more, that I am unaware of.
1. Indeed all abovementioned suffers.
2. There is a magical biological threshold.
3. The ability to suffer is determined in an anthropocentric way. I think that a cat suffers because I see it and I feel really sorry, I can relate to it. For octopus, I kind of can relate too, but less than to a cat. For code running on my laptop, I can't relate at all. An interesting corollary would be that a teddy bear suffers.
|
2f4d2bd8-89df-4c49-aef1-ca3cc0ad60f3 | trentmkelly/LessWrong-43k | LessWrong | Assuming Nails
Tangential followup to Defeating Ugh Fields in Practice.
Somewhat related to Privileging the Hypothesis.
Edited to add:
I'm surprised by negative/neutral reviews. This means that either I'm simply wrong about what counts as interesting, or I haven't expressed my point very well. Based on commenter response, I think the problem is the latter. In the next week or so, expect a much more concise version of this post that expresses my point about epistemology without the detour through a criticism of economics.
At the beginning of my last post, I was rather uncharitable to neoclassical economics:
> If I had to choose a single piece of evidence off of which to argue that the rationality assumption of neoclassical economics is totally, irretrievably incorrect, it's this article about financial incentives and medication compliance.... [to maintain that this theory is correct] is to crush reality into a theory that cannot hold it.
Some mistook this to mean that I believe neoclassical economists honestly, explicitly believe that all people are always totally rational. But, to quote Rick Moranis, "It's not what you think. It's far, far worse." The problem is that they often take the complex framework of neoclassical economics and believe that a valid deduction within this framework is a valid deduction about the real world. However, deductions within any given framework are entirely uninformative unless the framework corresponds to reality. But, because such deductions are internally valid, we often give them far more weight than they are due. Testing the fit of a theoretical framework to reality is hard, but a valid deduction within a framework feels so very satisfying. But even if you have a fantastically engineered hammer, you cannot go around assuming everything you want to use it on is a nail. It is all too common for experts to assume that their framework applies cleanly to the real world simply because it works so well in its own world.
If this concept doesn't |
24cf8a35-97fe-43cf-a0eb-c105c6f36e4b | trentmkelly/LessWrong-43k | LessWrong | How to Give Coming AGI's the Best Chance of Figuring Out Ethics for Us
[Note: this is a slightly edited version of an essay I entered into the AI Impacts essay contest on the Automation of Wisdom and Philosophy - entries due July 14, 2024. Crossposted to the EA Forum.]
TL;DR
A few possible scenarios are put forth to explore the likely ranges of time we may have from when the first AGI comes online to when an “ethics-bound” AGI would need to be ready to police against any malicious AGI. These ranges are compared against possible time ranges for an AGI to figure out an ethical system for it to operate under. Ethics aren’t expected to be solved “by default” because the first AGI's likely won’t have the ability to feel pain or emotions, have ethical intuitions of their own, or the ability to “try things on” and check them against their own experiences like humans can. Some potential pitfalls in creating a consistent system of ethics, and some “extreme” situations that may test its limits are presented. A list of recommendations is given for what we should do before the first AGI comes online so it can hone in as quickly as possible on a viable ethical system to operate under. Some possible prompts to get an AGI started to figure out ethics are included.
Introduction
It seems likely that Artificial General Intelligence (AGI) will be developed within the next 20 years if not significantly before. Once developed, people will want AGI’s to do things for them in the world, i.e., become agentic. AGI’s with limited agency can likely be relatively straightforwardly guard railed from doing significant harm, but as AGI’s are given more autonomy in decision making over a wider range of situations, it’ll become vital that they have a good handle on ethics so they can avoid bringing about potentially massive amounts of value destruction. This is under the assumption, of course, that alignment is “solved” such that an AGI can be made to align its actions with any ethical framework at all. Although there has been some work and progress in “ma |
07a91e1e-0e68-421e-b2e8-20a19d33a94e | trentmkelly/LessWrong-43k | LessWrong | Fear of centralized power vs. fear of misaligned AGI: Vitalik Buterin on 80,000 Hours
Vitalik Buterin wrote an impactful blog post, My techno-optimism. I found this discussion of one aspect on 80,00 hours much more interesting. The remainder of that interview is nicely covered in the host's EA Forum post.
My techno optimism apparently appealed to both sides, e/acc and doomers. Buterin's approach to bridging that polarization was interesting. I hadn't understood before the extent to which anti-AI regulation sentiment is driven by fear of centralized power. I hadn't thought about this risk before since it didn't seem relevant to AGI risk, but I've been updating to think it's highly relevant.
[this is automated transcription that's inaccurate and comically accurate by turns :)]
> Rob Wiblin (the host) (starting at 20:49):
>
> what is it about the way that you put the reasons to worry that that ensured that kind of everyone could get behind it
>
> Vitalik Buterin:
>
> [...] in addition to taking you know the case that AI is going to kill everyone seriously I the other thing that I do is I take the case that you know AI is going to take create a totalitarian World Government seriously [...]
>
> [...] then it's just going to go and kill everyone but on the other hand if you like take some of these uh you know like very naive default solutions to just say like hey you know let's create a powerful org and let's like put all the power into the org then yeah you know you are creating the most like most powerful big brother from which There Is No Escape and which has you know control over the Earth and and the expanding light cone and you can't get out right and yeah I mean this is something that like uh I think a lot of people find very deeply scary I mean I find it deeply scary um it's uh it is also something that I think realistically AI accelerates right
One simple takeaway is that recognizing and addressing that motivation for anti-regulation and pro-AGI sentiment when trying to work with or around the e/acc movement. But a second is whether |
f5ce067e-9d2b-48dd-916a-53b0ddaf11b8 | trentmkelly/LessWrong-43k | LessWrong | Launching the Forecasting AI Progress Tournament
I'm excited to let you know that Metaculus is launching the Forecasting AI Progress Tournament, a large-scale forecasting tournament with $50,000 in prizes. Participation is open to all.
The tournament is intended to provide a proof of concept for probabilistic forecasting for key questions related to the future of AI. The aim is to generate a massive dataset of AI forecasts over a range of time-frames, develop models to optimally aggregate those forecasts, and use these to build a map of the future of AI.
Those interested may pre-register here to stay up to date on the tournament. Participation will help contribute to elucidating key questions about the future of AI. The tournament starts next week and ends mid-April of 2021.
Prizes will be awarded for the most accurate predictions made about AI advances over 6 month, 1-year, 2-year timeframes. Forecasts over longer timeframes (5+ years) will also be made. Additional prizes are awarded for well-reasoned arguments and/or appropriate models for predictions on questions, and analyses of predictions that are made in the course of the tournament. |
3aef8140-5516-4e9b-88d7-b19082851b1c | trentmkelly/LessWrong-43k | LessWrong | Alienation and meta-ethics (or: is it possible you should maximize helium?)
(Cross-posted from Hands and Cities)
In a previous post, I tried to gesture at the possibility of a certain kind of wholeheartedness in ethical life. In this post, I want to examine a question about meta-ethics that seems to me important to this wholeheartedness: namely, whether you can be completely alienated from what you should do. By this I mean: is it possible for what you should do to hold no appeal whatsoever, even given arbitrarily deep and full understanding of reality and of the consequences of your actions? I discuss how three different meta-ethical questions approach this question, and why the differences matter for wholeheartedness.
(Note: this post plays fast and loose with a number of philosophical issues that it would take a lot of extra work to tease apart and pin down precisely; my aim is mostly just to gesture at a general picture that I think is important, not to set it out rigorously).
(Second note: the types of “shoulds” and ”normative facts” I have in mind here are the ones of all-things-considered practical deliberation – deliberation that encompasses both moral and non-moral considerations. Thus, the question of whether I should eat this cupcake, go back to grad school, clip my toenails, etc, are all normative questions in this sense, but not necessarily moral ones).
I.
On some versions of normative realism, the answer to the question above is yes: it might turn out that what you should do, according to the true normativity written into the fabric of the reality, is fully and forever alien from anything you or any other human cares about. It might be, for example, that there is an over-riding moral imperative to maximize the amount of helium gas, or to paint your mother a specific shade of pastel blue, or to eat a prime number of bricks every other Tuesday. Future philosophers might discover this fact (though given their current reliance on intuitions closely tied to actual patterns of human concern, their prospects for finding the trut |
6551c5ec-4d22-49f8-8f54-20225f11c32f | trentmkelly/LessWrong-43k | LessWrong | Lying to Save Humanity
You are an astronaut on an expedition to Mars when you encounter an Alien. Luckily, it speaks English, and you begin an exchange of cultures: history, art, technology, and food, whereupon the Alien interrupts:
“Our species has strong feelings about ice cream flavors. Tell me, what is your favorite flavor? If you answer chocolate, all is well, and our races can coexist. If you answer vanilla, we will annihilate every human.”
This is a dilemma, for you strongly prefer vanilla ice cream, but you have preferences against both lying and having humanity annihilated. You consider your moral framework.
Path 1
“I like chocolate ice cream,” you lie. One tiny moral lapse is worth saving all humanity.
“Aha! Liar!” screams the Alien. “The question was a test, for we have mind-reading technology. How do you think I learned English so quickly? In retribution for your treachery, we shall destroy all humans, starting with you.”
Outcome: Humanity annihilated.
Path 2
“I like vanilla ice cream,” you answer truthfully. You would never compromise your moral principles. Also, you aren’t convinced these Aliens could defeat your nukes.
“Very brave of you,” says the Alien. “What our species truly values is integrity. As a reward, I gift you this machine, which provides limitless energy, faster-than-light travel, immortality, and a few other things…”
Outcome: Humanity enters a trillion-year golden age.
Path 3
“I like…” you begin, then draw your phaser pistol and shoot the Alien in the face.
The Alien collapses, dead. You stand over its smoking corpse, not feeling any remorse, for you don’t think non-human species carry moral weight, and you are pleased at finding a way out of your ethical dilemma.
You wait to see if other Aliens might spring from behind the rocks, or if you are atomized by an orbital laser. Nothing happens.
Eventually, you decide there’s no point in wasting good protein, and you’re not vegetarian. You haul the Alien back to base, announce “Looks like |
2be0754c-9712-4d54-a0a7-04666eb3d7d4 | trentmkelly/LessWrong-43k | LessWrong | Arguments against constructivism (in education)?
Seems to me that constructivist education is already "in the water supply" of Less Wrong. My tweet-length definition of constructivism would be "teaching students so that they construct mental gears-level models of the subject, as opposed to just memorizing the teachers' passwords"; and I assume most people here would agree that this is the right thing to do.
My question is how much this is a fair characterization - or perhaps it is abstracting away some very important stuff - and what are the best arguments against constructivism. Because, per Litany of Tarski, if constructivism is wrong, I want to believe it is wrong; although frankly I do not expect it to be fundamentally wrong, more like constructivists having some typical blind spots, in which case I want to know where they are.
The motivation for the question is to help a (constructivist) teacher do their job well, or to help a teacher unfamiliar with constructivism learn to do their job better. Therefore if the objection is e.g. that Piaget was historically wrong about some technical detail, the importance of this objection depends on how likely today someone is to make a mistake in designing their lessons because of that fact.
Here are my attempts to steelman the opposition to constructivism:
* Excesses done in the name of constructivism, such as spending four years letting kids discover how addition of natural numbers works. (Either because according to the extreme interpretation, any guidance from the teacher would be a sin against the child's independent construction of mental models; or because there is always one more "weird trick" that according to internet some natives used to add two numbers, and because having more models is always better, it is absolutely necessary to also teach kids this trick. As a result, 10 years old kids know dozen different ways how to add 15+8, each of those takes them five minutes, they will make a mistake in half of them, and they haven't heard about multiplication ye |
c0c55095-8605-423a-93aa-709a0e24d9f4 | trentmkelly/LessWrong-43k | LessWrong | Covid 7/23: The Second Summit
Last week: Covid 7/16: Becoming the Mask
The news continues to be cautiously good.
Positive test rates are likely peaking. Deaths are increasing slower than one would have expected, and could well peak a few weeks from now without having risen much further.
This, in context, is what success looks like.
As Florida and Arizona go now, so goes the rest of the second wave later. If the hardest hit places are turning things around without much underlying regulatory change, and with continued huge testing delays, then that tells us that some combination of herd immunity and voluntary behavioral changes is getting us there already.
It was a quiet week by 2020 standards.
There was plenty of news, most importantly the whole thing where the Federal government graduating from banditry and piracy to kidnapping people off the streets in unmarked vans, which is outrageous and a dire threat to our freedom. That does not seem to be getting the proper level of alarm or attention.
There was also the explorations of GPT-3, which also does not seem to be getting the proper level of alarm or attention. People are literally talking about AI taking jobs away from programmers without stopping to think that maybe that might be burying the lead a little. We like to say There is No Fire Alarm for Artificial General Intelligence but let’s raise the possibility that the alarm is there and it’s working fine and it’s us that are the problem.
If you want the most fun improv experience I’ve ever seen that’s also kind of a harbinger of the end of the world, try the dragon model of AI dungeon here. It’s not as good as the pure prompt version, but it’s available to the public.
Also, Slate Star Codex is back! And there was much rejoicing. Woo-hoo.
In other great news, parents in San Francisco have realized that they can get into groups and hire teachers, and provide better educations in healthier and safer environments while spending less money than schools do, and can do so while hiring mo |
9a2e7574-6b29-4d7d-bd3b-e775e53deab4 | trentmkelly/LessWrong-43k | LessWrong | Speed & Performance is our current top priority
TLDR: Fewer bugfixes and support for the next week or two while we are focusing on making the site fast and stable. General shift in direction towards more stable and polished features over a large breadth of slightly buggy features.
It's been one week since the launch of the open beta. Since then we had some amazing posts, fixed tons of bugs, answered over 200 of your support questions on Intercom and added over 60 user-submitted issues to our Github. I think this week has been a good start for the new LessWrong, in large parts because of your continuous feedback and input. Thanks!
But despite the positives, and this became clear to us thanks to your input, there is a core problem with the site. LessWrong 2.0 is slow right now. Like, unbearably slow sometimes (I've experiences load times of 10+ seconds, which is totally unacceptable). This has probably caused many of you headaches, and has made the experience of using the site frustrating.
In addition to that, a good chunk of features are quite buggy. Sometimes when you create a post, it doesn't show up where it's supposed to, and you can't find it any other way than via a link. Or sometimes the page reloads and you lose all the things that you've written because we don't automatically save new posts and comments as drafts.
Here is a quick story of how we got to this state:
* Our top priority until now was to build an MVP that mostly has feature parity with reddit (so we can port over the data from the old site)
* We wanted to have a modular codebase that allows us to extend and change that functionality easily
* This means our development so far had been focused on developing features and getting the basic functionality going, not on making everything smooth and bug-free
And while it is tempting to continue to focus more on feature improvements and make the site more complete, I think it's time for LW 2.0 to focus its development on stability and speed. And only as soon as that has reached a high level, s |
9b3fd9a4-4327-472d-a206-033d47fa98dd | trentmkelly/LessWrong-43k | LessWrong | My attempt to explain Looking, insight meditation, and enlightenment in non-mysterious terms
Epistemic status: pretty confident. Based on several years of meditation experience combined with various pieces of Buddhist theory as popularized in various sources, including but not limited to books like The Mind Illuminated, Mastering the Core Teachings of the Buddha, and The Seeing That Frees; also discussions with other people who have practiced meditation, and scatterings of cognitive psychology papers that relate to the topic. The part that I’m the least confident of is the long-term nature of enlightenment; I’m speculating on what comes next based on what I’ve experienced, but have not actually had a full enlightenment. I also suspect that different kinds of traditions and practices may produce different kinds of enlightenment states.
While I liked Valentine’s recent post on kensho and its follow-ups a lot, one thing that I was annoyed by were the comments that the whole thing can’t be explained from a reductionist, third-person perspective. I agree that such an explanation can’t produce the necessary mental changes that the explanation is talking about. But it seemed wrong to me to claim that all of this would be somehow intrinsically mysterious and impossible to explain on such a level that would give people at least an intellectual understanding of what Looking and enlightenment and all that are. Especially not after I spoke to Val and realized that hey, I actually do know how to Look, and that thing he’s calling kensho, that’s happened to me too.
(Note however that kensho is a Zen term and I'm unfamiliar with Zen; I don't want to use a term which might imply that I was going with whatever theoretical assumptions Zen might have, so I will just talk about “my experience” when it comes up.)
So here is my attempt to give an explanation. I don’t know if I’ve succeeded, but here goes anyway.
----
One of my key concepts is going to be cognitive fusion.
Cognitive fusion is a term from Acceptance and Commitment Therapy (ACT), which refers to a person “fusi |
44cca6b3-40eb-4b4a-bba7-ab923ad639b4 | trentmkelly/LessWrong-43k | LessWrong | Chaotic era: avoid or survive?
In the world of the “Three-body problem” novel by Liu Cixin, stable eras, when the planet is rotating just around one sun, and everything is quite predictable, are changing to chaotic eras, when all three suns are involved, and nothing is possible to predict. In our history, we have relatively stable periods (when you are quite certain that your job or savings are not going to disappear tomorrow, for example). The levels of stability can be different, of course. The first few months of the pandemic were really less stable than life before the pandemic, but certainly more stable than life during WWII (at least in Europe). The question that I want to discuss here is how likely that we will get into a very chaotic era again, and what should we do with it. Notice, that by chaotic I don’t mean necessary “troublesome”. I mean only “hard to predict”.
Hard to predict for whom? We have many agents with different levels of expertise and different level of responsibility. If experts make a correct prediction, but many non-experts made a crucial mistake that strongly hurt them (loss of income or savings, for example), these non-experts can start acting based on emotions making the situation even less predictable, thus leading to the “domino effect”. Peculiar conspiracy theories spread among them can make the situation even worse. Another option is an irrational decision of a person with a huge level of responsibility, such as a dictator of a country with nuclear weapons, for example.
I can name a few scenarios that may increase the level of chaos in the system – just what came to my mind. Of course, there are more, most likely – those are just illustrations
-Due to AI, a lot of people may lose their jobs, and there may be no other job they have skills to do, so we get a huge amount unemployed and likely angry people, that can unpredictably vote or protest.
-The world financial system may transit to cryptoeconomics, which, in the process of transition, may lead to huge |
7b637aad-d118-4fd6-8f65-315f199e77ca | LDJnr/LessWrong-Amplify-Instruct | LessWrong | "The dominant model about status in LW seems to be one of relative influence. Necessarily, it's zero-sum. So we throw up our hands and accept that half the community is just going to run a deficit.Here's a different take: status in the sense of worth. Here's a set of things we like, or here's a set of problems for you to solve, and if you do, you will pass the bar and we will grant you personhood and take you seriously and allow you onto the ark when the world comes crumbling. Worth is positive-sum.I think both models are useful, but only one of these models underlies the emotional need we call status. I think it's the latter.Another assumption: humans are satisficers. Those that claim to the contrary have never been satisfied. An unsatisfied satisficer acts like a maximizer. I think that Maslov got pretty close with his hierarchy of needs. Not the specific set of needs, not necessarily their order, but the idea of humans working on one need at the time, until satisfaction, so that the next need comes up. It seems to me that many of us are stuck at the status level, and I think getting past it makes us surely happier and plausibly more effective.How is worth generated? Quite simply, by giving praise. You either find your behavior exceeding a standard that the community agreed on, or someone actually tells you you're doing well. The latter seems more powerful.I've asked around, and it even seems to be considered "normal" in most (non-systematizing) communities to habitually give praise. It's even apparently something people regard as necessary for proper psychological health. But honestly, apart from volunteering at CFAR, I can't recall getting much praise for anything I've done for the community. As a result I never quite feel like I'm doing enough, edging on burnout a few times. Reminds me of pica. Does working on AI Safety ever get me the sense of worth I'm looking for, or should I give up?So I'd like to suggest we try for Giving Praise as a rationalist virtue. It might just be a staple of group rationality."
The text above is a post found on LessWrong.
How can the act of giving praise be implemented effectively in a community without it losing its sincerity and value over time?
Implementing the act of giving praise effectively in a community without it losing its sincerity and value over time can be a delicate balance. Here are some strategies that could be employed:
1. **Specificity**: Instead of generic praise, specific compliments about a person's actions or contributions can make the praise feel more genuine and meaningful. This shows that you've noticed and appreciated the individual's unique efforts.
2. **Timeliness**: Giving praise soon after the action has occurred can make it more impactful. It shows that you're paying attention and that you care about acknowledging their efforts promptly.
3. **Balance**: While it's important to give praise, it's also crucial to maintain a balance. Overpraising can lead to the devaluation of praise. It's better to reserve praise for when it's truly deserved to maintain its value.
4. **Honesty**: Praise should be sincere and based on truth. False praise can be easily detected and can lead to mistrust.
5. **Encourage Peer-to-Peer Praise**: Encouraging members of the community to praise each other can help foster a positive and supportive environment. This can also help prevent the perception of praise as a top-down, hierarchical system.
6. **Personalization**: Tailor your praise to the individual. Different people value different forms of recognition. Some might prefer public acknowledgement, while others might prefer a private thank you.
7. **Constructive Feedback**: Alongside praise, constructive feedback is also essential for growth and improvement. This helps maintain a balance and shows that the aim is to help each other improve, not just to make each other feel good.
By following these strategies, a community can foster a culture of genuine and meaningful praise that maintains its value over time.
How can constructive feedback be effectively integrated with praise without undermining the positive impact of the latter?
Integrating constructive feedback with praise without undermining the positive impact of the latter can be achieved through the following strategies:
1. **The Sandwich Method**: This involves sandwiching the constructive feedback between two pieces of praise. Start with sincere praise, follow with the constructive feedback, and end with more praise. This method helps to soften the impact of the feedback and leaves the conversation on a positive note.
2. **Directness and Clarity**: Be clear and direct with your feedback. Avoid beating around the bush or being overly critical. Make sure the feedback is about the action or behavior, not the person.
3. **Focus on Improvement**: Frame the feedback in a way that focuses on future improvement rather than past mistakes. This can help the person see the feedback as a roadmap for success rather than a critique of failure.
4. **Balance**: Ensure that praise and feedback are balanced. If there is too much feedback compared to praise, it can feel like criticism. On the other hand, too much praise and not enough feedback can come off as insincere or unhelpful.
5. **Specificity**: Be specific in both your praise and feedback. Specific praise shows you are genuinely noticing and valuing their efforts, while specific feedback gives clear direction on what can be improved.
6. **Timeliness**: Give feedback as close to the event as possible. This makes it more relevant and easier for the person to remember the specifics of the situation.
7. **Dialogue, Not Monologue**: Make the feedback session a two-way conversation. Allow the person to respond to your feedback, ask questions, and provide their own perspective. This can help them feel more involved and less defensive.
By integrating these strategies, you can effectively combine praise with constructive feedback in a way that enhances the positive impact of both.
Reflecting on the strategies for integrating praise and constructive feedback, how might these methods be adapted for a remote working environment where non-verbal cues are absent and miscommunication can easily occur?
Adapting these strategies for a remote working environment does pose additional challenges due to the lack of non-verbal cues and potential for miscommunication. However, there are ways to effectively implement them:
1. **Use Video Calls**: Whenever possible, use video calls for giving feedback and praise. Seeing each other's facial expressions and body language can help convey tone and intent more effectively than text or voice alone.
2. **Written Communication**: When giving feedback or praise in written form, be extra careful with your wording. Make sure your message is clear and unambiguous. Using emojis or gifs can help convey tone and emotion, but use them judiciously and appropriately.
3. **Regular Check-ins**: Regularly scheduled check-ins can provide a structured opportunity for giving and receiving feedback and praise. This can help ensure that feedback and praise are given consistently and timely.
4. **Create a Feedback Culture**: Encourage a culture where feedback and praise are part of the norm. This can be done through regular team meetings, one-on-one sessions, or through dedicated feedback tools and platforms.
5. **Ask for Clarification**: If you're unsure about a piece of feedback or praise you've received, don't hesitate to ask for clarification. This can help avoid misunderstandings and ensure that the feedback or praise is understood as intended.
6. **Be Mindful of Time Zones**: If your team is spread across different time zones, be mindful of this when scheduling feedback sessions. Try to find a time that works for everyone involved.
7. **Personalize Your Approach**: Understand that everyone has different communication styles and preferences. Some people might prefer written feedback, while others might prefer a video call. Tailor your approach to suit the individual.
By adapting these strategies to the remote working environment, you can still effectively integrate praise and constructive feedback. |
41961f9c-d43e-45c9-8899-03aae78415ef | trentmkelly/LessWrong-43k | LessWrong | Sam Altman at SlateStarCodex Online Meetup
Our online meetup on November 8, at 10:30 AM Pacific Standard Time, 18:30 UTC, 20:30 Israel Standard Time, will feature Sam Altman, CEO of OpenAI, who will be answering questions. We will then have open video-chat rooms as usual.
We hope to focus the Q&A on AI, AGI, GPT-N and OpenAI, but feel free to ask questions on other topics if you like.
Please register here , and we'll send you an invitation. |
1462436c-b1f5-4ab2-9ed3-d6b66994140a | trentmkelly/LessWrong-43k | LessWrong | Weekly LW Meetups: Houston, Atlanta, Eugene OR, San Diego, Fort Collins CO, Waterloo, Melbourne
There are upcoming irregularly scheduled Less Wrong meetups in:
* Houston Meetup: Saturday, 11/19: 19 November 2011 02:25AM
* Atlanta Less Wrong Meetup: 19 November 2011 07:00PM
* First Eugene Meetup: 20 November 2011 01:00PM
* San Diego meetup: 20 November 2011 01:00PM
* Fort Collins, Colorado Meetup Wedneday 7pm: 23 November 2011 07:00PM
* First(New?) Waterloo Meetup: 24 November 2011 08:00PM
* Sheridan College - Oakville, ON: 25 November 2011 08:30PM
The following meetups take place in cities with regularly scheduled meetups, but involve a change in time or location, special meeting content, or simply a helpful reminder about the meetup:
* Monthly Bay Area meetup: 26 November 2011 07:00PM
* Melbourne practical rationality meetup: 02 December 2011 07:00PM
Cities with regularly scheduled meetups: Austin, Berkeley, Cambridge, MA, Irvine, London, Marin, CA (uses the Bay Area List), Melbourne, Mountain View, New York, Ottawa, Oxford, San Francisco, Toronto, Washington, DC, and West Los Angeles.
If you'd like to talk with other LW-ers face to face, and there is no meetup in your area, consider starting your own meetup; it's easy (more resources here). Check one out, stretch your rationality skills, and have fun!
In addition to the handy sidebar of upcoming meetups, a meetup overview will continue to be posted on the front page every Friday. These will be an attempt to collect information on all the meetups happening in the next weeks. The best way to get your meetup featured is still to use the Add New Meetup feature, but you'll now also have the benefit of having your meetup mentioned in a weekly overview. These overview posts will be moved to the discussion section when the new post goes up.
Please note that for your meetup to appear in the weekly meetups feature, you need to post your meetup before the Friday before your meetup!
If you check Less Wrong irregularly, consider subscribing to one or more city-specific mailing list in order to be notifi |
53b2e6e0-2dbc-4220-8437-e66026034698 | trentmkelly/LessWrong-43k | LessWrong | Book Launch: The Engines of Cognition
The Engines of Cognition is a newly published collection of essays by the LessWrong Community. They're currently available with Prime at Amazon US, where they'll arrive in time for Christmas, and you can read on to learn more about them.
About
In The Engines of Cognition, the writers set out to understand key elements of the art of rationality. Starting with the simple epistemic question of when and how to trust different sources of information, the essays in this collection move through understanding the lens of incentives, an exploration of when and why complex systems become modular, and finally into a discussions of failure, both personal and civilizational.
This book set is for people who want to read the most interesting ideas LessWrong has recently explored. It's for the people who read best away from screens, away from distractions. It's for people who do not check the site regularly, but would still like to get the ideas within. For such people, this is intended to be the ideal way to read LessWrong.
Essays in this book set take a variety of forms, from thought experiments to literature reviews, as well as book reviews, interviews, personal stories, microeconomic arguments, mathematical explanations, research advice, philosophical musings, published papers, disagreements-with-Robin-Hanson, forecasts for the future, survey data, cartoons, and more.
Authors featured include Eliezer Yudkowsky, Scott Alexander, Zvi Mowshowitz, and over 30 more LessWrong writers. The essays were originally published to LessWrong in 2019, and for the first time are available with editing and illustration in print form. In addition, given recent advances in machine learning art, each essay has a unique piece of ML-generated art based on the essay title, some of which are shown below.
Order now on Amazon Prime US ($30) and it will arrive in time for Christmas.
Book Spreads
Table of Contents
Trust
Rule Thinkers In, Not OutScott Alexander Gears vs BehaviorJohn S. Wentwort |
ca7643a6-d228-46f7-838a-c82ba9ce597a | trentmkelly/LessWrong-43k | LessWrong | The Handbook of Rationality (2021, MIT press) is now open access
Of particular interest here is section 4.1 on subjective accounts of probability and section 8.4 which explicitly addresses Knightian uncertainty and its challenges for Bayesian accounts.
https://direct.mit.edu/books/oa-edited-volume/5525/The-Handbook-of-Rationality |
90ea9b1f-e5c5-4860-b32d-a16a3cb30b20 | trentmkelly/LessWrong-43k | LessWrong | To first order, moral realism and moral anti-realism are the same thing
I've taken a somewhat caricaturist view of moral realism[1], describing it, essentially, as the random walk of a process defined by its "stopping" properties.
In this view, people start improving their morality according to certain criteria (self-consistency, simplicity, what they would believe if they were smarter, etc...) and continue on this until the criteria are finally met. Because there is no way of knowing how "far" this process can continue until the criteria are met, this can drift very far indeed from its starting point.
Now I would like to be able to argue, from a very anti-realist perspective, that:
* Argument A: I want to be able to judge that morality α is better than morality β, based on some personal intuition or judgement of correctness. I want to be able to judge that β is alien and evil, even if it is fully self-consistent according to formal criteria, while α is not fully self-consistent.
Moral realists look like moral anti-realists
Now, I maintain that this "random walk to stopping point" is an accurate description of many (most?) moral realist systems. But it's a terrible description of moral realists. In practice, most moral realists allow for the possibility of moral uncertainty, and hence that their preference approach might have a small chance of being wrong.
And how would they identify that wrongness? By looking outside the formal process, and checking if the path that the moral "self-improvement" is taking is plausible, and doesn't lead to obviously terrible outcomes.
So, to pick one example from Wei Dai (similar examples can be found in this post on self-deception, and in the "Senator Cruz" section of Scott Alexander's "debate questions" post):
> I’m envisioning that in the future there will also be systems where you can input any conclusion that you want to argue (including moral conclusions) and the target audience, and the system will give you the most convincing arguments for it. At that point people won’t be able to partic |
85b97544-5d4b-441c-9cbf-845534194b2e | trentmkelly/LessWrong-43k | LessWrong | Start taking drugs randomly
I wanted to try a drug. I also wanted to be able to distinguish the direct effects of the drug from the placebo effects of the drug. Most importantly, I wanted to isolate placebo side effects. It seemed there was a chance of real side effects, and that meant that it was basically guaranteed that I would imagine those side effects. However, if I could tell that I was imagining them, I figured I might stop doing it.
The usual way to distinguish placebo effects from real effects is to have a bunch of people, and give half of them the real drug and half the placebo drug. Then if five percent of both groups gets a headache, you know that it’s entirely placebo headache.
Of course, that had already been worked out for this drug. It probably causes a few real side effects, and a few placebo ones. But what I wanted to know was whether the side effects were real in my own particular case.
If you want to figure this out, a natural idea is to get someone else to randomly give you drug or placebo on different days (or on different few-day bouts, if the drug takes time to act). After enough times, you can tell if the drug periods are better or worse.
This is a hard enough experiment to run on yourself if it takes a day for the drug to work. But what if the drug might take a month to work or to have side effects? Someone could randomize whether you get the drug or placebo for whole several month long stints, but at this point it will take you a year to get a few data points, and each one will be as different from the others as several month long periods are wont to be, aside from any effect from the drug. And you will have missed out on taking the drug for half a year.
Here is a different way to do this: randomize the start date. I asked a friend to pick a random day within the next few months, and then give me placebo pills until that day and from then on, a real pill. He did this by putting all the pills inside big red capsules so that I couldn’t tell them apart, and then p |
a4bed85a-2172-436f-a92f-916f78154b67 | trentmkelly/LessWrong-43k | LessWrong | The Tragedy of Group Selectionism
Before 1966, it was not unusual to see serious biologists advocating evolutionary hypotheses that we would now regard as magical thinking. These muddled notions played an important historical role in the development of later evolutionary theory, error calling forth correction; like the folly of English kings provoking into existence the Magna Carta and constitutional democracy.
As an example of romance, Vero Wynne-Edwards, Warder Allee, and J. L. Brereton, among others, believed that predators would voluntarily restrain their breeding to avoid overpopulating their habitat and exhausting the prey population.
But evolution does not open the floodgates to arbitrary purposes. You cannot explain a rattlesnake's rattle by saying that it exists to benefit other animals who would otherwise be bitten. No outside Evolution Fairy decides when a gene ought to be promoted; the gene's effect must somehow directly cause the gene to be more prevalent in the next generation. It's clear why our human sense of aesthetics, witnessing a population crash of foxes who've eaten all the rabbits, cries "Something should've been done!" But how would a gene complex for restraining reproduction—of all things!—cause itself to become more frequent in the next generation?
A human being designing a neat little toy ecology—for entertainment purposes, like a model railroad—might be annoyed if their painstakingly constructed fox and rabbit populations self-destructed by the foxes eating all the rabbits and then dying of starvation themselves. So the human would tinker with the toy ecology—a fox-breeding-restrainer is the obvious solution that leaps to our human minds—until the ecology looked nice and neat. Nature has no human, of course, but that needn't stop us—now that we know what we want on aesthetic grounds, we just have to come up with a plausible argument that persuades Nature to want the same thing on evolutionary grounds.
Obviously, selection on the level of the individual won't produce |
5899b806-2b95-4bc7-a70f-329e3208fc09 | StampyAI/alignment-research-dataset/eaforum | Effective Altruism Forum | Update from Campaign for AI Safety
Hi!
*I am starting to share weekly updates from the Campaign for AI Safety on this forum. Please sign up on the* [*campaign website*](https://www.campaignforaisafety.org/) *to receive these by email.*
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🌐 The big global news this week is that Rishi Sunak [promised to look into AI existential risk](https://twitter.com/RishiSunak/status/1663838958558539776). Clearly, the collective efforts of the AI safety community are bearing some fruit. Congratulations to A/Prof Krueger and Center for AI Safety on this achievement. Now, let's go all the way to a moratorium!
---
🧑⚖️ We have [announced the judging panel for the student competition](https://www.campaignforaisafety.org/introducing-the-panel-of-judges-for-the-student-competition-on-drafting-a-treaty-on-the-moratorium-of-large-scale-ai-capabilities-r-d/) for drafting a treaty on moratorium of large-scale AI capabilities R&D:
1. Prof [John Zeleznikow](https://www.linkedin.com/in/john-zeleznikow-4249b41/?ref=campaignforaisafety.org) - Professor, Law School, Latrobe University
2. Dr [Neville Grant Rochow](https://www.linkedin.com/in/dr-neville-rochow-kc-4135481a1/?ref=campaignforaisafety.org) KC - Associate Professor (Adj), University of Adelaide Law School and Barrister
3. Dr [Guzyal Hill](https://www.linkedin.com/in/dr-guzyal-hill-68b69b80/?ref=campaignforaisafety.org) - Senior Lecturer, Charles Darwin University
4. [Jose-Miguel Bello Villarino](https://www.sydney.edu.au/law/about/our-people/academic-staff/jose-miguel-bellovillarino.html?ref=campaignforaisafety.org) - Research Fellow, ARC Centre of Excellence for Automated Decision-Making and Society, The University of Sydney
5. [Udomo Ali](https://www.linkedin.com/in/udomo-ali/?ref=campaignforaisafety.org) - Lawyer and researcher (AI & Law), University of Benin, and Nigerian Law School graduate
6. [Raymond Sun](https://www.linkedin.com/in/raymond-sun-64576a122/?ref=campaignforaisafety.org) - Technology Lawyer (AI) at Herbert Smith Freehills and Organiser at the Data Science and AI Association of Australia
Thank-you to all the judges for their involvement in this project. And thank-you to Nayanika Kundu for the initial work on promoting the competition.
Please do like and share posts about the competition on [LinkedIn](https://www.linkedin.com/feed/update/urn:li:activity:7062075977895452672/), [Facebook](https://www.facebook.com/campaignforaisafety/posts/pfbid0NPLzC9CchNMPEtzr8rmiRsWxMY4NzQKPgMN8HNYNM2oCzjpZgqhaDE3MSrWRAyv9l), [Instagram](https://www.instagram.com/p/CsIWz2jLKjP/), [Twitter](https://twitter.com/NikSamoylov/status/1660412669038563328). And please [donate](https://www.campaignforaisafety.org/donate/) or [become a paying campaign member](https://www.campaignforaisafety.org/#/portal/) to help advertise in paid media.
---
🤓 New messaging research came out in the past week on [alternative phrasing of "God-like AI"](https://www.campaignforaisafety.org/alternative-phrases-to-god-like-ai/). Previously we reported that this phrase did not resonate well, especially with older people. We revisited it and found that:
* No semantically similar phrases improved on "godlike AI" simultaneously on agreeableness and concern. The spelling "godlike AI" outperformed "God-like AI".
* The following phrases, while less concerning, are more agreeable: "superintelligent AI species", "AI that is smarter than us like we’re smarter than 2-year-olds".
* The following are more concerning, but less agreeable: "uncontrollable AI", "killer AI".
* Unless any better phrases are proposed, "godlike AI", alongside other terms, is a decent working term, especially when addressing younger audiences.
More research is underway. Please send requests if you want to know more about what the public thinks.
---
🟩 We have launched fresh billboards. We are focusing only on San Francisco at the moment. Thank you Dee Kathuria for contributing the wording and the design formats. Please [donate](https://www.campaignforaisafety.org/donate/) to help us reach more people in more cities.
---
🪧 PauseAI made headlines in Politico Pro for their recent protest in Brussels, raising concerns about the risks of artificial intelligence (AI). The passionate group gathered outside Microsoft's lobbying office, advocating for safe AI development with a placard that read: "Build AI safely or don't build AI at all."
[](https://www.linkedin.com/feed/update/urn:li:activity:7069915472141451264/?actorCompanyId=91097814)
---
📃 On policy front, we have just sent [our response to the UK CMA's information request on foundation models](https://www.campaignforaisafety.org/response-to-the-uk-cmas-information-request-on-foundation-models/). Thank you Miles Tidmarsh and Sue Anne Wong for the work on this submission.
There are three that we are going to work:
* [Request for comment launched by the National Telecommunications and Information Administration](https://www.federalregister.gov/documents/2023/04/13/2023-07776/ai-accountability-policy-request-for-comment) (USA. Due 12 June)
* [Policy paper "AI regulation: a pro-innovation approach"](https://www.gov.uk/government/publications/ai-regulation-a-pro-innovation-approach) (UK. Due 21 June 2023)
* [Supporting responsible AI: discussion paper](https://consult.industry.gov.au/supporting-responsible-ai) (Australia. Due 26 July)
Please respond to this email if you want to contribute.
---
✍️ There are two parliamentary petitions under review at the parliamentary offices:
* [Greg Colbourn’s petition](https://agi-moratorium-hq.slack.com/archives/C054NS9LZEG/p1685200846104569?thread_ts=1685194976.000699&cid=C054NS9LZEG) (UK).
* [My petition](https://agi-moratorium-hq.slack.com/archives/C054JSS05M4/p1685237460847269) (Australia).
---
Thank you for your support! Please share this email with friends.
Nik Samoylov from Campaign for AI Safety
[campaignforaisafety.org](https://www.campaignforaisafety.org/?e=4e1d644b75d254bce7bea1f45f5e7645&utm_source=campaignforaisafety&utm_medium=email&utm_campaign=donor_ask1&n=4) |
1ee05ff4-fb66-4ee2-ae3b-a140f15df708 | trentmkelly/LessWrong-43k | LessWrong | Ways that China is surpassing the US
[Edit: Changed the post title from the original article title to something more meaningful.]
I came across this article today, and I have to agree with it strongly based on my own recent trip to China. The update it triggered for me is the realization that China is genuinely doing better than the US on many fronts, most importantly on governance. How/why did that happen? Did anyone or any political theory predict this ahead of time? (In case it's not clear, this is not meant to be a rhetorical question. I'm surprised and confused and am wondering if someone or some theory can offer an explanation.) What does it imply for things like AI governance and global coordination on x-risks?
Below I'll quote two sections that summarize most of the article. Click the link above for the full text.
----------------------------------------
China is changing in a deep and visceral way, and it is changing fast , in a way that is almost incomprehensible without seeing it in person. In contrast to America’s stagnation, China’s culture, self-concept, and morale are being transformed at a rapid pace—mostly for the better.
For Americans, this transformation towards competence and prosperity should be the more worrying thing about China. We aren’t well-coordinated with China, and it is starting to surpass the U.S. in important ways. If we ignore this point by focusing only on the moral negatives, of which there are many, we risk underrating the competition until it’s too late, or failing to reflect on the importance of solving our own mounting problems of governance.
More optimistically, we should study Chinese development as a positive example of how to do better on the things that America has been fumbling recently: infrastructure, growth, industrial policy, and positive transformation of everyday life. America has been a shining beacon of this kind of development in the past, and can be again, if we’re willing to look closer at what China does right.
[...]
If I’m being honest, |
0aae1624-882a-4b0e-805d-538577dd9cb4 | trentmkelly/LessWrong-43k | LessWrong | Morality is Scary
I'm worried that many AI alignment researchers and other LWers have a view of how human morality works, that really only applies to a small fraction of all humans (notably moral philosophers and themselves). In this view, people know or at least suspect that they are confused about morality, and are eager or willing to apply reason and deliberation to find out what their real values are, or to correct their moral beliefs. Here's an example of someone who fits this view:
> I’ve written, in the past, about a “ghost” version of myself — that is, one that can float free from my body; which travel anywhere in all space and time, with unlimited time, energy, and patience; and which can also make changes to different variables, and play forward/rewind different counterfactual timelines (the ghost’s activity somehow doesn’t have any moral significance).
>
> I sometimes treat such a ghost kind of like an idealized self. It can see much that I cannot. It can see directly what a small part of the world I truly am; what my actions truly mean. The lives of others are real and vivid for it, even when hazy and out of mind for me. I trust such a perspective a lot. If the ghost would say “don’t,” I’d be inclined to listen.
I'm currently reading The Status Game by Will Storr (highly recommended BTW), and found in it the following description of how morality works in most people, which matches my own understanding of history and my observations of humans around me:
> The moral reality we live in is a virtue game. We use our displays of morality to manufacture status. It’s good that we do this. It’s functional. It’s why billionaires fund libraries, university scholarships and scientific endeavours; it’s why a study of 11,672 organ donations in the USA found only thirty-one were made anonymously. It’s why we feel good when we commit moral acts and thoughts privately and enjoy the approval of our imaginary audience. Virtue status is the bribe that nudges us into putting the interests |
625a82dc-f210-4ddf-9fde-ffd2c5cd3385 | trentmkelly/LessWrong-43k | LessWrong | Intuitions about goal-directed behavior
One broad argument for AI risk is the Misspecified Goal argument:
> The Misspecified Goal Argument for AI Risk: Very intelligent AI systems will be able to make long-term plans in order to achieve their goals, and if their goals are even slightly misspecified then the AI system will become adversarial and work against us.
My main goal in this post is to make conceptual clarifications and suggest how they affect the Misspecified Goal argument, without making any recommendations about what we should actually do. Future posts will argue more directly for a particular position. As a result, I will not be considering other arguments for focusing on AI risk even though I find some of them more compelling.
I think of this as a concern about long-term goal-directed behavior. Unfortunately, it’s not clear how to categorize behavior as goal-directed vs. not. Intuitively, any agent that searches over actions and chooses the one that best achieves some measure of “goodness” is goal-directed (though there are exceptions, such as the agent that selects actions that begin with the letter “A”). (ETA: I also think that agents that show goal-directed behavior because they are looking at some other agent are not goal-directed themselves -- see this comment.) However, this is not a necessary condition: many humans are goal-directed, but there is no goal baked into the brain that they are using to choose actions.
This is related to the concept of optimization, though with intuitions around optimization we typically assume that we know the agent’s preference ordering, which I don’t want to assume here. (In fact, I don’t want to assume that the agent even has a preference ordering.)
One potential formalization is to say that goal-directed behavior is any behavior that can be modelled as maximizing expected utility for some utility function; in the next post I will argue that this does not properly capture the behaviors we are worried about. In this post I’ll give some intuitions abou |
144026a5-cc61-4b38-b1d7-d982c6443896 | trentmkelly/LessWrong-43k | LessWrong | Ethical Intuitionism, part 1
I read Michael Huemer’s book Ethical Intuitionism at Bryan Caplan’s suggestion. Before reading it I thought that Ethical Intuitionism seemed an unlikely position and that Bryan and Michael seemed like smart guys, so I hoped I might be persuaded to significantly change my mind. I still believe both of the premises, but did not get as far as changing my mind, so here I’ll report back on some of my reasons. I actually read it a while ago, and am reconstructing some of this post from memories and scribbles in margins, so apologies if this causes any inaccurate criticism.
Ethical intuitionism is the position that there exist real, irreducible, moral properties, and that these can be known about through intuition. i.e. some things are ‘good’ or ‘right’ independent of anyone’s feelings or preferences, and you can learn about this through finding such ideas in your head. I found Huemer’s case for it well written, thought provoking, and for the most part valid. However I thought it had some problems, and was overall not compelling.
Huemer divides metaethics as into five types of theories then argues against non-cognitivism, relativism, nihilism, and naturalism. The last theory standing is intuitionism, which Humer goes on to defend from the criticisms it has been attacked with before. Note that these metaethical theories are both theories of what ethics is and theories of how we can know about it.
I
The first problem is related to intuitionism being primarily a theory of how we can know about ethics (through intuitions), while being hazier on what exactly ethics is, in terms of anything else. It’s something real that we can learn about through our intuitions. It is normative. If the theory contained any more concrete specification of the nature of ethics or how it got there, I think it would become more obviously subject to the same criticisms made of other metaethical theories. For instance, this is one of Huemer’s criticisms of subjectivism:
> Fourth, consider the que |
d0356774-d603-4fa2-8484-0618592cf5e6 | trentmkelly/LessWrong-43k | LessWrong | Can You Give Support or Feedback for My Program to Alleviate Poverty?
Hi LessWrong,
Two years ago, when I travelled to Belize, I came up with an idea for a self-sufficient scalable program to address poverty. I saw how many people in Belize were unemployed or getting paid very low wages, but I also saw how skilled they were, a result of English being the national language and a mandatory education system. Many Belizeans have a secondary/high school education in Belize, and the vast majority have at least a primary school education and can speak English. I thought to myself, "it's too bad I can't teleport Belizeans to the United States, because in the U.S., they would automatically be able to earn many times more the minimum wage in Belize with their existing skills."
But I knew there was a way to do it: "virtual teleportation." My solution involves using computer and internet access in conjunction with training and support to connect the poor with high paying international work opportunities. My tests of virtual employment using Upwork and Amazon Mechanical Turk show that it is possible to earn at least twice the minimum wage in Belize, around $3 an hour, working with flexible hours. This solution is scalable because there is a consistent international demand for very low wage work (relatively speaking) from competent English speakers, and in other countries around the world like South Africa, many people matching that description can be found and lifted out of poverty. The solution could become self-sufficient because running a virtual employment enterprise or taking a cut of the earnings of members using virtual employment services (as bad as that sounds) can generate enough income to pay for the relatively low costs of monthly internet and the one-time costs of technology upgrades.
If you have any feedback, comments, suggestions, I would love to hear about it in the comments section. Feedback on my fundraising campaign at igg.me/at/bvep is also greatly appreciated.
If you are thinking about supporting the idea, my team and I ne |
ecc4fb2d-be5e-4d62-82e6-5abede975976 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | Military AI as a Convergent Goal of Self-Improving AI
This is our accepted chapter in the edited volume Forthcoming as a chapter in Artificial Safety And Security ([Roman V. Yampolskiy](https://www.facebook.com/roman.yampolskiy?fref=gs&hc_location=group) , ed.), CRC Press.
>Abstract Better instruments to predict the future evolution of artificial intelligence (AI) are needed, as the destiny of our civilization depends on it. One of the ways to such prediction is the analysis of the convergent drives of any future AI, started by Omohundro. We show that one of the convergent drives of AI is a militarization drive, arising from AI’s need to wage a war against its potential rivals by either physical or software means, or to increase its bargaining power. This militarization trend increases global catastrophic risk or even existential risk during AI takeoff, which includes the use of nuclear weapons against rival AIs, blackmail by the threat of creating a global catastrophe, and the consequences of a war between two AIs. As a result, even benevolent AI may evolve into potentially dangerous military AI. The type and intensity of militarization drive depend on the relative speed of the AI takeoff and the number of potential rivals. We show that AI militarization drive and evolution of national defense will merge, as a superintelligence created in the defense environment will have quicker takeoff speeds, but a distorted value system. We conclude with peaceful alternatives.
https://www.academia.edu/35130825/Military*AI* as*a* Convergent*Goal* of*Self-Improving* AI
Link without registration and opened for commenting: https://docs.google.com/document/d/15D71qhhY-ZsAY7syzZsr1lKopTODbdeXVPElaPaIqyA/edit |
8b19f5ac-ff01-43d2-82a3-f737e95e36eb | trentmkelly/LessWrong-43k | LessWrong | Ambitious Altruistic Software Engineering Efforts: Opportunities and Benefits
LessWrong Note
I recently published this on the EA Forum, then was recommended to also cross-post here. It's written a bit more for the EA community, but I think of course much of it applies and would be relevant for the communities here too. The LessWrong community has a long and deep history with software engineering.
I don't plan on cross-posting the future posts in this sequence to LessWrong, because it seems messier on multiple sides, but do let me know if you'd prefer that.
----------------------------------------
This is part 1 of what I expect to be a short series. The next parts will focus on the costs of these technical projects and more specific ways to kickstart them.
Meta
Rigor: Writing based on my experiences and education in the tech industry. I spent around 50 hours on this series plus some sporadic thinking and discussion. I focused more on idea quantity than neatness or data.
Intended Audience: Effective altruists interested in either funding or working on ambitious software projects.
History: This series was originally written as one Google Doc in March 2021 for private conversations. I've since done some updating to post it publicly. One important update since this document was written is that the CEA tech team has seemed to really up hiring considerably. This seems great, though I'm sure we could have even more, so it still seemed useful to post this.
About me (Ozzie Gooen):
I’ve been thinking about this a lot over the last several years. I joined 80,000 Hours as a web developer in 2013 hoping to do major/ambitious work, but there I realized that the funding situation at the time wasn’t large enough for most ambitious software efforts. I’ve since spent time in other startups, making my own startups, and doing consulting for small and large companies.
Recently I’ve been planning what work QURI should do. We’re considering focusing on engineering. Some of the motivation for this topic is to help make decisions for QURI.
Acknowledgments: |
495fefee-5e59-436d-a711-539ae64c9b9f | trentmkelly/LessWrong-43k | LessWrong | Accuracy of arguments that are seen as ridiculous and intuitively false but don't have good counter-arguments
I think one of the biggest reasons I am worried about AI doom is that there doesn't seem to be very good counter-arguments. Most of them are just bad, and the ones that aren't a bit vague (usually something along the lines of "humanity will figure it out" or "maybe LLMs will scale in nice ways").
However, I'm curious as to how accurate this heuristic is. My question: What examples in the past are there of "argument is widespreadly seen as ridiculous and intuitively false, but the argument is pretty solid and the counter-arguments aren't". (Sorry if that's a bit vague, use your best judgement. I'm looking specifically for examples that are similar to the AI x-risk debate.) And did they turn out true or false? Try to include reasons why the argument was so strong, a typical counter-argument, and the strongest counter-argument.
Please use spoiler text for the final answer so that I can try to predict it before seeing it! |
b38268c1-9d32-40b4-852a-6e7365dfc203 | trentmkelly/LessWrong-43k | LessWrong | Developing AI Safety: Bridging the Power-Ethics Gap (Introducing New Concepts)
TLDR
* One of the most significant challenges facing humanity is the widening gap between our rapidly increasing power and our slower progress (if there is any) in ethical understanding and application. I term this the power-ethics gap, and this has been creating increasing suffering and death (to animals and humans) from human actions as history progresses.
* AI threatens to dramatically expand the power-ethics gap.
* This creates an urgent imperative for humanity to cultivate better ethics.
* Currently, the AI safety field predominantly focuses on the critical task of maintaining human control over powerful AI systems, often overlooking the vital ethical dimension. Consequently, philosophical inquiry is essential to re-view the field, introducing new concepts (or refining existing ones) that properly emphasize this ethical component.
* I suggest several new concepts in this post.
* More resources can be dedicated to the crucial question of AI value selection, and related work that aims to ensure the values embedded within AI systems align with an encompassing ethical view.
This post can be seen as a continuation of the this post.
(To further explore this topic, you can watch a 34-minute video outlining a concept map of the AI space and potential additions. Recommended viewing speed: 1.25x).
This post drew some insights from the Sentientism podcast and the Buddhism for AI course.
My Point of View
I am looking at the AI safety space mainly through the three fundamental questions: What is? What is good? How do we get there?
* "What is?" relates to understanding reality through data and intelligence. I define intelligence as the capacity to make accurate predictions across increasingly complex counterfactual scenarios, given specific data and environmental context. The combination of data and intelligence constitutes power. Greater power enables a deeper understanding of "what is" and improved predictive capabilities across various scenarios.
* "What is |
86f86cb9-af95-41c9-88fe-d3dd0bf6a211 | StampyAI/alignment-research-dataset/special_docs | Other | How to Explain Individual Classification Decisions
Journalof Machine LearningResearch 11(2010)1803-1831 Su bmitted 4/09;Revised 11/09;Published6/10
How toExplain Individual Classification Decisions
David Baehrens∗BAEHRENS @CS.TU-BERLIN.DE
Timon Schroeter∗TIMON@CS.TU-BERLIN.DE
Technische Universit ¨at Berlin
Franklinstr. 28/29, FR6-9
10587 Berlin, Germany
StefanHarmeling∗STEFAN.HARMELING @TUEBINGEN .MPG.DE
MPI forBiological Cybernetics
Spemannstr. 38
72076 T¨ubingen, Germany
MotoakiKawanabe†MOTOAKI .KAWANABE @FIRST.FRAUNHOFER .DE
Fraunhofer InstituteFIRST.IDA
Kekulestr.7
12489 Berlin, Germany
KatjaHansen KHANSEN @CS.TU-BERLIN.DE
Klaus-RobertM ¨uller KLAUS-ROBERT.MUELLER @TU-BERLIN.DE
Technische Universit ¨at Berlin
Franklinstr. 28/29, FR6-9
10587 Berlin, Germany
Editor:Carl Edward Rasmussen
Abstract
After building a classifier with modern tools of machine lear ning we typically have a black box at
handthatisabletopredictwellforunseendata. Thus,weget ananswertothequestion whatisthe
mostlikelylabelofagivenunseendatapoint. However,most methodswillprovidenoanswer why
the model predicted a particular label for a single instance and what features were most influential
forthatparticularinstance. Theonlymethodthatiscurren tlyabletoprovidesuchexplanationsare
decision trees. This paper proposes a procedure which (base d on a set of assumptions) allows to
explain thedecisions of anyclassification method.
Keywords: explaining, nonlinear, black box model, kernel methods, Am es mutagenicity
1. Introduction
Automatic nonlinear classification is a common and powerful tool in data analysis . Machine learn-
ing research has created methods that are practically useful and that ca n classify unseen data after
beingtrainedon alimitedtrainingsetof labeled examples.
Nevertheless, most of the algorithms do not explaintheir decision. However in practical data
analysis it is essential to obtain an instance-based explanation, that is, we w ould like to gain an
∗. The firstthreeauthors contributed equally.
†. Alsoat Technische Universit ¨at Berlin,Franklinstr. 28/29, FR6-9,10587 Berlin,Germany.
c/circleco√yrt2010David Baehrens, TimonSchroeter, Stefan Harmeling, Moto akiKawanabe, Katja Hansen and Klaus-RobertM ¨uller.
BAEHRENS , SCHROETER , HARMELING , KAWANABE , HANSEN AND M¨ULLER
understanding what input features made the nonlinear machine give its ans wer for each individual
datapoint.
Typically, explanations are provided jointly for all instances of the training s et, for example
feature selection methods (including Automatic Relevance Determination) find out which inputs
are salient for a good generalization (see Guyon and Elisseeff,2003, for a review). While this can
giveacoarseimpressionabouttheglobalusefulnessofeachinputdime nsion,itisstillanensemble
view and does not provide an answer on an instance basis.1In the neural network literature also
solely an ensemble view was taken in algorithms like input pruning (e.g., Bishop, 1995; LeCun
et al., 1998). The only classification that does provide individual explan ations are decision trees
(e.g.,Hastieetal.,2001).
This paper proposes a simple framework that provides local explanation v ectors applicable to
anyclassificationmethodinordertohelpunderstandingpredictionresultsforsin gledatainstances.
The local explanation yields the features relevant for the prediction at the very points of interest
in the data space, and is able to spot local peculiarities that are neglected in th e global view, for
example, duetocancellation effects.
Thepaperisorganizedasfollows: Wedefinelocalexplanationvectorsa sclassprobabilitygradi-
entsinSection2andgiveanillustrationforGaussianProcessClassification (GPC).Somemethods
output a prediction without a direct probability interpretation. For these we p ropose in Section 3 a
way to estimate local explanations. In Section 4 we apply our methodology to lear n distinguishing
propertiesofIrisflowersbyestimatingexplanationvectorsforak-NNclas sifierappliedtotheclas-
sic Iris data set. In Section 5 we discuss how our approach applied to a SVM classifier allows us to
explainhowdigit“2”isdistinguishedfromdigit“8”intheUSPSdataset. InSe ction6wefocuson
a more real-world application scenario where the proposed explanation ca pabilities prove useful in
drug discovery: Human experts regularly decide how to modify existing lead compounds in order
to obtain new compounds with improved properties. Models capable of explain ing predictions can
helpintheprocessofchoosingpromisingmodifications. Ourautomaticallygene ratedexplanations
match with chemical domain knowledge about toxifying functional groups of th e compounds in
question. Section 7 contrasts our approach with related work and Section 8 discusses characteristic
propertiesand limitationsof our approach,beforeweconclude thepaper inSection 9.
2. Definitions ofExplanation Vectors
In this Section we will give definitions for our approach of local explanatio n vectors in the classi-
fication setting. We start with a theoretical definition for multi-class Bayes classifi cation and then
giveaspecializeddefinition beingmorepracticalfor thebinarycase.
Forthemulti-classcase,supposewearegivendatapoints x1,...,xn∈ℜdwithlabels y1,...,yn∈
{1,...,C}and we intend to learn a function that predicts the labels of unlabeled data poin ts. As-
suming that the data is IID-sampled from some unknown joint distribution P(X,Y), we define the
1. This point is illustrated in Figure 1 (Section 2). Applying feature selection m ethods to the training set (a) will lead
to the (correct) conclusion that both dimensions are equally important for accurate classification. As an alternative
to this ensemble view, one may ask: Which features (or combinations there of) are most influential in the vicinity
of each particular instance. As can be seen in Figure 1 (c), the answer d epends on where the respective instance is
located. On the hypotenuse and at the corners of the triangle, both featur es contribute jointly, whereas along each of
the remaining twoedges theclassification depends almostcompletely onjust one of thefeatures.
1804
HOW TOEXPLAININDIVIDUAL CLASSIFICATION DECISIONS
Bayes classifier,
g∗(x)=arg min
c∈{1,...,C}P(Y/\e}atio\slash=c|X=x)
whichis optimalforthe 0-1loss function(seeDevroyeetal.,1996).
For the Bayes classifier we define the explanation vector of a data point x0to be the derivative
withrespectto xatx=x0oftheconditional probabilityof Y/\e}atio\slash=g∗(x0)givenX=x,or formally,
Definition 1
ζ(x0):=∂
∂xP(Y/\e}atio\slash=g∗(x)|X=x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
x=x0.
Notethat ζ(x0)isad-dimensionalvectorjustlike x0is. Theclassifier g∗partitionsthedataspace ℜd
intoupto Cpartsonwhich g∗isconstant. Weassumethattheconditionaldistribution P(Y=c|X=
x)is first-order differentiable w.r.t. xfor all classes cand over the entire input space. For instance,
this assumption holds if P(X=x|Y=c)is for allcfirst-order differentiable in xand the supports
of the class densities overlap around the border for all neighboring pair s in the partition by the
Bayesclassifier. Thevector ζ(x0)definesoneachofthosepartsavectorfieldthatcharacterizesthe
flow away from the corresponding class. Thus entries in ζ(x0)with large absolute values highlight
features that will influence the class label decision of x0. A positive sign of such an entry implies
that increasing that feature would lower the probability that x0is assigned to g∗(x0). Ignoring the
orientations of the explanation vectors, ζforms a continuously changing (orientation-less) vector
field along which the class labels change. This vector field lets us locallyunderstand the Bayes
classifier.
We remark that ζ(x0)becomes a zero vector, for example, when P(Y/\e}atio\slash=g∗(x)|X=x)|x=x0is
equal to one in some neighborhood of x0. Our explanation vector fits well to classifiers where the
conditional distribution P(Y=c|X=x)is usually not completely flat in some regions. In the
case of deterministic classifiers, despite of this issue, Parzen window estimato rs with appropriate
widths (Section 3) can provide meaningful explanation vectors for many sa mples in practice (see
alsoSection8).
Inthecaseofbinaryclassificationwedirectlydefinelocalexplanationvec torsaslocalgradients
of theprobabilityfunction p(x)=P(Y=1|X=x)of thelearnedmodel forthepositiveclass.
For a probability function p:ℜd→[0,1]of a classification model learned from examples
{(x1,y1),...,(xn,yn)} ∈ℜd×{−1,+1}the explanation vector for a classified test point x0is the
localgradientof patx0:
Definition 2
ηp(x0):=∇p(x)|x=x0.
Bythisdefinitiontheexplanation ηisagaina d-dimensionalvectorjustlikethetestpoint x0is.
Thesignofeachofitsindividualentriesindicateswhetherthepredictionwo uldincreaseordecrease
when the corresponding feature of x0is increased locally and each entry’s absolute value gives the
amount of influence in the change in prediction. The vector ηgives the direction of the steepest
ascent from the test point to higher probabilities for the positive class. Fo r binary classification the
negative version −ηp(x0)indicates the changes in features needed to increase the probability for
thenegativeclasswhich maybeespecially usefulfor x0predictedinthepositiveclass.
1805
BAEHRENS , SCHROETER , HARMELING , KAWANABE , HANSEN AND M¨ULLER
For an example we apply Definition 2 to model predictions learned by Gaussian Process Clas-
sification(GPC),seeRasmussenandWilliams (2006). GPC isusedherefor threereasons:
(i) In our real-world application we are interested in classifying data from d rug discovery, which
is an area where Gaussian processes have proven to show state-of-th e-art performance, see, for
example, Obrezanova and Segall (2010), Obrezanova et al., Schroe ter et al. (2007c), Schroeter
et al. (2007a), Schroeter et al. (2007b), Schwaighofer et al. (200 7), Schwaighofer et al. (2008) and
Obrezanovaetal.(2008). Itisnaturaltoexpectamodelwithhighpred ictionaccuracyonacomplex
problem to capture relevant structure of the data which is worth explaining a nd may give domain
specific insights in addition to the values predicted. For an evaluation of the ex plaining capabilities
of ourapproach onacomplex problemfromchemoinformatics seeSection 6.
(ii) GPC does model the class probability function used in Definition 2 directly. F or other classi-
fication methods such as Support Vector Machines that do not provide a p robability function as its
outputwe givean exampleforan estimationmethodstartingfromDefinition1inSe ction3.
(iii) The local gradients of the probability function can be calculated analytica lly for differentiable
kernelsas we discussnext.
Letf(x)=∑n
i=1αik(x,xi)beaGaussianProcess(GP)modeltrainedonsamplepoints x1,...,xn∈
ℜdwherekisakernelfunctionand αiarethelearnedweightsofeachsamplepoint. Foratestpoint
x0∈ℜdlet varf(x0)be the variance of f(x0)under the GP posterior of f. Because the posterior
cannot be calculated analytically for GP classification models, we used an ap proximation by ex-
pectation propagation (EP) (Kuss and Ramussen, 2005). In the case of the probit likelihood term
defined by the error function, the probability for being of the positive clas sp(x0)can be computed
easilyfromthis approximatedposterioras
p(x0)=1
2erfc/parenleftBigg
−f(x0)√
2·/radicalbig
1+varf(x0)/parenrightBigg
,
whereerfcdenotesthecomplementaryerrorfunction(seeEquation6inSc hwaighoferetal.,2008).
Thenthelocalgradient of p(x0)is givenby2
∇p(x)|x=x0=exp/parenleftBig
−f(x0)2
2(1+varf(x0))/parenrightBig
√
2π/parenleftBigg
∇f(x)|x=x0/radicalbig
1+varf(x0)−1
2f(x0)
(1+varf(x0))3
2∇varf(x)|x=x0/parenrightBigg
.(1)
Asakernelfunctionchoose,forexample,theRBF-kernel k(x0,x1)=exp(−w(x0−x1)2),whichhas
the derivative (∂/∂x0,j)k(x0,x1)=−2wexp(−w(x0−x1)2)(x0,j−x1,j)forj∈{1,...,d}. Then the
elements of thelocalgradient ∇f(x)|x=x0are
∂f
∂x0,j=−2wn
∑
i=1αiexp(−w(x0−xi)2)(x0,j−xi,j)forj∈{1,...,d}.
2. For a detailed derivation, seeAppendix A.1.
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HOW TOEXPLAININDIVIDUAL CLASSIFICATION DECISIONS
Forvar f(x0)=k(x0,x0)−kT
∗(K+Σ)−1k∗thederivativeis givenby3
∇varf(x)|x=x0=∂varf
∂x0,j=/parenleftbigg∂
∂x0,jk(x0,x0)/parenrightbigg
−2∗kT
∗(K+Σ)−1∂
∂x0,jk∗forj∈{1,...,d}.
0
0.2
0.4
0.6
0.8
100.20.40.60.81−1−0.500.51
(a) Object0
0.2
0.4
0.6
0.8
100.20.40.60.8100.20.40.60.81
(b) Model
0
0.2
0.4
0.6
0.8
100.10.20.30.40.50.60.70.80.91
(c) Local explanation vectors0
0.2
0.4
0.6
0.8
100.10.20.30.40.50.60.70.80.91
(d) Directionof explanation vectors
Figure1: ExplainingsimpleobjectclassificationwithGaussianProcesses
Panel (a) of Figure 1 shows the training data of a simple object classification task and panel (b)
showsthemodellearnedusingGPC.4Thedataislabeled −1forthebluepointsand +1forthered
points. As illustrated in panel (b) the model is a probability function for the pos itive class which
gives every data point a probability of being in this class. Panel (c) show s the probability gradient
of the model together with the local gradient explanation vectors. Along the h ypotenuse and at the
cornersofthetriangleexplanationsfrombothfeaturesinteracttowards thetriangleclasswhilealong
3. Herek∗= (k(x0,x1),...,k(x0,xn))Tis the evaluation of the kernel function between the test point x0and every
training point. Σis the diagonal matrix of the variance parameter. For details see Rasmuss en and Williams (2006,
Chapter 3).
4. Hyperparameters weretuned by agradient ascend ontheevidence .
1807
BAEHRENS , SCHROETER , HARMELING , KAWANABE , HANSEN AND M¨ULLER
the edges the importance of one of the two feature dimensions dominates. At the transition from
the negative to the positive class the length of the local gradient vectors re presents the increased
importance of the relevant features. In panel (d) we see that explanatio ns close to the edges of the
plot (especially in the right hand side corner) point away from the positiv e class. However, panel
(c)showsthat theirmagnitude isverysmall. Fordiscussionof thisissueseeS ection 8.
3. Estimating Explanation Vectors
Several classification methods directly estimate the decision rule, which often h as no interpretation
astheprobabilityfunctionintermsofDefinition2. ForexampleSupportVector Machinesestimate
adecisionfunctionof theform
f(x)=n
∑
i=1αik(xi,x)+b,
αi,b∈ℜ. Suppose we have two classes (each with one cluster) in one dimension (se e Figure 2)
andtrainaSVMwithRBFkernel. Forpointsoutsidethedataclusters f(x)tendstozero. Thus,the
derivativeof f(x)(shownasarrowsabovethecurves)forpointsontheveryleftoronth everyright
sideoftheaxiswillpointtothewrongside. Inthefollowing,wewillexplainhow explanationscan
0
xp( | )y=1 x1
0.5
0xclassifier output
Figure2: Classifier outputof anSVM (top)comparedto p(y=1|x)(bottom).
beobtained forsuchclassifiers.
In practice, we do not have access to the true underlying distribution P(X,Y). Consequently,
we have no access to the Bayes classifier as defined in Section 2. Instead , we can apply sophis-
ticated learning machinery like Support Vector Machines (Vapnik, 1995; S ch¨olkopf and Smola,
2002;M¨ulleretal.,2001)thatestimatessomeclassifier gthattriestomimic g∗. Fortestdatapoints
z1,...,zm∈ℜdwhich are assumed to be sampled from the same unknown distribution as the train -
ing data,gestimates labels g(z1),...,g(zm). Now, instead of trying to explain g∗, to which we have
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HOW TOEXPLAININDIVIDUAL CLASSIFICATION DECISIONS
noaccess,wewilldefineexplanationvectorsthathelpusunderstandthe classifier gonthetestdata
points.
Since we do not assume that we have access to some intermediate real-valued classifier output
here(ofwhich gmightbeathresholdedversionandwhichfurthermightnotbeanestimateof P(Y=
c|X=x)),wesuggesttoapproximate gbyanotherclassifier ˆ g,theactualformofwhichresembles
the Bayes classifier. There are several choices for ˆ g, for example, GPC, logistic regression, and
Parzen windows.5In this paper we apply Parzen windows to the training points to estimate the
weighted classdensities P(Y=c)·P(X|Y=c),for theindexset Ic={i|g(xi)=c}
ˆpσ(x,y=c)=1
n∑
i∈Ickσ(x−xi) (2)
andwithkσ(z)beingaGaussiankernel kσ(z)=exp(−0.5z⊤z/σ2)/√
2πσ2(asalwaysotherkernels
arealsopossible). This estimates P(Y=c|X=x)forallc,
ˆpσ(y=c|x)=ˆpσ(x,y=c)
ˆpσ(x,y=c)+ˆpσ(x,y/\e}atio\slash=c)≈∑i∈Ickσ(x−xi)
∑ikσ(x−xi)(3)
andthus isan estimateofthe Bayes classifier(thatmimics g),
ˆgσ(x)=arg min
c∈{1,...,C}ˆpσ(y/\e}atio\slash=c|x).
This approach has the advantage that we can use our estimated classifier gto generate any amount
oflabeleddataforconstructing ˆ g. Thesinglehyper-parameter σischosensuchthat ˆ gapproximates
g(whichwewant toexplain),thatis,
ˆσ:=argmin
σm
∑
j=1I/braceleftbig
g(zj)/\e}atio\slash=ˆgσ(zj)/bracerightbig
,
whereI{···}is the indicator function. σis assigned the constant value ˆσfrom here on and omitted
as asubscript. For ˆ gitis straightforwardtodefine explanationvectors:
Definition 3
ˆζ(z):=∂
∂xˆp(y/\e}atio\slash=g(z)|x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
x=z=/parenleftBig
∑i/∈Ig(z)k(z−xi)/parenrightBig/parenleftBig
∑i∈Ig(z)k(z−xi)(z−xi)/parenrightBig
σ2/parenleftbig
∑n
i=1k(z−xi)/parenrightbig2
−/parenleftBig
∑i/∈Ig(z)k(z−xi)(z−xi)/parenrightBig/parenleftBig
∑i∈Ig(z)k(z−xi)/parenrightBig
σ2/parenleftbig
∑n
i=1k(z−xi)/parenrightbig2.
This is easily derived using Equation (3) and the derivative of Equation ( 2), see Appendix A.3.1.
Note that we use ginstead of ˆ g. This choice ensures that the orientation ofˆζ(z)fits to the labels
assignedby g,whichallows better interpretations.
In summary, we imitate the classifier gwhich we would like to explain locally by a Parzen
windowclassifier ˆ gthathasthesameformastheBayesestimatorandforwhichwecanestimatethe
5. ForSupportVectorMachinesPlatt(1999)fitsasigmoidfunctiontomap theoutputstoprobabilities. Inthefollowing,
we willpresent amoregeneral method for estimatingexplanation vectors .
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BAEHRENS , SCHROETER , HARMELING , KAWANABE , HANSEN AND M¨ULLER
explanationvectorsusingDefinition3. Practicallytherearesomecaveats: T hemimickingclassifier
ˆghas to be estimated from geven in high dimensions; this needs to be done with care. However,
in principle we have an arbitrary amount of training data available for constr ucting ˆgsince we may
useour estimatedclassifier gtogeneratelabeled data.
4. ExplainingIris FlowerClassification by k-NearestNeighbors
TheIrisflowerdataset(introducedinFisher,1936)describes150flo wersfromthegenusIrisbyfour
features: sepal length, sepal width, petal length, and petal width, all of which are easily measured
properties of certain leaves of the corolla of the flower. There are three clusters in this data set that
correspondtothree differentspecies: Irissetosa ,Irisvirginica ,andIrisversicolor .
Letusconsidertheproblemofclassifyingthedatapointsof Irisversicolor (class0)againstthe
othertwospecies(class1). Weappliedstandardclassificationmachineryto thisproblemasfollows:
•Class 0consistsofall examples of Irisversicolor .
•Class 1consistsofall examples of Irissetosa andIrisvirginica .
•Randomly split150datapoints into100trainingand50 testexamples.
•Normalize trainingandtestsetusingthemeanand varianceof thetrainingset.
•Applyk-nearest neighbor classification with k=4 (chosen by leave-one-out cross-validation
onthe trainingdata).
•Trainingerror is3% (i.e.,3mistakesin100).
•Testerroris8% (i.e.,4mistakesin50).
In order to estimate explanation vectors we mimic the classification results with a Par zen window
classifier. The best fit (3% error) is obtained with a kernel width of σ=0.26(chosen by leave-one-
outcross-validationonthetrainingdata).
Since the explanation vectors live in the input space we can visualize them with scatter plots of
the initially measured features. The resulting explanations (i.e., vectors) for the test set are shown
in Figure 3. The blue markers correspond to explanation vectors of Iris setosa and the red markers
correspond to those of of Iris virginica (both class 1). Both groups of markers point to the green
markers of Iris versicolor . The most important feature is the combination of petal length and petal
width (see the corresponding panel), the product of which correspon ds roughly to the area of the
petals. However,theresultingexplanations for thetwospecies inclass1ar edifferent:
•Irissetosa (class1)isdifferentfrom Irisversicolor (class0)becauseitspetalareais smaller.
•Irisvirginica (class1)isdifferentfrom Irisversicolor (class0)becauseitspetalareais larger.
Alsothedimensionsofthesepal(anotherpartoftheblossom)arerelevan t,butnotasdistinguishing.
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HOW TOEXPLAININDIVIDUAL CLASSIFICATION DECISIONS
Figure3: Scatterplotsoftheexplanationvectorsforthetestdata. Showna reallexplanationvectors
for both classes: class 1 containing Iris setosa (shown in blue) and Iris virginica (shown
in red) versus class 0 containing only the species Iris versicolor (shown in green). Note
thattheexplanationswhyanIrisflowerisnotan Irisversicolor isdifferentfor Irissetosa
andIrisvirginica .
5. ExplainingUSPSDigit Classification bySupportVector Mach ine
We now apply the framework of estimating explanation vectors to a high dimension al data set, the
USPSdigits. Theclassificationproblemthatwedesignedforillustrationpurp osesisdetailedinthe
followinglist:
•digits: 16 ×16images thatarereshapedto256 ×1dimensionalcolumnvectors
•classifier: SVM from Schwaighofer (2002) with RBF kernel width σ=1 and regularization
constantC=10(chosenbygridsearchincross-validationon thetrainingdata).
•trainingset: 47“twos”,53“eights”;trainingerror0.00
•testset: 48“twos”,52 “eights”;testerror0.05
WeapproximatedtheestimatedclasslabelsobtainedbytheSVMwiththeParzenwin dowclassifier
(Parzen window size σ=10.2505, chosen by grid search in cross-validation on the training data).
The SVM and the Parzen window classifier only disagreed on 2% of the test examples, so a good
1811
BAEHRENS , SCHROETER , HARMELING , KAWANABE , HANSEN AND M¨ULLER
Figure4: USPS digits (training set): “twos” (left) and “eights” (right) with co rrect classification.
For each digit from left to right: (i) explanation vector (with black being neg ative, white
beingpositive),(ii)theoriginaldigit,(iii-end)artificialdigitsalongtheexplan ationvector
towardstheother class.
Figure5: USPS digits(testsetbottompart): “twos”(left)and“eights”(rig ht)withcorrectclassifi-
cation. Foreachdigitfromlefttoright: (i)explanationvector(withblackbe ingnegative,
whitebeingpositive),(ii)theoriginaldigit,(iii-end)artificialdigitsalongtheex planation
vector towardstheother class.
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HOW TOEXPLAININDIVIDUAL CLASSIFICATION DECISIONS
fit was achieved. Figures 4 and 5 show our results. All parts show three examples per row. For
eachexamplewedisplayfromlefttoright: (i)theexplanationvector,(ii)theo riginaldigit,(iii-end)
artificial digits along the explanation vector towards the other class.6These artificial digits should
help to understand and interpret the explanation vector. Let us first hav e a look at the results on the
trainingset:
Figure4(leftpanel): Letusfocusonthetopexampleframedinred. Thelinethatformsthe“two”
is part of some “eight” from the data set. Thus the parts of the lines that are miss ing show
up in the explanation vector: if the dark parts (which correspond to the missin g lines) are
added to the “two” digit then it will be classified as an “eight”. In other words , because of
the lack of those parts the digit was classified as a “two” and not as an “eigh t”. A similar
explanationholdsforthemiddleexampleframedinredinthesameFigure. Notall examples
transform easily to “eights”: Besides adding parts of black lines, some exis ting black spots
(that make the digit be a “two”) must be removed. This is reflected in the explan ation vector
bywhitespots/lines. Thebottom“two”,framedinred,isactuallyadashandisin thedataset
by mistake. However, its explanation vector shows nicely which parts have to be added and
which havetoberemoved.
Figure4(right panel): Weseesimilarresultsforthe“eights”class. Theexplanationvectorsagain
tell us how the “eights” have to change to become classified as “twos”. Howe ver, sometimes
the transformation does not reach the “twos”. This is probably due to the fa ct that some of
the “eights”areinsidethe cloudof “eights”.
On the test set the explanation vectors are not as pronounced as on the tr aining set. However, they
showsimilar tendencies:
Figure5(leftpanel): Weseethecorrectlyclassified“twos”. Let’sfocusontheexampleframedin
red. Againtheexplanationvectorshowsushowtoedittheimageofthe“two”to transformit
into an “eights”, that is, exactly which parts of the digit were important for the classification
result. For several other “twos” the explanation vectors do not directly le ad to the “eights”
butweight thedifferentpartsof thedigits thatwererelevantfortheclas sification.
Figure5(right panel): Similarly to the training data, we see that also these explanation vectors
are not bringing all “eights” to “twos”. Their explanation vectors mainly sug gest to remove
mostofthe“eights”(blackpixels)andaddsomeblackinthelowerpart(thelig htparts,which
looklike awhiteshadow).
Overall, the explanation vectors tell us how to edit our example digits to change the assigned class
label. Hereby, we get a better understanding of the reasons why the cho sen classifier classified the
wayitdid.
6. ExplainingMutagenicity Classification byGaussianProces ses
InthefollowingSectionwedescribeanapplicationofourlocalgradientex planationmethodologyto
acomplexrealworlddataset. Ouraimistofindstructurespecifictotheproble mdomainthathas not
6. For the sake of simplicity, no intermediate updates were performed, tha t is, artificial digits were generated by taking
equal-sized steps inthe directiongiven by theoriginalexplanation vector c alculated forthe originaldigit.
1813
BAEHRENS , SCHROETER , HARMELING , KAWANABE , HANSEN AND M¨ULLER
beenfedintotrainingexplicitlybutiscapturedimplicitlybytheGPCmodelinthehigh- dimensional
featurespaceusedtodetermineitsprediction. Weinvestigatethetaskofpred ictingAmesmutagenic
activityofchemicalcompounds. Notbeingmutagenic(i.e.,notabletocausemu tationsintheDNA)
is an important requirement for compounds under investigation in drug disco very and design. The
Ames test (Ames et al., 1972) is a standard experimental setup for measuring mutagenicity. The
followingexperimentsarebasedonasetofAmestestresultsfor6512che micalcompoundsthatwe
publishedpreviously.7
GPCwas applied asfollows:
•Class 0consistsofnon-mutagenic compounds.
•Class 1consistsofmutagenic compounds.
•Randomly split6512datapoints into2000trainingand4512 testexamples such that:
–Thetrainingsetconsistsof equallymanyclass 0andclass 1examples.
–Forthe steroidcompound classthebalance inthetrainingandtestsetisenfo rced.
•10 additional random splits were investigated individually. This confirmed the results pre-
sentedbelow.
•Each example (chemical compound) is represented by a vector of counts o f 142 molecular
substructurescalculatedusingthe D RAGONsoftware(Todeschiniet al.,2006).
•Normalize trainingandtestsetusingthemeanand varianceof thetrainingset.
•Apply GPCmodelwith RBF kernel.
•Performance (84 % area under curve) confirms our previous results ( Hansen et al., 2009).
Errorratescan beobtained fromFigure6.
Togetherwiththepredictionwecalculatedtheexplanationvector(asintroduc edinDefinition2)for
each testpoint. Theremainder of thisSectionis anevaluationof theselocale xplanations.
In Figures 7 and 8 we show the distribution of the local importance of selected features across
thetestset: Foreachinputfeaturewegenerateahistogramoflocalimpor tancevalues,asindicated
by its corresponding entry in the explanation vector of each of the 4512 tes t compounds. The
features examined in Figure 7 are counts of substructures known to caus e mutagenicity. We show
all approved “specific toxicophores” introduced by Kazius et al. (200 5) that are also represented
in the D RAGONset of features. The features shown in Figure 8 are known to detoxify c ertain
toxicophores (again see Kazius et al., 2005). With the exception of 7(e) th e toxicophores also have
a toxifying influence according to our GPC prediction model. Feature 7(e) s eems to be mostly
irrelevant for the prediction of the GPC model on the test points. In contras t the detoxicophores
show overall negative influence on the prediction outcome of the GPC model. Modifying the test
compoundsbyaddingtoxicophoreswillincreasetheprobabilityofbeingmuta genicaspredictedby
theGPC modelwhileadding detoxicophores willdecreasethispredicted pro bability.
7. See Hansen et al. (2009) for results of modeling this set using differ ent machine learning methods. The data itself is
available online at http://ml.cs.tu-berlin.de/toxbenchmark .
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HOW TOEXPLAININDIVIDUAL CLASSIFICATION DECISIONS
00.20.40.60.8100.20.40.60.81
false positive ratetrue positive rate
AUC = 0.84
Figure6: Receiver operatingcharacteristiccurveofGPC modelfor muta genicityprediction
We have seen that the conclusions drawn from our explanation vectors a gree with established
knowledgeabouttoxicophoresanddetoxicophores. Whilethisisreassu ring,suchasanitycheckre-
quiredexistingknowledgeaboutwhichcompoundsaretoxicophoresand detoxicophoresandwhich
arenot. Thusitisinterestingtoaskwhetherwealsocouldhave discovered thatknowledgefromthe
explanation vectors. To answer this question we ranked all 142 features by the means of their local
gradients.8Clear trends result: 9 out of 10 known toxicophores can be found close to the top of the
list(meanrankof19). Theonlyexception(rank81)isthearomaticnitrosamin efeature.9Thistrend
is even stronger for the detoxicophores: The mean rank of these five fe atures is 138 (out of 142),
that is, they consistently exhibit the largest negative local gradients. Con sequently, the established
knowledge about toxicophores and detoxicophores could indeed have beendiscovered using our
methodology.
In the following paragraph we will discuss steroids10as an example of an important compound
class for which the meaning of features differs from this global trend, so that local explanation
vectorsareneeded tocorrectlyidentifyrelevantfeatures.
Figure 9 displays the difference in relevance of epoxide (a) and aliphatic n itrosamine (c) sub-
structures for the predicted mutagenicity of steroids and non-steroid compo unds. For compari-
son we also show the distributions for compounds chosen at random from the test set (b,d). Each
subfigure contains two measures of (dis-)similarity for each pair of distribu tions. The p-value of
the Kolmogorov-Smirnoff test (KS) gives the probability of error when re jecting the hypothesis
that both relative frequencies are drawn from the same underlying distrib ution. The symmetrized
8. Tables resulting from this ranking are made available as a supplement to this paper and can be downloaded from the
journals website.
9. This finding agrees with the result obtained by visually inspecting Figure 7 (e). We found that only very few com-
pounds withthisfeature arepresentinthedata set. Consequently,detectio n of this featureis onlypossibleif enough
of these few compounds are included in the training data. This was not the c ase in the random split used to produce
the resultspresented above.
10. Steroidsarenaturalproductsandoccurinhumans,animals,andp lants. Theyhaveacharacteristicbackbonecontain-
ingfourfusedcarbon-rings. Manyhormonesimportanttothedevelop mentofthehumanbodyaresteroids,including
androgens, estrogens, progestagens, cholesterol and natural an abolics. These have been used as starting points for
the development of many differentdrugs,including themostreliable co ntraceptives currently onthe market.
1815
BAEHRENS , SCHROETER , HARMELING , KAWANABE , HANSEN AND M¨ULLER
−1−0.5 00.5100.020.040.060.080.10.120.14
local gradient DR17:nArNO2relative frequency
(a) aromatic nitro−1−0.5 00.5100.020.040.060.080.1
local gradient DR17:nArNH2relative frequency
(b) aromatic amine−1−0.5 00.5100.010.020.030.040.05
local gradient DR17:nArNOrelative frequency
(c) aromatic nitroso
−1−0.5 00.5100.020.040.060.08
local gradient DR17:nRNNOxrelative frequency
(d) aliphatic nitrosamine−1−0.5 00.5100.010.020.030.040.050.060.07
local gradient DR17:nArNNOxrelative frequency
(e) aromatic nitrosamine−1−0.5 00.5100.010.020.030.040.05
local gradient DR17:nOxiranesrelative frequency
(f) epoxide
−1−0.5 00.5100.010.020.030.040.050.060.07
local gradient DR17:nAziridinesrelative frequency
(g) aziridine−1−0.5 00.5100.020.040.060.080.10.12
local gradient DR17:nN=Nrelative frequency
(h) azide−1−0.5 00.5100.020.040.060.080.1
local gradient DR17:nArNHOrelative frequency
(i) aromatic hydroxylamine
−1−0.5 00.5100.010.020.030.040.050.060.07
local gradient X:AliphaticHaliderelative frequency
(j) aliphatic halide
Figure7: Distribution of local importance of selected features across the te st set of 4512 com-
pounds. Nineoutoftenknowntoxicophores(Kaziusetal.,2005)indee dexhibitpositive
localgradients.
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HOW TOEXPLAININDIVIDUAL CLASSIFICATION DECISIONS
−1−0.5 00.5100.020.040.060.080.1
local gradient DR17:nSO2Nrelative frequency
(a) sulfonamide−1−0.5 00.5100.0050.010.0150.020.0250.030.035
local gradient DR17:nSO2OHrelative frequency
(b) sulfonicacid−1−0.5 00.5100.010.020.030.040.05
local gradient DR17:nS(=O)2relative frequency
(c) arylsulfonyl
−1−0.5 00.5100.010.020.030.040.05
local gradient DR17:nRCOOHrelative frequency
(d) aliphatic carboxylic acid−1−0.5 00.5100.020.040.060.080.1
local gradient DR17:nArCOOHrelative frequency
(e) aromatic carboxylic acid
Figure8: Distribution of local importance of selected features across the te st set of 4512 com-
pounds. Allfiveknowndetoxicophores exhibit negativelocal gradien ts
Kullback-Leibler divergence (KLD) gives a metric of the distance betwee n the two distributions.11
Whilecontainingepoxidesgenerallytendstomakemoleculesmutagenic(seedisc ussionabove),we
donotobservethiseffectforsteroids: InFigure9(a),almostallepo xidecontainingnon-steroidsex-
hibitpositivegradients,therebyfollowingtheglobaldistributionofepoxide containingcompounds
as shown in Figure 7(f). In contrast, almost all epoxide containing steroid s exhibit gradients just
below zero. “Immunity” of steroids to the epoxide toxicophore is an establishe d fact and has first
been discussed by Glatt et al. (1983). This peculiarity in chemical space is clearly exhibited by
the local explanation given by our approach. For aliphatic nitrosamine, the situation in the GPC
model is less clear but still the toxifying influence seems to be less in steroids tha n in many other
compounds. To our knowledge, this phenomenon has not yet been discu ssed in the pharmaceutical
literature.
Inconclusion,wecanlearnfromtheexplanation vectorsthat:
•Toxicophores tendtomakecompounds mutagenic(class1).
•Detoxicophores tend tomakecompounds non-mutagenic (class0).
•Steroids are immune to the presence of some toxicophores (epoxide, possib ly also aliphatic
nitrosamine).
11. SymmetryisachievedbyaveragingthetwoKullback-Leiblerdiverge nces:KL(P1,P2)+KL(P2,P1)
2,comparetoJohnson
and Sinanovic (2000). To prevent zero-values in the histograms which would lead to infinite KL distances, an ε>0
has been added toeach bincount.
1817
BAEHRENS , SCHROETER , HARMELING , KAWANABE , HANSEN AND M¨ULLER
−0.5 0 0.5 100.050.10.150.2KS p−value = 0.00
symm. KLD = 23.14
local gradient DR17:nOxiranesrelative frequency
steroid
non−steroid
(a) epoxide feature: steroidvs. non-steroid−0.5 0 0.5 100.020.040.060.08KS p−value = 0.12
symm. KLD = 11.08
local gradient DR17:nOxiranesrelative frequency
random
others
(b) epoxide feature: randomcompounds vs. therest
−0.2 00.2 0.4 0.600.050.10.150.20.25KS p−value = 0.00
symm. KLD = 12.51
local gradient DR17:nRNNOxrelative frequency
steroid
non−steroid
(c) aliphatic nitrosaminefeature: steroidvs. non-steroid−0.2 00.2 0.4 0.600.020.040.060.080.1KS p−value = 0.16
symm. KLD = 5.07
local gradient DR17:nRNNOxrelative frequency
random
others
(d) aliphatic nitrosamine feature: random compounds vs.
therest
Figure9: The local distribution of feature importance to steroids and rando m non-steroid com-
poundssignificantlydiffersfortwoknowntoxicophores. Thesmalllocal gradientsfound
forthesteroids(showninblue)indicatethatthepresenceofeachtoxicop horeisirrelevant
tothemoleculestoxicity. Fornon-steroids(showninred)theknowntoxicoph oresindeed
exhibitpositivelocal gradients.
7. Related Work
Assigning potentially different explanations to individual data points distingu ishes our approach
from conventional feature extraction methods that extract global featur es that are relevant for all
data points, that is, those features that allow to achieve a small overall pred iction error. Our notion
of explanation is not related to the prediction error, but only to the label pro vided by the prediction
algorithm. Eveniftheerrorislarge,ourframeworkisabletoanswertheque stionwhythealgorithm
has decidedon adatapointthe wayitdid.
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HOW TOEXPLAININDIVIDUAL CLASSIFICATION DECISIONS
Theexplanationvectorproposedhereissimilarinspirittosensitivityanalysis whichiscommon
invariousareasofinformationscience. Aclassicalexampleisoutliersensiti vityinstatistics(Ham-
pel et al., 1986). In this case, the effects of removing single data points on estimated parameters
are evaluated by an influence function. If the influence for a data point is significantly large, it is
detected as an outlier and should be removed for the following analysis. In r egression problems,
leverage analysis is a procedure along similar lines. It detects leverage po ints which have potential
to give large impact on the estimate of the regression function. In contrast to the influential points
(outliers),removingaleveragesamplemaynotactuallychangetheregress or,ifitsresponseisvery
close to the predicted value. E.g., for linear regression the samples whose in puts are far from the
mean are the leverage points. Our framework of explanation vectors cons iders a different view. It
describes the influence of movingsingle data points locally and it thus answers the question which
directions are locally most influential to the prediction. The explanation vector s are used to extract
sensitive features that are relevant to the prediction results, rather than detecting/eliminating the
influentialsamples.
In recent decades, explanation of results by expert systems has been an important topic in the
ArtificialIntelligencecommunity. EspeciallyforexpertsystemsbasedonBay esianbeliefnetworks,
such explanation is crucial in practical use. In this context sensitivity ana lysis has also been used
as a guiding principle (Horvitz et al., 1988). There the influence is evaluate d by removing a set of
variables (features) from the evidence and the explanation is constructe d from those variables that
affect inference (relevant variables). For example, Suermondt (19 92) measures the cost of omitting
asinglefeature Eibythecross-entropy
H−(Ei)=H(p(D|E);P(D|E\Ei))=N
∑
j=1P(dj|E)logP(dj|E)
p(dj|E\Ei),
whereEdenotes the evidence and D= (d1,...,dN)Tis the target variable. The cost of a subset
F⊂Ecan be defined similarly. This line of research is more connected to our work , because
explanation candepend ontheassignedvalues of theevidence E,andis thuslocal.
Similarly Robnik-Sikonja and Kononenko (2008) and Strumbelj and Konone nko (2008) try to
explain the decision of trained kNN-, SVM-, and ANN-models for individua l instances by measur-
ing the difference in their prediction with sets of features omitted. The cost of omitting features
is evaluated as the information difference, the log-odds ratio, or the differ ence of probabilities be-
tweenthemodelwithknowledgeaboutallfeaturesandwithomissions,respe ctively. Toknowwhat
the prediction would be without the knowledge of a certain feature the model is r etrained for every
choice of features whose influence is to be explained. To save the time of co mbinatorial training
Robnik-Sikonja and Kononenko (2008) propose to use neutral values which have to be estimated
by a known prior distribution of all possible parameter values. As a theoretic al framework for con-
sidering feature interactions, Strumbelj and Kononenko (2008) propos e to calculate the differences
between modelpredictions foreverychoiceof featuresubset.
For multi-layer perceptrons Fraud and Clrot (2002) measure the importanc e of individual in-
put variables on clusters of test points. Therefore the change in the model output is evaluated for
the change of a single input variable in a chosen interval while all other inpu t variables are fixed.
LemaireandFeraud(2007)useasimilarapproachonaninstancebyinstan cebasis. Byconsidering
eachinputvariableinturnthereisnowaytomeasureinputfeatureinteractio nsonthemodeloutput
(seeLeCunetal.,1998).
1819
BAEHRENS , SCHROETER , HARMELING , KAWANABE , HANSEN AND M¨ULLER
p( | )y=1 x
x1x1
x20.5
Figure10: ζ(x)isthe zerovector inthemiddleof theclusterin themiddle.
The principal differences between our approach and these framewor ks are: (i) We consider
continuous features and no structure among them is required, while some oth er frameworks start
from binary features and may require discretization steps with the need to es timate parameters for
it. (ii) We allow changes in any direction, that is, any weighted combination of va riables, while
other approaches onlyconsider onefeatureatatimeor theomissionofase tof variables.
8. Discussion
We have shown that our methods for calculating / estimating explanation vector s are useful in a
varietyofsituations. Inthefollowingwediscusstheir limitations.
8.1 What Can We Doif theDerivativeis Zero?
This situation is depicted in Figure 10. In the lower panel we see a two-dimensio nal data set con-
sisting of three clusters. The middle cluster has a different class than the clu sters on the left and
on the right. Only the horizontal coordinate (i.e., x1) is relevant for the classification. The upper
panel shows the projected data and a representative slice of ζ(x). However, the explanation ζ(x)
for the center point of the middle cluster is the zero vector, because at that pointp(Y=1|X=x)
is maximal. What can we do in such situations? Actually, the (normalized) explana tion vector
is derived from the following optimization problem for finding the locally most infl uential direc-
tion: argmax/bardblε/bardbl=1{p(Y/\e}atio\slash=g∗(x0)|X=x0+ε)−p(Y/\e}atio\slash=g∗(x0)|X=x0)}. In case that the first deriva-
tive of the above criterion is zero, its Taylor expansion starts from the sec ond order term, which
is a quadratic form of its Hessian matrix. In the example data set with three cluste rs, the explana-
tion vector is constant along the second dimension. The most interesting direction is given by the
eigenvector corresponding to the largest eigenvalue of the Hessian. Th is direction will be in our
example along the first dimension. Thus, we can learn from the Hessian that the first coordinate
is relevant for the classification, but we do not obtain an orientation for it. I nstead it means that
both directions (left and right) will influence the classification. However, if the conditional distri-
1820
HOW TOEXPLAININDIVIDUAL CLASSIFICATION DECISIONS
butionP(Y=1|X=x)is flat in some regions, no meaningful explanation can be obtained by the
gradient-based approach with the remedy mentioned above. Practically, by using Parzen window
estimators with larger widths, the explanation vector can capture coarse stru ctures of the classifier
atthepointsthatarenotsofarfromtheborders. InA.3.2wegiveanillus trationofthispoint. Inthe
future,wewouldliketoworkonglobalapproaches,forexample,based ondistancestotheborders,
or extensions of the approach by Robnik-Sikonja and Kononenko (200 8). Since these procedures
are expected to be computationally demanding, our proposal is useful in pr actice, in particular for
probabilisticclassifiers.
8.2 Does OurFramework Generate DifferentExplanations forDiffere nt Prediction Models?
When using the local gradient of the model prediction directly as in Definition 2 and Section 6,
the explanation follows the given model precisely by definition. For the estimatio n framework this
dependsonwhetherthedifferentclassifiersclassifythedatadifferen tly. Inthatcasetheexplanation
vectors will be different, which makes sense, since they should explain the classifier at hand, even
if its estimated labels were not all correct. On the other hand, if the different classifiersagree on all
labels,theexplanation willbeexactlyequal.
8.3 Which ImplicitLimitationsDoAnalyticalGradients InheritFrom Gaussia nProcess
Models?
A particular phenomenon can be observed at the boundaries of the trainin g data: Far from the
training data, Gaussian Process Classification models predict a probability o f 0.5 for the positive
class. When querying the model in an area of the feature space where pr edictions are negative,
and one approaches the boundaries of the space populated with training d ata, explanation vectors
will point away from any training data and therefore also away from area s of positive prediction.
This behavior can be observed in Figure 1(d), where unit length vector s indicate the direction of
explanation vectors. In the right hand side corner, arrows point away from the triangle. However,
we can see that the length of these vectors is so small that they are not even visible in Figure
1(c). Consequently, this property of GPC models does not pose a restric tion for identifying the
locallymostinfluentialfeaturesbyinvestigatingthefeatureswiththehighestab solutevaluesinthe
respectivepartialderivatives,as showninSection 6.
8.4 Stationarityof theData
Since explanation vectors are defined as local gradients of the model pre diction (see Definition 2),
no assumption on the data is made: The local gradients follow the predictive mod el in any case. If,
however, the model to be explained assumes stationarity of the data, the expla nation vectors will
inherit this limitation and reflect any shortcomings of the model (e.g., when the mod el is applied
to non-stationary data). Our method for estimating explanation vectors, on the other hand, assumes
stationarityof thedata.
When modeling data that is in fact non-stationary, appropriate measures to d eal with such data
sets should be taken. One option is to separate the feature space into stationa ry and non-stationary
parts using Stationary Subspace Analysis as introduced by von B ¨unau et al. (2009). For further
approaches to data set shift see Sugiyama et al. (2007b), Sugiyama et a l. (2007a), and the book by
Quionero-Candela etal.(2009).
1821
BAEHRENS , SCHROETER , HARMELING , KAWANABE , HANSEN AND M¨ULLER
9. Conclusion
This paper proposes a method that sheds light on the black boxes of nonlin ear classifiers. In other
words, we introduce a method that can explain the local decisions taken by a rbitrary (possibly)
nonlinearclassificationalgorithms. Inanutshell,theestimatedexplanationsare localgradientsthat
characterizehowadatapointhastobemovedtochangeitspredictedlabel. F ormodelswheresuch
gradient information cannot be calculated explicitly, we employ a probabilistic a pproximate mimic
of thelearningmachine tobeexplained.
To validate our methodology we show how it can be used to draw new conclus ions on how the
various Iris flowers in Fisher’s famous data set are different from ea ch other and how to identify
the features with which certain types of digits 2 and 8 in the USPS data set can b e distinguished.
Furthermore, we applied our method to a challenging drug discovery proble m. The results on that
data fully agree with existing domain knowledge, which was not available to our method. Even
local peculiarities in chemical space (the extraordinary behavior of stero ids) was discovered using
thelocal explanations givenby ourapproach.
Future directions are two-fold: First we believe that our method will find its wa y into the tool
boxes of practitioners who not only want to automatically classify their data bu t who also would
like to understand the learned classifier. Thus using our explanation frame work in computational
biology (see Sonnenburg et al., 2008) and in decision making experiments in psychophysics (e.g.,
Kienzle et al., 2009) seems most promising. The second direction is to genera lize our approach to
other predictionproblems suchas regression.
Acknowledgments
This work was supported in part by the FP7-ICT Programme of the Europe an Community, under
thePASCAL2NetworkofExcellence,ICT-216886andbyDFGGrantMU 987/4-1. Wewouldlike
to thank Andreas Sutter, Antonius Ter Laak, Thomas Steger-Hartmann and Nikolaus Heinrich for
publishingtheAmes mutagenicitydata set(Hansenetal., 2009).
AppendixA.
In the following we present the derivation of direct local gradients and illu strate aspects like the
effect of different kernel functions, outliers and local non-linearitie s. Furthermore we present the
derivation of explanation vectors based on the parzen window estimation an d illustrate how the
qualityofthefitoftheParzenwindowapproximationaffectsthequalityofthee stimatedexplanation
vectors.
A.1 Derivation of Direct Local Gradients
Equation(1)is derivedby thefollowingsteps:
1822
HOW TOEXPLAININDIVIDUAL CLASSIFICATION DECISIONS
∇p(x)|x=x0
=∇1
2erfc/parenleftBigg
−f(x)√
2∗/radicalbig
1+varf(x)/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
x=x0
=∇1
2/parenleftBigg
1−erf/parenleftBigg
−f(x)√
2∗/radicalbig
1+varf(x)/parenrightBigg/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
x=x0
=−1
2∇erf/parenleftBigg
−f(x)√
2∗/radicalbig
1+varf(x)/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
x=x0
=−exp/parenleftBig
−f(x0)2
2(1+varf(x0))/parenrightBig
√π∇/parenleftBigg
−f(x)√
2∗/radicalbig
1+varf(x)/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
x=x0
=−exp/parenleftBig
−f(x0)2
2(1+varf(x0))/parenrightBig
√π/parenleftBigg
−1√
2∇/parenleftBigg
f(x)/radicalbig
1+varf(x)/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
x=x0/parenrightBigg
=exp/parenleftBig
−f(x0)2
2(1+varf(x0))/parenrightBig
√
2π/parenleftBigg
∇f(x)|x=x0/radicalbig
1+varf(x0)+f(x0)/parenleftbigg
∇varf(x)/vextendsingle/vextendsingle
x=x0∗−1
2(1+varf(x0))−3
2/parenrightbigg/parenrightbigg
=exp/parenleftBig
−f(x0)2
2(1+varf(x0))/parenrightBig
√
2π/parenleftBigg
∇f(x)|x=x0/radicalbig
1+varf(x0)−1
2f(x0)
(1+varf(x0))3
2∇varf(x)|x=x0/parenrightBigg
.
A.2 IllustrationofDirect Local Gradients
In the following we give some illustrative examples of our method to explain models using local
gradients. Since the explanation is derived directly from the respective mo del, it is interesting to
investigate its acurateness depending on different model parameters and in instructive scenarios.
We examine the effects that local gradients exhibit when choosing differe nt kernel functions, when
introducingoutliers,and whentheclassesarenot linearlyseparableloca lly.
A.2.1 C HOICE OF KERNELFUNCTION
Figure 11 shows the effect of different kernel functions on the triang le toy data from Figure 1. The
followingobservationscanbemade:
•In any case note that the local gradients explain the model, which in turn may or may not
capture thetruesituation.
•InSubfigure11(a)thelinearkernelleadstoamodelwhichfailstocapture thenon-linearclass
separation. This model misspecification is reflected by the explanations give n for this model
inSubfigure11(b).
•The rational quadratic kernel is able to more accurately model the non-linea r separation. In
Subfigure 11(c) a non-optimal degree parameter has been chosen for illustrative purposes.
1823
BAEHRENS , SCHROETER , HARMELING , KAWANABE , HANSEN AND M¨ULLER
0
0.2
0.4
0.6
0.8
100.20.40.60.810.10.150.20.250.30.350.40.450.5
(a) linear model00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91
(b) linear explanation
0
0.2
0.4
0.6
0.8
100.20.40.60.810.3030.30320.30340.30360.30380.3040.3042
(c) rationalquadratic model00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91
(d) rational quadratic explanation
Figure11: Theeffectofdifferentkernelfunctions tothelocalgrad ientexplanations
For other parameter values the rational quadratic kernel leads to similar res ults as the RBF
kernelfunctionusedinFigure1.
•The explanations in Subfigure 11(d) obtained for this model show local pe rturbations at the
small “bumps” of the model but the trends towards the positive class are still c lear. As pre-
viously observed in Figure 1, the explanations make clear that both feature s interact at the
corners andonthehypotenuseof thetriangleclass.
A.2.2 O UTLIERS
InFigure12theeffectsoftwooutliersintheclassificationdatatoGPCwithRBFk ernelareshown.
Oncemore,notethatthelocalgradientsexplainthemodel,whichinturnmayorma ynotcapturethe
truesituation. Thesizeoftheregionaffectedbytheoutliersdependsonthe kernelwidthparameter.
Weconsider thefollowingitems:
1824
HOW TOEXPLAININDIVIDUAL CLASSIFICATION DECISIONS
0
0.2
0.4
0.6
0.8
100.20.40.60.81−1−0.500.51
(a) outliers inclasses0
0.2
0.4
0.6
0.8
100.20.40.60.8100.20.40.60.81
(b) outliers inmodel
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91
(c) outlier explanation
Figure12: Theeffectof outlierstothelocal gradientexplanations
•Local gradients are in the same way sensitive to outliers as the model which the y try to
explain. Here a single outlier deforms the model and with it the explanation which may be
extracted fromit.
•Being derivatives the sensitivity of local gradients to a nearby outlier is inc reased over the
sensitivityof themodelprediction itself.
•Thus the local gradient of a point near an outlier may not reflect a true ex planation of the
features important in reality. Nevertheless it is the model here which is wrong around an
outlier inthefirstplace.
•The histograms in the Figures 7, 8, and 9 in Section 6 show the trends of the re spective
features inthedistributionof alltestpoints andarethus notaffectedbysin gleoutliers.
To compensate for the effect of outliers to the local gradients of points in the affected region
we propose to use a sliding window method to smooth the gradients around each point of interest.
1825
BAEHRENS , SCHROETER , HARMELING , KAWANABE , HANSEN AND M¨ULLER
Thus for each point use the mean of all local gradients in the hypercube c entered at this point and
of appropriate size. This way the disrupting effect of an outlier is averag ed out for an appropriately
chosenwindowsize.
A.2.3 L OCALNON-LINEARITY
0
0.2
0.4
0.6
0.8
100.20.40.60.81−1−0.500.51
(a) locallynon-linear object0
0.2
0.4
0.6
0.8
100.20.40.60.8100.20.40.60.81
(b) locally non-linear model
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91
(c) locally non-linear explanation
Figure13: Theeffectoflocal non-linearitytothelocalgradientexplan ations
Theeffectoflocallynon-linearclassboundariesinthedataisshowninFig ure13againforGPC
withanRBF kernel. Thefollowingpointscan beobserved:
•Allthe non-linearclass boundaries areaccuratelyfollowedbythelocal g radients.
•The circle shaped region of negative examples surrounded by positive ones shows the full
rangeof featureinteractions towardsthepositiveclass.
•On the ridge of single positive instances the model introduces small valleys wh ich are re-
flected bythelocal gradients.
1826
HOW TOEXPLAININDIVIDUAL CLASSIFICATION DECISIONS
A.3 EstimatingbyParzen Window
Finallyweelaborateonsomedetailsofourestimationapproachoflocalgradie ntsbyParzenwindow
approximation. Firstwegivethederivationtoobtaintheexplanationvectora ndsecondweexamine
howtheexplanation varieswiththegoodness of fitof theParzenwindow meth od.
A.3.1 D ERIVATION OF EXPLANATION VECTORS
Thesearemoredetails onthe derivationof Definition3. We usetheindex set Ic={i|g(xi)=c}:
∂
∂xkσ(x)=−x
σ2kσ(x)
∂
∂xˆpσ(x,y/\e}atio\slash=c)=1
n∑
i/∈Ickσ(x−xi)−(x−xi)
σ2
∂
∂xˆpσ(y/\e}atio\slash=c|x)
=/parenleftBig
∑i/∈Ick(x−xi)/parenrightBig/parenleftBig
∑n
i=1k(x−xi)(x−xi)/parenrightBig
σ2/parenleftbig
∑n
i=1k(z−xi)/parenrightbig2
−/parenleftBig
∑i/∈Ick(x−xi)(x−xi)/parenrightBig/parenleftBig
∑n
i=1k(x−xi)/parenrightBig
σ2/parenleftbig
∑n
i=1k(z−xi)/parenrightbig2
=/parenleftBig
∑i/∈Ick(x−xi)/parenrightBig/parenleftBig
∑i∈Ick(x−xi)(x−xi)/parenrightBig
σ2/parenleftbig
∑n
i=1k(z−xi)/parenrightbig2
−/parenleftBig
∑i/∈Ick(x−xi)(x−xi)/parenrightBig/parenleftBig
∑i∈Ick(x−xi)/parenrightBig
σ2/parenleftbig
∑n
i=1k(z−xi)/parenrightbig2
andthus fortheindex set Ig(z)={i|g(xi)=g(z)}
ˆζ(z)=∂
∂xˆp(y/\e}atio\slash=g(z)|x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
x=z
=/parenleftBig
∑i/∈Ig(z)k(z−xi)/parenrightBig/parenleftBig
∑i∈Ig(z)k(z−xi)(z−xi)/parenrightBig
σ2/parenleftbig
∑n
i=1k(z−xi)/parenrightbig2
−/parenleftBig
∑i/∈Ig(z)k(z−xi)(z−xi)/parenrightBig/parenleftBig
∑i∈Ig(z)k(z−xi)/parenrightBig
σ2/parenleftbig
∑n
i=1k(z−xi)/parenrightbig2.
A.3.2 G OODNESS OF FIT BYPARZENWINDOW
In our estimation framework the quality of the local gradients depends on the a pproximation of the
classifierwewanttoexplainbyParzenwindowsforwhichwecancalculate theexplanationvectors
as givenbyDefinition3.
Figure 14(a) shows an SVM model trained on the classification data from Fig ure 13(a). The
localgradientsestimatedforthismodelbydifferentParzenwindowappro ximationsaredepictedin
Subfigures14(b),14(c),and14(d). We observethefollowingpoints :
1827
BAEHRENS , SCHROETER , HARMELING , KAWANABE , HANSEN AND M¨ULLER
0
0.2
0.4
0.6
0.8
100.20.40.60.8100.20.40.60.81
(a) SVM model00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91
(b) estimated explanation with σ=0.00069
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91
(c) estimated explanation with σ=0.100.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91
(d) estimated explanation with σ=1.0
Figure14: Good fit of Parzen window approximation affects the quality of th e estimated explana-
tionvectors
•TheSVM modelwas trainedwith C=10and usingan RBF kernelof width σ=0.01.
•In Subfigure 14(b) a small window width has been chosen by minimizing the mean absolute
error over the validation set of labels predicted by the SVM classifier. Thu s we obtain ex-
plaining local gradients on the class boundaries but zero vectors in the inn er class regions.
While this resembles the piecewise flat SVM model most accurately it may be more u seful
practically to choose a larger width to obtain non-zero gradients pointing to the borders in
this regionsas well. Fora moredetailed discussionof zerogradientsseeSe ction8.
•A larger width practically useful in this example is shown in Subfigure 14(c). Here the local
gradients intheinner classregions pointtotheother classas well.
1828
HOW TOEXPLAININDIVIDUAL CLASSIFICATION DECISIONS
•For a too large window width in Subfigure 14(d) the approximation fails to obtain local gra-
dientswhichcloselyfollowthemodel. Hereonlytwodirectionsareleftandtheg radientsfor
the blueclassonthe leftand onthebottompointinthewrongdirection.
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9f9a8a9f-b871-427a-bf14-cc43eaffb446 | trentmkelly/LessWrong-43k | LessWrong | State of play of AI progress (and related brakes on an intelligence explosion) [Linkpost]
This time around, I'm sharing a post on Interconnects on why he doesn't believe that the AI 2027 scenario by @Daniel Kokotajlo and many others will come true, and he has 4 sections on this plus a bonus section below:
1. How labs make progress on evaluations
2. Current AI is broad, not narrow intelligence
3. Data research is the foundation of algorithmic AI progress
4. Over-optimism of RL training
Bonus: A world with compute shifting to inference
I'll quote a couple paragraphs of each section below, but you will want to go to the article to get the full experience:
> (1st section) One of the hardest things to communicate in AI is talking down the various interpretations of evaluation progress looking vertical over time. If the evals are going from 0 to 1 in one year, doesn’t that indicate the AI models are getting better at everything super fast? No, this is all about how evaluations are scoped as “reasonable” in AI development over time.
>
> None of the popular evaluations, such as MMLU, GPQA, MATH, SWE-Bench, etc., that are getting released in a paper and then solved 18 months later are truly held out by the laboratories. They’re training goals. If these evaluations were unseen tests and going vertical, you should be much more optimistic about AI progress, but they aren’t.
>
> Instead of thinking of stacking rapid evaluation progress on one line in a cumulative, rapid improvement in intelligence, the above plots should make one think that AI is getting better at many tasks, rather than being superhuman in narrow tasks.
>
> (2nd section) In a few years, we’ll look back and see that AI is now 95% robust on a lot of things that only worked 1-5% of the time today. A bunch of new use cases will surprise us as well. We won’t see AI systems that are so intelligent that they cause seismic shifts in the nature of certain domains. Software will still be software. AI will be way better than us at completing a code task and finding a bug, but the stacks we are workin |
1e4dbf4a-55ab-4895-b5fb-fe5b2c7f6830 | trentmkelly/LessWrong-43k | LessWrong | Why personal experimentation?
Cross posted from Overcoming Bias. Comments there.
***
I experiment with many things, as do those around me. Some of this is randomization and explicit records, more is just trying different things, muttering ‘VOI‘ and repeating what felt good. I refer here to everything between a cautious banana-mustard-ham sandwich and polyamory.
Robin has suggested that I over-invest in such exploration. That most new things should be bad, and so most experimentation a private loss for public gain. What’s more, there shouldn’t be lots of low hanging fruit in trying things out. Most of the things humans frequently want to do (eat, sleep, change moods, organize time, learn, interact with others) should have been well figured out in ancient times. And anything that does still need checking out should be divided between many people.
Nonetheless, it looks to me like experimentation is worth it. Lots of the things we do seem barely satisfactory, there seem likely to be better alternatives, it seems hard to learn what has been tried for what ends, or what is good from listening to others or reading, and I and my friends seem to actually find good things by looking. e.g. Beeminder, explicit charity evaluation, unusual degrees of honesty, workflowy and explicit organization seem to often add value over the defaults, not to mention many tiny things, like banana-mustard-ham sandwiches.
If it is true that a lot of experimentation is worth it, we have a slight puzzle: if there is valuable information I might glean by experimentation, why hasn’t it been worth it for others in the past to collect it and put it where I can see it?
I will try to answer this over the next few posts. Before that, what do you think?
|
7cc648cb-ef85-4ac2-9a0f-61c9a96c8cdd | trentmkelly/LessWrong-43k | LessWrong | A policy guaranteed to increase AI timelines
The number of years until the creation of powerful AI is a major input to our thinking about risk from AI and which approaches are most promising for mitigating that risk. While there are downsides to transformative AI arriving many years from now, rather than few years from now, most people seem to agree that it is safer for AI to arrive in 2060 than in 2030. Given this, there is a lot of discussion about what we can do to increase the number of years until we see such powerful systems. While existing proposals have their merits, none of them can ensure that AI will arrive later than 2030, much less 2060.
There is a policy that is guaranteed to increase the number of years between now and the arrival of transformative AI. The General Conference on Weights and Measures defines one second to be 9,192,631,770 cycles of the optical radiation emitted during a hyperfine transition in the ground state of a cesium 133 atom. Redefining the second to instead be 919,263,177 cycles of this radiation will increase the number of years between now and transformative AI by a factor of ten. The reason this policy works is the same reason that defining a time standard works–the microscopic behavior of atoms and photons is ultimately governed by the same physical laws as everything else, including computers, AI labs, and financial markets, and those laws are unaffected by our time standards. Thus fewer cycles of cesium radiation per year implies proportionately fewer other things happening per year.
Making such a change might not sound politically tractable, but there is already precedent for making radical changes to the definition of a second. Previously it was defined in terms of Earth’s solar orbit, and before that in terms of Earth’s rotation. These physical processes and their implementations as time standards bear little resemblance to the present-day quantum mechanical standard. In contrast, a change that preserves nearly the entire standard, including all significant figur |
a2b4b2bf-9a8f-43d5-9d02-a8e595ac0037 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | Superintelligence 27: Pathways and enablers
*This is part of a weekly reading group on [Nick Bostrom](http://www.nickbostrom.com/)'s book, [Superintelligence](http://www.amazon.com/Superintelligence-Dangers-Strategies-Nick-Bostrom/dp/0199678111). For more information about the group, and an index of posts so far see the [announcement post](/lw/kw4/superintelligence_reading_group/). For the schedule of future topics, see [MIRI's reading guide](https://intelligence.org/wp-content/uploads/2014/08/Superintelligence-Readers-Guide-early-version.pdf).*
---
Welcome. This week we discuss the twenty-seventh section in the [reading guide](https://intelligence.org/wp-content/uploads/2014/08/Superintelligence-Readers-Guide-early-version.pdf): ***Pathways and enablers***.
This post summarizes the section, and offers a few relevant notes, and ideas for further investigation. Some of my own thoughts and questions for discussion are in the comments.
There is no need to proceed in order through this post, or to look at everything. Feel free to jump straight to the discussion. Where applicable, page numbers indicate the rough part of the chapter that is most related (not necessarily that the chapter is being cited for the specific claim).
**Reading**: “Pathways and enablers” from Chapter 14
---
Summary
=======
1. Is hardware progress good?
1. Hardware progress means machine intelligence will arrive sooner, which is probably bad.
2. More hardware at a given point means less understanding is likely to be needed to build machine intelligence, and brute-force techniques are more likely to be used. These probably increase danger.
3. More hardware progress suggests there will be more hardware overhang when machine intelligence is developed, and thus a faster intelligence explosion. This seems good inasmuch as it brings a higher chance of a singleton, but bad in other ways:
1. Less opportunity to respond during the transition
2. Less possibility of constraining how much hardware an AI can reach
3. Flattens the playing field, allowing small projects a better chance. These are less likely to be safety-conscious.
4. Hardware has other indirect effects, e.g. it allowed the internet, which contributes substantially to work like this. But perhaps we have enough hardware now for such things.
5. On balance, more hardware seems bad, on the impersonal perspective.
2. Would brain emulation be a good thing to happen?
1. Brain emulation is coupled with 'neuromorphic' AI: if we try to build the former, we may get the latter. This is probably bad.
2. If we achieved brain emulations, would this be safer than AI? Three putative benefits:
1. "The performance of brain emulations is better understood"
1. However we have less idea how modified emulations would behave
2. Also, AI can be carefully designed to be understood
2. "Emulations would inherit human values"
1. This might require higher fidelity than making an economically functional agent
2. Humans are not that nice, often. It's not clear that human nature is a desirable template.
3. "Emulations might produce a slower take-off"
1. It isn't clear why it would be slower. Perhaps emulations would be less efficient, and so there would be less hardware overhang. Or perhaps because emulations would not be qualitatively much better than humans, just faster and more populous of them
2. A slower takeoff may lead to better control
3. However it also means more chance of a multipolar outcome, and that seems bad.
3. If brain emulations are developed before AI, there may be a second transition to AI later.
1. A second transition should be less explosive, because emulations are already many and fast relative to the new AI.
2. The control problem is probably easier if the cognitive differences are smaller between the controlling entities and the AI.
3. If emulations are smarter than humans, this would have some of the same benefits as cognitive enhancement, in the second transition.
4. Emulations would extend the lead of the frontrunner in developing emulation technology, potentially allowing that group to develop AI with little disturbance from others.
5. On balance, brain emulation probably reduces the risk from the first transition, but added to a second transition this is unclear.
4. Promoting brain emulation is better if:
1. You are pessimistic about human resolution of control problem
2. You are less concerned about neuromorphic AI, a second transition, and multipolar outcomes
3. You expect the timing of brain emulations and AI development to be close
4. You prefer superintelligence to arrive neither very early nor very late
3. The person affecting perspective favors speed: present people are at risk of dying in the next century, and may be saved by advanced technology
Another view
============
I talked to Kenzi Amodei about her thoughts on this section. Here is a summary of her disagreements:
>
> Bostrom argues that we probably shouldn't celebrate advances in computer hardware. This seems probably right, but here are counter-considerations to a couple of his arguments.
>
> **The great filter**
> A big reason Bostrom finds fast hardware progress to be broadly undesirable is that he judges the state risks from sitting around in our pre-AI situation to be low, relative to the step risk from AI. But the so called '[Great Filter](http://en.wikipedia.org/wiki/Great_Filter)' gives us reason to question this assessment.
>
> The argument goes like this. Observe that there are a lot of stars (we can detect about ~10^22 of them). Next, note that we have never seen any alien civilizations, or distant suggestions of them. There might be aliens out there somewhere, but they certainly haven't gone out and colonized the universe enough that we would notice them (see '[The Eerie Silence](http://www.amazon.com/The-Eerie-Silence-Renewing-Intelligence/dp/B005OHT0WS)' for further discussion of how we might observe aliens).
>
> This implies that somewhere on the path between a star existing, and it being home to a civilization that ventures out and colonizes much of space, there is a 'Great Filter': at least one step that is hard to get past. 1/10^22 hard to get past. We know of somewhat hard steps at the start: a star might not have planets, or the planets may not be suitable for life. We don't know how hard it is for life to start: this step could be most of the filter for all we know.
>
> If the filter is a step we have passed, there is nothing to worry about. But if it is a step in our future, then probably we will fail at it, like everyone else. And things that stop us from visibly colonizing the stars are may well be existential risks.
>
> At least [one way](https://meteuphoric.wordpress.com/2010/03/23/sia-doomsday-the-filter-is-ahead/) of understanding anthropic reasoning suggests the filter is much more likely to be at a step in our future. Put simply, one is much more likely to find oneself in our current situation if being killed off on the way here is unlikely.
>
> So what could this filter be? One thing we know is that it probably isn't AI risk, at least of the powerful, tile-the-universe-with-optimal-computations, sort that Bostrom describes. A rogue singleton colonizing the universe would be just as visible as its alien forebears colonizing the universe. From the perspective of the Great Filter, either one would be a 'success'. But there are no successes that we can see.
>
> What's more, if we expect to be fairly safe once we have a successful superintelligent singleton, then this points at risks arising before AI.
>
> So overall this argument suggests that AI is less concerning than we think and that other risks (especially early ones) are more concerning than we think. It also suggests that AI is harder than we think.
>
> Which means that if we buy this argument, we should put a lot more weight on the category of 'everything else', and especially the bits of it that come before AI. To the extent that known risks like biotechnology and ecological destruction don't seem plausible, we should more fear unknown unknowns that we aren't even preparing for.
>
> **How much progress is enough?**
>
> Bostrom points to positive changes hardware has made to society so far. For instance, hardware allowed personal computers, bringing the internet, and with it the accretion of an AI risk community, producing the ideas in Superintelligence. But then he says probably we have enough: "hardware is already good enough for a great many applications that could facilitate human communication and deliberation, and it is not clear that the pace of progress in these areas is strongly bottlenecked by the rate of hardware improvement."
>
> This seems intuitively plausible. However one could probably have erroneously made such assessments in all kinds of progress, all over history. Accepting them all would lead to madness, and we have no obvious way of telling them apart.
>
> In the 1800s it probably seemed like we had enough machines to be getting on with, perhaps too many. In the 1800s people probably felt overwhelmingly rich. If the sixties too, it probably seemed like we had plenty of computation, and that hardware wasn't a great bottleneck to social progress.
>
> If a trend has brought progress so far, and the progress would have been hard to predict in advance, then it seems hard to conclude from one's present vantage point that progress is basically done.
>
>
>
Notes
=====
**1. How is hardware progressing?**
I've been looking into this lately, at [AI Impacts](http://aiimpacts.org/trends-in-the-cost-of-computing/). Here's a figure of [MIPS](http://en.wikipedia.org/wiki/Instructions_per_second#Millions_of_instructions_per_second)/$ growing, from [Muehlhauser and Rieber](https://intelligence.org/2014/05/12/exponential-and-non-exponential/#footnote_7_11027).
(Note: I edited the vertical axis, to remove a typo)
**2. Hardware-software indifference curves**
It was brought up in this chapter that hardware and software can substitute for each other: if there is endless hardware, you can run worse algorithms, and vice versa. I find it useful to picture this as indifference curves, something like this:
*(Image: Hypothetical curves of hardware-software combinations producing the same performance at Go ([source](http://aiimpacts.org/how-ai-timelines-are-estimated/)).)*
I wrote about predicting AI given this kind of model [here](http://aiimpacts.org/how-ai-timelines-are-estimated/).
**3. The potential for discontinuous AI progress**
While we are on the topic of relevant stuff at [AI Impacts](http://aiimpacts.org/), I've been [investigating](http://aiimpacts.org/discontinuous-progress-investigation/) and quantifying the claim that AI might suddenly undergo huge amounts of abrupt progress (unlike brain emulations, according to Bostrom). As a step, we are finding other things that have undergone huge amounts of progress, such as [nuclear](http://aiimpacts.org/discontinuity-from-nuclear-weapons/) [weapons](http://aiimpacts.org/were-nuclear-weapons-cost-effective-explosives/) and [high temperature superconductors](http://aiimpacts.org/cases-of-discontinuous-technological-progress/):
(Figure originally from [here](http://www.doitpoms.ac.uk/tlplib/superconductivity/discovery.php))
**4. The person-affecting perspective favors speed less as other prospects improve**
I agree with Bostrom that the person-affecting perspective probably favors speeding many technologies, in the status quo. However I think it's worth noting that people with the person-affecting view should be scared of existential risk again as soon as society has achieved some modest chance of greatly extending life via specific technologies. So if you take the person-affecting view, and think there's a reasonable chance of very long life extension within the lifetimes of many existing humans, you should be careful about trading off speed and risk of catastrophe.
**5. It seems unclear that an emulation transition would be slower than an AI transition.**
One reason to expect an emulation transition to proceed faster is that there is [an unusual reason](https://meteuphoric.wordpress.com/2014/09/29/discontinuous-paths/) to expect abrupt progress there.
**6. Beware of brittle arguments**
This chapter presented a large number of detailed lines of reasoning for evaluating hardware and brain emulations. [This kind of concern](http://rationalaltruist.com/2013/05/08/beware-brittle-arguments/) might apply.
In-depth investigations
=======================
If you are particularly interested in these topics, and want to do further research, these are a few plausible directions, some inspired by Luke Muehlhauser's [list](http://lukemuehlhauser.com/some-studies-which-could-improve-our-strategic-picture-of-superintelligence/), which contains many suggestions related to parts of *Superintelligence.*These projects could be attempted at various levels of depth.
1. Investigate in more depth how hardware progress affects factors of interest
2. Assess in more depth the likely implications of whole brain emulation
3. Measure better the hardware and software progress that we see (e.g. some efforts at [AI Impacts](http://aiimpacts.org/trends-in-the-cost-of-computing/), [MIRI](https://intelligence.org/2014/02/28/the-worlds-distribution-of-computation-initial-findings/), [MIRI](https://intelligence.org/2014/05/12/exponential-and-non-exponential/) and [MIRI](http://intelligence.org/files/AlgorithmicProgress.pdf))
4. Investigate the extent to which hardware and software can substitute (I describe more projects [here](http://aiimpacts.org/research-topic-hardware-software-and-ai/))
5. Investigate the likely timing of whole brain emulation (the [Whole Brain Emulation Roadmap](http://www.fhi.ox.ac.uk/brain-emulation-roadmap-report.pdf) is the main work on this)
If you are interested in anything like this, you might want to mention it in the comments, and see whether other people have useful thoughts.
How to proceed
==============
This has been a collection of notes on the chapter. **The most important part of the reading group though is discussion**, which is in the comments section. I pose some questions for you there, and I invite you to add your own. Please remember that this group contains a variety of levels of expertise: if a line of discussion seems too basic or too incomprehensible, look around for one that suits you better!
Next week, we will talk about how collaboration and competition affect the strategic picture. To prepare, **read** “Collaboration” from Chapter 14The discussion will go live at 6pm Pacific time next Monday 23 March. Sign up to be notified [here](http://intelligence.us5.list-manage.com/subscribe?u=353906382677fa789a483ba9e&id=28cb982f40). |
7006ef93-0172-470b-b419-67c473a1047c | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Factored Cognition
*Note: This post (originally published [here](https://ought.org/presentations/factored-cognition-2018-05)) is the transcript of a presentation about a project worked on at the non-profit Ought. It is included in the sequence because it contains a very clear explanation of some of the key ideas behind iterated amplification.*
---
The presentation below motivates our [Factored Cognition](https://ought.org/projects/factored-cognition) project from an AI alignment angle and describes the state of our work as of May 2018. Andreas gave versions of this presentation at CHAI (4/25), a Deepmind-FHI seminar (5/24) and FHI (5/25).
I'll talk about *Factored Cognition*, our current main project at Ought. This is joint work with Ozzie Gooen, Ben Rachbach, Andrew Schreiber, Ben Weinstein-Raun, and (as board members) Paul Christiano and Owain Evans.
Before I get into the details of the project, I want to talk about the broader research program that it is part of. And to do that, I want to talk about research programs for AGI more generally.
Right now, the dominant paradigm for researchers who explicitly work towards AGI is what you could call "scalable learning and planning in complex environments". This paradigm substantially relies on training agents in simulated physical environments to solve tasks that are similar to the sorts of tasks animals and humans can solve, sometimes in isolation and sometimes in competitive multi-agent settings.
To be clear, not all tasks are physical tasks. There's also interest in more abstract environments as in the case of playing Go, proving theorems, or participating in goal-based dialog.
For our purposes, the key characteristic of this research paradigm is that agents are optimized for success at particular tasks. To the extent that they learn particular decision-making strategies, those are learned implicitly. We only provide external supervision, and it wouldn't be entirely wrong to call this sort of approach "recapitulating evolution", even if this isn't exactly what is going on most of the time.
As many people have pointed out, it could be difficult to become confident that a system produced through this sort of process is aligned - that is, that all its cognitive work is actually directed towards solving the tasks it is intended to help with. The reason for this is that alignment is a property of the decision-making process (what the system is "trying to do"), but that is unobserved and only implicitly controlled.
Aside: Could more advanced approaches to transparency and interpretability help here? They'd certainly be useful in diagnosing failures, but unless we can also leverage them for training, we might still be stuck with architectures that are difficult to align.
What's the alternative? It is what we could call *internal supervision* - supervising not just input-output behavior, but also cognitive processes. There is some prior work, with [Neural Programmer-Interpreters](https://arxiv.org/abs/1511.06279?context=cs) perhaps being the most notable instance of that class. However, depending on how you look at it, there is currently much less interest in such approaches than in end-to-end training, which isn't surprising: A big part of the appeal of AI over traditional programming is that you don't need to specify how exactly problems are solved.
In this talk, I'll discuss an alternative research program for AGI based on internal supervision. This program is based on imitating human reasoning and meta-reasoning, and will be much less developed than the one based on external supervision and training in complex environments.
The goal for this alternative program is to codify reasoning that people consider "good" ("helpful", "corrigible", "conservative"). This could include some principles of good reasoning that we know how to formalize (such as probability theory and expected value calculations), but could also include heuristics and sanity checks that are only locally valid.
For a system built this way, it could be substantially easier to become confident that it is aligned. Any bad outcomes would need to be produced through a sequence of human-endorsed reasoning steps. This is far from a guarantee that the resulting behavior is good, but seems like a much better starting point. (See e.g. [Dewey 2017](http://effective-altruism.com/ea/1ca/my_current_thoughts_on_miris_highly_reliable/#s6).)
The hope would be to (wherever possible) punt on solving hard problems such as what decision theory agents should use, and how to approach epistemology and value learning, and instead to build AI systems that inherit our epistemic situation, i.e. that are uncertain about those topics to the extent that we are uncertain.
I've described external and internal supervision as different approaches, but in reality there is a spectrum, and it is likely that practical systems will combine both.
However, the right end of the spectrum - and especially approaches based on learning to reason from humans - are more neglected right now. Ought aims to specifically make progress on automating human-like or human-endorsed deliberation.
A key challenge for these approaches is scalability: Even if we could learn to imitate how humans solve particular cognitive tasks, that wouldn't be enough. In most cases where we figured out how to automate cognition, we didn't just match human ability, but exceeded it, sometimes by a large margin. Therefore, one of the features we'd want an approach to AI based on imitating human metareasoning to have is a story for how we could use that approach to eventually exceed human ability.
Aside: Usually, I fold "aligned" into the definition of ["scalable"](https://ought.org/projects/factored-cognition/scalability) and describe [Ought's mission](https://ought.org/mission) as "finding scalable ways to leverage ML for deliberation".
What does it mean to "automate deliberation"? Unlike in more concrete settings such as playing a game of Go, this is not immediately clear.
For Go, there's a clear task (choose moves based on a game state), there's relevant data (recorded human games), and there's an obvious objective (to win the game). For deliberation, none of these are obvious.
As a task, we'll choose question-answering. This encompasses basically all other tasks, at least if questions are allowed to be big (i.e. can point to external data).
The data we'll train on will be recorded human actions in *cognitive workspaces*. I'll show an example in a couple of slides. The basic idea is to make thinking explicit by requiring people to break it down into small reasoning steps, to limit contextual information, and to record what information is available at each step.
An important point here is that our goal is not to capture human reasoning exactly as it is in day-to-day life, but to capture *a* way of reasoning that people would endorse. This is important, because the strategies we need to use to make thinking explicit will necessarily change how people think.
Finally, the objective will be to choose cognitive actions that people would endorse after deliberation.
Note the weird loop - since our task is automating deliberation, the objective is partially defined in terms of the behavior that we are aiming to improve throughout the training process. This suggests that we might be able to set up training dynamics where the supervision signal always stays a step ahead of the current best system, analogous to GANs and self-play.
We can decompose the problem of automating deliberation into two parts:
1. How can we make deliberation sufficiently explicit that we could in principle replicate it using machine learning? In other words, how do we generate the appropriate kind of training data?
2. How do we actually automate it?
In case you're familiar with [Iterated Distillation and Amplification](https://ai-alignment.com/iterated-distillation-and-amplification-157debfd1616): The two parts roughly correspond to amplification (first part) and distillation (second part).
The core concept behind our approach is that of a *cognitive workspace*. A workspace is associated with a question and a human user is tasked with making progress on thinking through that question. To do so, they have multiple actions available:
* They can reply to the question.
* They can edit a scratchpad, writing down notes about intermediate results and ideas on how to make progress on this question.
* They can ask sub-questions that help them answer the overall question.
Sub-questions are answered in the same way, each by a different user. This gives rise to a tree of questions and answers. The size of this tree is controlled by a budget that is associated with each workspace and that the corresponding user can distribute over sub-questions.
The approach we'll take to automating cognition is based on recording and imitating actions in such workspaces. Apart from information passed in through the question and through answers to sub-questions, each workspace is isolated from the others. If we show each workspace to a different user and limit the total time for each workspace to be short, e.g. 15 minutes, we *factor* the problem-solving process in a way that guarantees that there is no unobserved latent state that is accumulated over time.
There are a few more technicalities that are important to making this work in practice.
The most important one is probably the use of *pointers*. If we can only ask plain-text questions and sub-questions, the bandwidth of the question-answering system is severely limited. For example, we can't ask "Why did the protagonist crash the car in book X" because the book X would be too large to pass in as a literal question. Similarly, we can't delegate "Write an inspiring essay about architecture", because the essay would be too large to pass back.
We can lift this restriction by allowing users to create and pass around pointers to datastructures. A simple approach for doing this is to replace plain text everywhere with *messages* that consist of text interspersed with references to other messages.
The combination of pointers and short per-workspace time limits leads to a system where many problems are best tackled in an algorithmic manner. For example, in many situations all a workspace may be doing is mapping a function (represented as a natural language message) over a list (a message with linked list structure), without the user knowing or caring about the content of the function and list.
Now let's try to be a bit more precise about the parts of the system we've seen.
One component is the *human policy*, which we treat as a stateless map from contexts (immutable versions of workspaces) to actions (such as asking a particular sub-question).
Coming up with a single such actions should take the human at most a few minutes.
The other main component is the transition function, which consumes a context and an action and generates a set of new contexts.
For example, if the action is to ask a sub-question, there will be two new contexts:
1. The successor of the parent context that now has an additional reference to a sub-question.
2. The initial context for the newly generated sub-question workspace.
Composed together, the human policy and the transition function define a kind of evaluator: A map from a context to a set of new contexts.
In what follows, nodes (depicted as circles) refer to workspaces. Note that both inputs and outputs of workspaces can be messages with pointers, i.e. can be very large objects.
I'll mostly collapse workspaces to just questions and answers, so that we can draw entire trees of workspaces more easily.
By iteratively applying the evaluator, we generate increasingly large trees of workspaces. Over the course of this process, the root question will become increasingly informed by answers to sub-computations, and should thus become increasingly correct and helpful. (What exactly happens depends on how the transition function is set up, and what instructions we give to the human users.)
This process is essentially identical to what Paul Christiano refers to as [amplification](https://ai-alignment.com/policy-amplification-6a70cbee4f34): A single amplification step augments an agent (in our case, a human question-answerer) by giving it access to calls to itself. Multiple amplification steps generate trees of agents assisting each other.
I'll now walk through a few examples of different types of thinking by recursive decomposition.
The longer-term goal behind these examples is to understand: How decomposable is cognitive work? That is, can amplification work - in general, or for specific problems, with or without strong bounds on the capability of the resulting system?
Perhaps the easiest non-trivial case is arithmetic: To multiply two numbers, we can use the rules of addition and multiplication to break down the multiplication into a few multiplications of smaller numbers and add up the results.
If we wanted to scale to very large numbers, we'd have to represent each number as a nested pointer structure instead of plain text as shown here.
We can also implement other kinds of algorithms. Here, we're given a sequence of numbers as a linked list and we sum it up one by one. This ends up looking pretty much the same as how you'd sum up a list of numbers in a purely functional programming language such as Lisp or Scheme.
Indeed, we can implement *any* algorithm using this framework - it is computationally universal. One way to see this is to implement an evaluator for a programming language, e.g. following the example of the meta-circular evaluator in SICP.
As a consequence, if there's a problem we can't solve using this sort of framework, it's not because the framework can't run the program required to solve it. It's because the framework can't *come up with* the program by composing short-term tasks.
Let's start moving away from obviously algorithmic examples. This example shows how one could generate a Fermi estimate of a quantity by combining upper and lower bounds for the estimates of component quantities.
This example hints at how one might implement conditioning for probability distributions. We could first generate a list of all possible outcomes together with their associated probabilities, then filter the list of outcomes to only include those that satisfy our condition, and renormalize the resulting (sub-)distribution such that the probabilities of all outcomes sum to one again.
The general principle here is that we're happy to run very expensive computations as long as they're semantically correct. What I've described for conditioning is more or less the textbook definition of exact inference, but in general that is computationally intractable for distributions with many variables. The reason we're happy with expensive computations is that eventually we won't instantiate them explicitly, but rather emulate them using cheap ML-based function approximators.
If we want to use this framework to implement agents that can eventually exceed human capability, we can't use most human object-level knowledge, but rather need to set up a process that can learn human-like abilities from data in a more scalable way.
Consider the example of understanding natural language: If we wanted to determine whether a pair of sentences is a contradiction, entailment, or neutral (as in the SNLI dataset), we could simply ask the human to judge - but this won't scale to languages that none of the human judges know.
Alternatively, we can break down natural language understanding into (very) many small component tasks and try to solve the task without leveraging the humans' native language understanding facilities much. For example, we might start by computing the meaning of a sentence as a function of the meanings of all possible pairs of sub-phrases.
As in the case of probabilistic inference, this will be computationally intractable, and getting the decomposition right in the first place is substantially harder than solving the object-level task.
Here's a class of problems that seems particularly challenging for factored cognition: Problems where people would usually learn concepts over an extended period of time.
Consider solving a problem that is posed halfway through a math textbook. Usually, the textbook reader would have solved many simpler problems up to this point and would have built up conceptual structures and heuristics that then allow them to solve this new problem. If we need to solve the problem by composing work done by a large collection of humans, none of which can spend more than 15 minutes on the task, we'll have to replace this intuitive, implicit process with an externalized, explicit alternative.
It's not entirely clear to me how to do that, but one way to start would be to build up knowledge about the propositions and entities that are part of the problem statement by effectively applying semantic parsing to the relevant parts of the textbook, so that we can later ask whether (e.g.) a proposition with meaning X implies a proposition with meaning Y, where both X and Y are large nested pointer structures that encode detailed meaning representations.
If this reminds you of Good Old-Fashioned AI, it is not by accident. We're essentially trying to succeed where GOFAI failed, and our primary advantage is that we're okay with exponentially expensive computations, because we're not planning to ever run them directly. More on that soon.
So far, the workspaces we've looked at were quite sparse. All questions and answers were limited to a sentence or two. This "low-bandwidth" setting is not the only way to use the system - we could alternatively instruct the human users to provide more detail in their questions and to write longer answers.
For the purpose of automation, low bandwidth has advantages, both in the short term (where it makes automation easier) and in the long term (where it reduces a particular class of [potential security vulnerabilities](https://ai-alignment.com/security-amplification-f4931419f903)).
Empirical evidence from experiments with humans will need to inform this choice as well, and the correct answer is probably at least slightly more high-bandwidth than the examples shown so far.
Here's a kind of reasoning that I feel relatively optimistic that we can implement using factored cognition: Causal reasoning, both learning causal structures from data as well as computing the results of interventions and counterfactuals.
The particular tree of workspaces shown here doesn't really illustrate this, but I can imagine implementing Pearl-style algorithms for causal inference in a way where each step locally makes sense and slightly simplifies the overall problem.
The final example, meta-reasoning, is in some ways the most important one: If we want factored cognition to eventually produce *very* good solutions to problems - perhaps being competitive with any other systematic approach - then it's not enough to rely on the users directly choosing a good object-level decomposition for the problem at hand. Instead, they'll need to go meta and use the system to reason about what decompositions would work well, and how to find them.
One kind of general pattern for this is that users can ask something like "What approach should we take to problem `#1`?" as a first sub-problem, get back an answer `#2`, and then ask "What is the result of executing approach `#2` to question `#1`?" as a second sub-question. As we increase the budget for the meta-question, the object-level approach can change radically.
And, of course, we could also go meta twice, ask about approaches to solving the first meta-level problem, and the same consideration applies: Our meta-level approach to finding good object-level approaches could improve substantially as we invest more budget in meta-meta.
So far, I've shown one particular instantiation of factored cognition: a way to structure workspaces, a certain set of actions, and a corresponding implementation of the transition function that generates new workspace versions.
By varying each of these components, we can generate other ways to build systems in this space. For example, we might include actions for asking clarifying questions. I've written about these degrees of freedom on [our taxonomy page](https://ought.org/projects/factored-cognition/taxonomy).
Here's one example of an alternate system. This is a straightforward Javascript port of parts of Paul Christiano's [ALBA implementation](https://github.com/paulfchristiano/alba).
Workspaces are structured as sequences of observations and actions. All actions are commands that the user types, including `ask`, `reply`, `view` (for expanding a pointer), and `reflect` (for getting a pointer to the current context).
The command-line version is [available on Github](https://github.com/oughtinc/hch).
A few days ago, we open-sourced [Patchwork](https://github.com/oughtinc/patchwork), a new command-line app for recursive question-answering where we paid particular attention to build it in a way that is a good basis for multiple users and automation. To see a brief screencast, take a look at [the README](https://github.com/oughtinc/patchwork#patchwork).
Suppose decomposition worked and we could solve difficult problems using factored cognition - how could we transition from only using human labor to partial automation and eventually full automation? I'll discuss a few approaches, starting from very basic ideas that we can implement now and progressing to ones that will not be tractable using present-day ML.
Let's again consider a tree of workspaces, and in each workspace, one or more humans taking one or more actions.
For simplicity, I'll pretend that there is just a single action per workspace. This allows me to equivocate nodes and actions below. Nothing substantial changes if there are multiple actions.
I'll also pretend that all humans are essentially identical, which is obviously false, but allows me to consider the simpler problem of learning a single human policy from data.
As a first step towards automation, we'll *memoize* the human H. That is, whenever we would show a context to H, we first check whether we've shown this context to some other H before, and if so, we directly reuse the action that was taken previously.
This is a big win if many contexts are simple. For example, it may be very common to want to map a function over a list, and this operation will always involve the same kinds of sub-questions (check if the list is empty, if not get the first element, apply the function to the first element, etc). Ideally, we only do this sort of work once and then reuse it in the future. Memoization gets us part of the way there.
A significant step up in difficulty, we can try to imitate the behavior of H even in cases where the new context doesn't match any element of our "training set" exactly.
Of course, for general question-answering, we won't be able to fully automate the human policy any time soon. This means that any imitation algorithms we do apply will need to make choices about whether a context under consideration is the sort of situation where they can make good predictions about what a human would do, or whether to abstain.
If algorithms can make multiple choices in sequence, we need algorithms that are well-calibrated about when their actions are appropriate, and that in particular have very few false positives. Otherwise, even a relatively low probability of false positives could cascade into sequences of inappropriate actions.
We've tried to isolate and study this particular problem - making well-calibrated predictions in AI-complete domains - in a separate project called [Predicting Slow Judgments](https://ought.org/projects/judgments). So far, we've found it challenging to make non-trivial predictions about human responses for the dataset we've collected there.
How useful would it be to be able to automate some fraction of human actions? If the total number of actions needed to solve a task is exponentially large (e.g. because we're enumerating all potential sub-phrases of a paragraph of text), even being able to automate 90% of all actions wouldn't be enough to make this approach computationally tractable. To get to tractability in that regime, we need to automate entire subtrees. (And we need to do so using an amount of training data that is not itself exponentially large - an important aspect that this talk won't address at all.)
Let's reconsider amplification. Recall that in this context, each node represents the question-answer behavior implemented by a workspace operated on by some agent (to start with, a human). This agent can pose sub-questions to other agents who may or may not themselves get to ask such sub-questions, as indicated by whether they have nodes below them or not.
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O(2n). This process will become intractable before long.
(The next few slides describe Paul Christiano's [Iterated Distillation and Amplification](https://ai-alignment.com/iterated-distillation-and-amplification-157debfd1616) approach to training ML systems.)
Instead of iterating amplification, let's pause after one step. We started out with a single agent (a human) and then built a composite system using multiple agents (also all humans). This composite system is slower than the one we started out with. This slowdown perhaps isn't too bad for a single step, but it will add up over the course of multiple steps. To iterate amplification many times, we need to avoid this slowdown. What can we do?
The basic idea is to train an ML-based agent to imitate the behavior of the composite system. A simple (but insufficient!) approach would be to generate training data - questions and answers - based on the behavior of the composite system, and to train a supervised learner using this dataset.
In practice, this sort of training ("distillation") would probably need to involve not just imitation, but more advanced techniques, including adversarial training and approaches to interpretability that allow the composite system (the "overseer") to reason about the internals of its fast ML-based successor.
If we wanted to implement this training step in rich domains, we'd need ML techniques that are substantially better than the state of the art as of May 2018, and even then, some domains would almost certainly resist efficient distillation.
But, hypothetically, if we could implement faithful distillation, we would have a much better starting point for the next amplification step: We could compose together multiple instances of the fast ML-based learner, and the result would be a tree of agents that is only as large as the one we built in the first step (3 nodes, say), but exhibits the question-answer behavior of an agent that has multiple advisors, *each of which* as capable as the entire tree at the first step.
We can repeat whatever training process we used in the first step to get a yet better distilled system that "imitates" the behavior of the overseer composed of the systems trained in the previous step.
Through repeated amplification and distillation, we could hope to eventually satiate the representational and computational abilities of whatever ML system we're using in the distillation step, while guiding it towards implementing question-answer behavior that corresponds to what H would do if they had a large number of well-resourced assistants.
In practice, we might not want to implement this process as a series of distinct systems, and instead run self-play where a single system serves both as the overseer and the system-to-be-trained.
If Iterated Amplification and Distillation can work, we might be able to approximate the results of running some computations that would naively take exponential time: after *n* steps of amplification and distillation, we'd use a fast ML-based approximator to run computations that would take O(2n) time if we instantiated them explicitly.
As a particularly interesting special case, this might include the kinds of human-guided computations that arise from people take actions in cognitive workspaces.
There are many open questions for the scheme described above, both on whether we can make reasoning explicit, and on whether we can automate it efficiently even if it is made explicit. While I've talked a bit about automation, anything beyond basic automation is out of scope for Ought right now, so I'll focus on open questions related to decomposition.
For decomposition, the two main questions we ask ourselves are:
1. Can factored cognition recover the ability of a single human working over time for essentially all important tasks?
2. If so, can we exceed the capability of other systematic approaches to problem-solving if we just use sufficiently large budgets, i.e. compose sufficiently many small workspaces in sufficiently large trees? Equivalently, can we reach essentially arbitrarily high capability if we execute sufficiently many amplifications steps?
Our plan is to study both of these questions using a set of challenge problems.
The idea behind these challenge problems is to pick problems that are particularly likely to stretch the capabilities of problem solving by decomposition:
1. When people tackle tricky *math or programming puzzles*, they sometimes give up, go to bed, and the next day in the shower they suddenly know how to solve it. Can we solve such puzzles even if no single individual spends more than 15 minutes on the problem?
2. We've already seen a math textbook example earlier. We want to know more generally whether we can replicate the effects of learning over time, and are planning to study this using different kinds of *textbook problems*.
3. Similarly, when people reason about evidence, e.g. about whether a statement that a politician made is true, they seem to make incremental updates to opaque internal models and may use heuristics that they find difficult to verbalize. If we instead require all evidence to be aggregated explicitly, can we still match or exceed their *fact-checking* capabilities?
4. All examples of problems we've seen are one-off problems. However, ultimately we want to use automated systems to interact with a stateful world, e.g. through *dialog*. Abstractly, we know [how to approach this situation](https://ought.org/projects/factored-cognition/taxonomy#interaction), but we'd like to try it in practice e.g. on personal questions such as "Where should I go on vacation?".
5. For systems to scale to high capability, we've noted earlier that they will need to reason about cognitive strategies, not just object-level facts. *Prioritizing tasks* for a user might be a domain particularly suitable for testing this, since the same kind of reasoning (what to work on next) could be used on both object- and meta-level.
If we make progress on the feasibility of factored cognition and come to believe that it might be able to match and eventually exceed "normal" thinking, we'd like to move towards learning more about how this process would play out.
What would the human policy - the map from contexts to actions - look like that would have these properties? What concepts would be part of this policy? For scaling to high capability, it probably can't leverage most of the object-level knowledge people have. But what else? Abstract knowledge about how to reason? About causality, evidence, agents, logic? And how big would this policy be - could we effectively treat it as a lookup table, or are there many questions and answers in distinct domains that we could only really learn to imitate using sophisticated ML?
What would happen if we scaled up by iterating this learned human policy many times? What instructions would the humans that generate our training data need to follow for the resulting system to remain corrigible, even if run with extremely large amounts of computation (as might be the case if distillation works)? Would the behavior of the resulting system be chaotic, strongly dependent on its initial conditions, or could we be confident that there is a [basin of attraction](https://ai-alignment.com/corrigibility-3039e668638) that all careful ways of setting up such a system converge to? |
6f6223b3-a453-4772-8d3e-54e1f9c38a37 | trentmkelly/LessWrong-43k | LessWrong | Machines vs. Memes 2: Memetically-Motivated Model Extensions
[This is post 2 of 3 in a series written as the output for the 2022 AI Safety Camp, for the ‘Impact of Memetics on Alignment’ team, coached by Daniel Kokotajlo and comprising Harriet Farlow, Nate Rush and Claudio Ceruti. You can view the first post in this series here. Feedback so appreciated I'll do a jig for every comment.]
Overview
This post is a beginner’s attempt at extending the generally-accepted model of reinforcement learning agents (see this post).
This extension encourages us to think about artificial agents in a more suggestive and (potentially) more productive way. Specifically, this extensions captures an agent’s ability to change the policies it uses over time, while abstracting the decision to change policies away from the agent itself.
This encourages thinking about agents as more dynamic creatures than the standard RL formalization. It also allows for some interesting arguments about constraints on these agents' behaviors that are relevant to AI safety. This post is not optimistic, as it seems these more dynamic agents are less likely to be well-aligned.
This post was written as a culmination of my time at AI Safety camp. I am not an AI researcher, nor am I particularly good at math. Please treat these suggestions as a beginner taking a shot in a specific direction. Of course, any and all feedback is greatly appreciated!
Motivation
The model extensions I suggest are motivated by the study of memetics and memes.
By memes, I’m not talking about Grumpy Cat - I’m talking more generally about memes as a unit of “cultural replication.” The theory of memetics argues that humans are a uniquely big system of cultural replicators, and as such a well-developed theory of memes might be able to explain much of our behavior.
Although memetic theory is far from mainstream, its relevance to AI Safety is immediately obvious. Humans are the most intelligent agents we know of, and we are also by far the most memetic. As such, it’s natural to wonder how th |
d1cff7f0-9465-4a7f-952e-04d8d0d78b7e | trentmkelly/LessWrong-43k | LessWrong | Bell's Theorem: No EPR "Reality"
Previously in series: Entangled Photons
(Note: So that this post can be read by people who haven't followed the whole series, I shall temporarily adopt some more standard and less accurate terms; for example, talking about "many worlds" instead of "decoherent blobs of amplitude".)
The legendary Bayesian, E. T. Jaynes, began his life as a physicist. In some of his writings, you can find Jaynes railing against the idea that, because we have not yet found any way to predict quantum outcomes, they must be "truly random" or "inherently random".
Sure, today you don't know how to predict quantum measurements. But how do you know, asks Jaynes, that you won't find a way to predict the process tomorrow? How can any mere experiments tell us that we'll never be able to predict something—that it is "inherently unknowable" or "truly random"?
As far I can tell, Jaynes never heard about decoherence aka Many-Worlds, which is a great pity. If you belonged to a species with a brain like a flat sheet of paper that sometimes split down its thickness, you could reasonably conclude that you'd never be able to "predict" whether you'd "end up" in the left half or the right half. Yet is this really ignorance? It is a deterministic fact that different versions of you will experience different outcomes.
But even if you don't know about Many-Worlds, there's still an excellent reply for "Why do you think you'll never be able to predict what you'll see when you measure a quantum event?" This reply is known as Bell's Theorem.
In 1935, Einstein, Podolsky, and Rosen once argued roughly as follows:
Suppose we have a pair of entangled particles, light-years or at least light-minutes apart, so that no signal can possibly travel between them over the timespan of the experiment. We can suppose these are polarized photons with opposite polarizations.
Polarized filters block some photons, and absorb others; this lets us measure a photon's polarization in a given orientation. Entangled |
2630b616-ddcb-448e-b705-88594db7d254 | trentmkelly/LessWrong-43k | LessWrong | The case for "Generous Tit for Tat" as the ultimate game theory strategy
I wish I could title this How “Pacifism” wins at life, since it’d immediately make everything much easier (especially writing this post), but as most well-meaning kids swiftly learn on the playground - the world is not gentle.
It’s always bothered me how impossible it is to negate the need for strength in life (whether physical, emotional, individual or collective), especially considering how unevenly and unfairly that stat is usually distributed.
What bothers me even more however, is how many people I’ve seen gleefully misapply and overfit this fact, falsely concluding that since one cannot survive without strength, it is the only meaningful form of power in the world. I’m sure you’ve seen these people too.
There is another side to this coin however. One that I care deeply about, and that carries just as much weight and impact as anything mentioned above: if you ever want to get anywhere as a person, society, or species, you can’t afford to negate gentleness either.
Karate is the perfect intuitive manifestation of this principle. Respect and conflict aversion are its core philosophy, but not because of a lack of viable alternatives.
Quick note: We’re gonna dive into game theory here, so if you’re new to/rusty on the terminology, the single best thing you can do is click here for a brilliant interactive crash course on all the meaningful bits. If you’re short on time you’re better off prioritizing this click rather than reading the rest of this post (Nicky Case is amazing)!
Many smartass variations of the prisoner’s dilemma have been cooked up over the years, with each change in the parameters of the game bringing along with it a new balance of incentives, and different winning strategies accordingly.
Should you make like Jesus and always turn the other cheek, or should you play like Lucifer himself and ruthlessly exploit everyone you meet? Maybe you need a complex web of conditions defining how you act towards others based on your relationship so far, or m |
6ed16e40-0d0d-4c5d-ad99-daebe01a1010 | trentmkelly/LessWrong-43k | LessWrong | The AI Revolution in Biology
An expert in the field (Amelie Schreiber) predicts the convergence of AI models for biology to the state where designing drugs, targeted genetic mutations (which then could be enacted through genetic therapy), or viruses for specific purposes becomes really easy in a few years. (Note: apparently, the podcast was recorded before the release of AlphaFold 3.)
Unlike AGI safety, where OpenAI's argument of "iterative deployment" merits more weight[1], in biology I think we can already map the capabilities that will be unlocked. In fact, this podcast does it pretty well, it seems to me. So, it should also be possible to develop effective regulation and controls ahead of the creation of the bio-capabilities.
This raises the questions of the relative speeds of bio-capabilities, regulation, and controls development. And the ethics of pushing bio-capabilities if regulation and controls are lagging behind.
1. ^
Even though many people, including me still don't think "iterative development" is justified the way OpenAI does it. |
3c3dc66d-0f7e-4581-96f6-fd976cefb48b | trentmkelly/LessWrong-43k | LessWrong | Alignment from equivariance
Epistemic status: research direction
Equivariance is a way of enforcing a user-specified sense of consistency across inputs for a NN. It's a concept from computer vision which I think could be used to align AIs, or rather for building an interface/framework for aligning them.
So far it has been applied exclusively in CV - so it's speculation (by me) that you could apply it beyond CV. But I think I can convince you it's a reasonable speculation because, as you'll see, equivariance and consistency, for neural networks, are deeply linked mathematically.
Like me I imagine you expect (/hope!) that "consistency with the law" or "moral consistency" will turn out to be formalizable-enough to be part of the design of a computer programs. And I could almost go as far as to say that those vague senses of the word "consistency", when you formalize them, simply are equivariance. So it makes sense to me that a reasonable design program aiming towards moral or legal consistency could be started by trying to generalize equivariance.
Equivariance as we currently have it
Here's the abstract picture you should, and usually do, see on the first page of any paper about equivariance - don't worry if you don't get it based on this:
Let's see examples - here's one from equivariance king Maurice Weiler:
If I'm right, you who read alignment papers will be seeing lots more of these diagrams in future, so you might as well get used to them! Here are two more from Divyanshu Mishra and Mark Lukacs:
These are called commutative diagrams, and a neural network is equivariant to X if for all inputs that differ by X, its outputs differ by X'. On the left there is translation equivariance, on the right is scale equivariance.
Do you always want translation equivariance, or scale equivariance in computer vision? No. If you were training one on satellite photography that was all guaranteed to be at the same scale, you (probably) wouldn't want scale equivariance - otherwise your NN might not be |
54cdb7cd-ad48-4d5b-a303-5e5f4085a2a7 | trentmkelly/LessWrong-43k | LessWrong | Hufflepuff Cynicism on Hypocrisy
Epistemic Status: I've been thinking about this for a number of years, looking for steelmen of the position against hypocrisy. I haven't found anything satisfying yet, but maybe you can tell me why hypocrisy is actually bad? Barring that, I'm rather confident in the view expressed here.
"Hypocrisy is bad" is a deeply-rooted assumption in our culture. If I'm trying to give someone advice, I'll flinch away from saying things I don't do myself. For example, I don't have a driver's license, so I would hesitate before suggesting that someone else get one. If I do suggest it, I'll get kind of apologetic, and hedge my statement with "but I haven't gotten one yet". This, despite the fact that whether I've gotten one is almost irrelevant to whether they should get one. It seems to me that this habit is universal in American culture, and I'd be surprised (and intrigued!) to hear about any culture where it isn't.
I think the anti-hypocrisy norm is likely based on a blame/praise model of advice. If advice is always norm-enforcing criticism, and is tied to how many "social points" you score, then the anti-hypocrisy norm prevents a particular type of social exploit. A hypocrite can enforce, and claim to believe in, norms which they themselves break. They score social points, gaining status and power, based on an unfair self-favoring application of rules. Rulers who are above the laws they themselves enforce violate our fairness norms; they are the very image of corruption.
However, this problem can be addressed in a more specific way by calling out unfairness, building power structures with transparency and accountability in mind, and deserving trust as a community. The cost of the anti-hypocrisy norm is too high; it throws out too much useful advice, constraining the directions in which we think when we're trying to help one another.
Rob Bensinger already wrote a whole post on why hypocrisy is a bad concept, which I also endorse. Ironically, though, he has a different concep |
137cce10-cce9-457a-b5b2-58ca581adcaa | trentmkelly/LessWrong-43k | LessWrong | Keltham's Lectures in Project Lawful
(Not an April fools' joke)
If anyone wants to return to one of Keltham's lectures, here's a list with links to all the lectures. In principle, one could also read these without having read the rest of Project Lawful aka Planecrash, I am not sure how easy it would be to follow though. I included short excerpts, so you can also find the sections in the e-book.
Happy reading/listening:
Validity
Podcast Episode 9, minute 26
Keltham takes their scrambling-time as a pause in which to think. It's been a long time, at least by the standards of his total life lived so far, since the very basics were explained to him.
Evolution
Podcast Episode 15, minute 1
To make sure everybody starts out on the same page, Keltham will quickly summarize earlier conversations for the benefit of any harem members who might not have been in hearing range for it.
Negotiation
Podcast Episode 39, minute 8
"Never mind. For a start, today, I'm going to try to quickly review the way that dath ilani learn about negotiation -"
Probability 1
Podcast Episode 73, minute 12
So now, hopefully, they're getting to the good part! Previous stuff was just sort of trying to show you what Law is at all, like with Validity.
Probability 2
Podcast Episode 82
"Greetings, my esteemed co-researchers.
Utility
Podcast Episode 107, minute 24
After that, he's got the time for another hour of Law before dinner, and after a Lesser Restoration the energy too, so next up is Utility.
Probable utility
Podcast Episode 110, minute 13
In his morning lecture, Keltham makes a further run at Utility
Cooperation 1
Podcast Episode 112, minute 10
Keltham, then, will jump ahead of Decision, to a lesser fragment of Coordination.
Cooperation 2
Podcast Episode 117, minute 6
So today he's going to try a lecture in Baseline, approaching some of the mathematics of Decision as is more complicated than the simple cases.
References: The Cartoon Guide to Löb's Theorem
Governance
Podcast Episode 120, minute 1
Keltham |
3311a001-559f-4dde-9e00-5f6e2c22949a | trentmkelly/LessWrong-43k | LessWrong | Looking and the no-self
(This post is a reply to Kaj's "My attempt to explain Looking" and Valentine's Kenshō. The target audience is people who feel like they have a good grasp on the underlying concepts but of course the post might also be interesting for other people)
I have the feeling that both you and Valentine mix the concept of Looking together with no-self even through they are independent for each other. In the Zen Buddhist theoretical landscape they are linked together but linking them together is a specific position of that tradition.
My main approach and the concept I have to think about this space comes from Danis Bois. He is an interesting Frenchman, who discovered mediation while learning to be an osteopath. The next ten years he spent 8 hours per day meditating, traveled to guru's in India and taught what he called at the time Fascia therapy. At a time where he was successful with his life in the spiritual community he thought they were too dogmatic and decided to go to university. He studied education as his major and become a professor in Portugal and recently retired from being a professor.
There's no renunciation of the self in Bois work. He grants that it often makes sense to think about having multiple 'I' but neither of them is false. On the other hand having a strong feeling of existence is a key in his way.
David Chapman wrote a lot about the conflict between Consensus Buddhism and Tantra Buddhism. Zen Buddhism is part of what Chapman calls consensus Buddhism. One of the key differences is that consensus Buddhism seeks the renunciation of the self while Tantra doesn't.
While the path of renunciation exists, I consider it to be in conflict with the standard rational goals of having an impact on the world. Given the predominance of Zen in popular culture it might seem like it's the only valid path, but it's not the only spiritual path that enables the ability to Look.
Both Danis Bois paradigm and Tantra care a lot about relationships and consider relating to |
b044e9b3-8d2d-4963-b90e-292688b82a93 | StampyAI/alignment-research-dataset/youtube | Youtube Transcripts | Governing Transformative AI
hello
my name is mauricio and i'm excited to
welcome you to the talk
governing transformative ai i'm thrilled
to introduce you to marcus andreyan
andrew young is head of operations and
policy engagement
focusing on growing the centre for the
governance of ai at the future of
humanity institute which is based at
oxford university
andrew young focuses on making the
centre's research relevant to important
stakeholders
acting as an enabler for research and
contributing to fhi research
he has a background in history and
philosophy of science with a focus on
the philosophy of economics and
evidence-based policy
today he'll be discussing the field of
ai governance with an emphasis on
long-term risks and opportunities
we invite you as always to use the
software platform during the session
on the right side panel feel free to
share your thoughts on the live
discussion board
and leave an upvote to any questions for
a speaker to answer at the end of the
presentation
after the session we invite you to use
swapcart to set up meetings with fellow
conference attendees
without further ado welcome marcus hello
everyone
good to see you all or i hope rather
it's good for you to see me
i'm going to be talking today about the
field of the governance of ai and
particularly with a focus on more
transformative possibilities
and giving you sort of a showcase of
some of the work going on at the center
for the governance of ai at the future
of madness institute
where i work um
firstly you might start by sort of
thinking a little bit about what vi
governance is
um ensure it sort of refers to how
humanity
can best sort of navigate the transition
to a world with advanced ai systems
um it sort of focuses in particular on
sort of the ethical
economic political military decisions on
ai development and deployment that's
going to sort of
shape the impacts of this this
technology around the world
but it's not just about sort of this
kind of descriptive set of questions is
also about the normative so it's trying
to
understand how these decisions will be
made how this technology will play out
with an eye towards figuring out what
actions we should be taking now
um to sort of make sure that its impacts
are as good as possible
another thing that sort of another kind
of definition i quite like
is more sort of a negative definition so
this is trying to
basically just saying something like um
even if we solve the challenges of
technology
of sort of aligning ai systems to the
intentions of their users for example
we have a whole bunch of problems still
still left to go
and ai governance is trying to
understand what those problems are and
trying to see what we could
could be doing about them
i think one thing that's particularly
useful when thinking about this uh
this field is sort of dividing up risk
into a few different clusters
um one is sort of accident risks um
where
something happens uh despite the sort of
best intentions of their
of their user or the of the creator and
causes some kind of accident that causes
harm
another one is sort of malicious use so
the system does what it's supposed to do
but it's actor or sort of user tries to
have it to do something
something bad something that's harmful
to the world i think one other sort of
category risk that i think is
particularly important to pay attention
to especially if you're sort of
coming out this problem from a a
governance perspective
is sort of structural risk and so these
are risks that sort of
either might basically risk that might
change the incentives of actors
such that bad outcomes happen one
example of this might be sort of
um you uh another sort of important set
of risks that people often think about
is sort of
how strategic or sort of competitive
dynamics
might lead actors to for example under
invest in things that would make sure
that they
their systems are safe and that might
then then cause
to sort of increase the risk of
potentially bad accidents happening
another sort of cluster of things might
be sort of
changing the incentive structure for for
actors in a more more sort of concrete
way and so one one sort of
interesting example of this i think is
thinking about the invention of machine
gun
which was sort of the early ones were
invented by a man called gatling
who actually when he invented it he was
sort of thought this would be a good
thing for the world
um because he reasoned something like um
well
we have a certain amount of bullets
going out on the field
and if i sort of figure out a machine
that could sort of shoot
more bullets per person of soldier that
i have on the field then
surely we could just have fewer soldiers
on the field and then
but then sort of lo and behold this
turned out to be some some fairly poor
reasoning because he he forgot that
uh this sort of efficiency gain might
have just been taken out by having more
bullets in the field
uh which is uh what happened and which
also led to this
sort of uh quite uh defensive dominant
sort of technological suite that we
ended up having
in that in the world war one um in that
legislative trench warfare
uh which led to a whole a whole bunch of
deaths um which i think interestingly
was also partly because
um the actors didn't quite understand
the structure that they were in
they didn't quite understand how
defensive dominant this this technology
suite that they had of a sort of barbed
wire of um
machine gun etc was and so they sort of
kept sending troops and kept hoping that
they'd be able to to break through when
that was going to be very very difficult
i think another thing that's sort of
important to to think about in this
space
is to sort of realize or reflect on all
the ways in which
ai governance is going to be a very very
difficult problem
i think one way to to get that is to
think about ai technology
as a general purpose technology i.e a
technology that's going to have impacts
across
a whole bunch of economic social
military processes
similar to those of electricity for
example and
if that's the case then this is going to
be a very very difficult technology to
to govern more exclusive control
primarily because this sort of harms are
going to be very very diffuse
and also there are going to be benefits
and it's going to be very difficult to
separate these benefits
from the potential harms and it's gonna
be difficult to sort of
see which things are harmful which
things are beneficial and it's also
gonna be difficult to
make sure that actors sort of only do
the beneficial thing
uh rather than the harmful thing as well
um i think this
means um this it's sort of unclear to me
what this means in terms of whether this
is
an important problem to work on on the
one hand it might make it
more even more important to work on if
this is sort of a um
a problem that sort of we end up needing
having to solve uh to be able to get to
uh safe and beneficial outcomes from ai
um or it might be able to uh might just
mean that sort of this we should be
focusing on
other problems instead we should be
focused on sort of
complementary goods to air governments
such as better safety for example
uh yeah i'll go through some examples
that come from
the research group at the center for the
governance of ai
um there are also a bunch of other uh
important and interesting research
groups in this in this space i'll focus
on our
research because i that's the one that i
know best um i'll just sort of
hop through sort of a romp of like
different kinds of uh research parts
that we've been doing but there's a
whole bunch more out there a whole bunch
more that's being done
and a whole bunch more that has been
done in the past that you might find
find really interesting i'm mainly
highlighting things that are sort of
more recent that you might not have
heard of and that i find i find sort of
more interesting at the moment the way
that we sort of try to divide up the
space of research uh mainly comes from
this
research agenda that helen defoe our
director wrote in 2018
which tries to divide up the space in
sort of these four categories so first
is sort of the technical landscape
trying to understand what are the kinds
of capabilities that we're going to have
when so trying to do forecasting and
also trying to understand what actors
are going to have access to
to what capabilities then there's sort
of politics so this is trying to
understand how political actors how
politicians governments uh maybe even
sort of companies
and other actors with a significant
amount of power how they're going to
interact with us with this technology
and what kinds of decisions we should be
expecting them to make um
uh in this and this domain then there's
sort of this other cluster of sort of
ideal governance
so this is trying to understand um what
are sort of the the
kinds of uh places that we want to go so
what are the kinds of norms that we want
to be put in place
what's the kind of institutional design
that we want companies to put in place
so for example
um what are sort of um yeah if
if you were to design an ethics board
for example what should that look like
uh to be able to sort of make sure that
this technology is used for
uh use for good purposes and then so the
fourth question is trying to take all
these
uh these these three different clusters
these three previous clusters
trying to translate them into what do we
do today to try to lead to
some of these sort of more long-term
beneficial outcomes that we're
that we're looking for and
starting with the technical landscape
two people that i find very interesting
uh one is sort of this
um piece or this this work on this
concept of structure transparency
uh which has come out of um a bunch of
different
directions for example a bunch of work
done at fhi over the past few years a
lot of the thinking done by
uh drexler for example and the thing
that you're trying to do here is you're
trying to understand
what are ways in which we might be able
to what are the sort of potential
benefits of the potential effects of
being able to
unbundle information uh more from each
other
um and so one way to sort of exemplify
what we're talking about here is to
think about
bomb sniffer dogs that sort of when they
walk around the airport
they have this really really nice
feature where they they'll be able to
tell a guard
whether a bomb is in a particular bag
and so then we only need to search
that particular bag that the dog thinks
that the mom is in and so this means
that the privacy infringement
for the benefit that we get i.e fewer
bombs on airplanes etc
is is really is really quite small
if we had a different technological
suite for example if there we
didn't know how to train dogs to be able
to sniff
sniff for for bombs then we might end up
in a very different system where maybe
the way that we need to be able to
guarantee that there weren't any
any bombs in the airplane might be to
look inside of everyone's bags
for example um and
this is the kind of thing that we're
thinking about in this space
it's trying to understand um how can we
use new technologies or maybe even just
use
um sort of technologies and sort of uh
and sort of processes that we that we
have today
and that we can we can sort of unbundle
um sort of privacy infringing
information being spread um from the
sort of potential beneficial information
to be spread so you can imagine for
example
um you might set up a system where a
police officer might only be able to
access
a certain kind of um footage
certain kind of studio footage um by
maybe putting in the face
of someone who has been approved as
being a
likely suspect of a crime and in that
case maybe they they
the only thing that they get access to
might be then get the get the videos
where that face
shows up or sort of the time around that
that time and then the thought is that
that might be a lot less privacy
infringing
and so therefore it might come with a
bunch of benefits i think there's a lot
of
interesting uh interesting work here and
in particular uh work that i'm excited
about is sort of thinking about
what the potential effects of having
more technologies
like this might be in the world for
example you might imagine
um sort of one question that
particularly interesting is sort of how
these
potential benefits will be taken out so
will we take this
sort of efficiency gain out in terms of
uh
simply having the sort of roughly the
same amount of privacy infringement
happening in the world
um but we just get more benefit out of
it or is this
these are these kinds of technologies
that on net will be privacy preserving
or not
and what might this kind of technology
do to a democracy versus an
authoritarian state for example
seems like very very important exciting
questions to me
another set of questions looks at this
uh concept of cooperative ai
which is something that alan d for our
director has been working on a bunch
i think one way to motivate it is just
to think about um
the distinction between horizontal and
vertical coordination
and so vertical coordination here is
sort of um
the an ai system doing what what its uh
principle or what its user wants it to
do so this is
at least historically a lot of the work
uh that we've been talking about in the
sort of
technically a safety space focuses on on
this particular question which seems
very important if we don't solve it it
seems like things are going to be bad
but the thought here is that even if we
solve that problem
we're going to have additional problems
which consist of how do we make sure
that there is
cooperation or coordination between
between these actors or between sort of
maybe human ai system pairs
and and other human ai system pairs
and one one sort of way to pull out that
instinct might be that
today you can imagine that sort of uh my
actions are at least
to a pretty large extent um aligned with
with my will
or sort of what i what i want want to
happen or what i want to do
and even if that's the case even though
it's case with with most of humanity we
end up with a lot of conflict
and we end up with wars we end up with
all kinds of kinds of bad stuff
and so if we don't solve this sort of
cooperation problems between ai systems
as sort of
their ability to get things done in the
world increases and then we might be
leaving a bunch of value on the table or
we might be creating sort of unexpected
risks that we weren't uh thinking about
before
and so this is sort of a sort of line of
work that's pushing in the direction of
trying to
trying to build up a field that tries to
solve these kinds of problems so it's
trying to understand how
ai systems can be cooperating with each
other how ais and humans can cooperate
with each other
but also trying to understand how we can
build ai systems that might be able to
help
human to human cooperation so for
example are there ways in which we could
bootstrap or enhance democracy if we
find ways to be able to
extract people's preferences in a more
efficient way in a more
sort of high fidelity way than sort of
someone voting
once every four years for example
i think there's a lot of future
interesting work going on here um
one in particular a bit of work that i'm
excited about is
trying to understand questions that are
important to what the impact of doing
this this work
uh might be in particular if you're
trying to sort of shape the world in
particular russian and so one question
that seems particularly important is
this question of
uh whether we should expect these
capabilities of a sort of ais and be
able to cooperate
uh to be developed sort of by default uh
or whether they're something that we
sort of should put extra energy into
making happen in the water
in the policy in security space um
one sort of set of work that i'm
particularly excited about looks like
sort of similar to this this report that
we recently published called
ai policy levers which looks at what the
u.s government
what their tools might be to try to
shape ai research
development and deployment and this is
basically just a mapping of what are
they what are
all are the different tools what are
those which ones are they most likely to
use
given given what they currently look
like and i think that seems like a very
very important set of questions
primarily because the us government
might be one of the most sort of
important actors
in this in this whole ai governance
problem um so the future work that i'm
very excited about
sort of relates to this is one is just
looking at other actors
um and uh looking at what kinds of tools
they might be
be using in the future so looking at the
eu looking at china seems important
um and sort of having writing these
pieces that people can use that sort of
reference
uh reference work to sort of get a brief
overview of what's what's going on
i think another bit of work that seems
very important to me is trying to
understand
um how the tools that these
actors have access to might change uh if
we're gonna
if we end up in a world where ai systems
end up being very uh very transformative
and so you could imagine that if air
systems end up being very transformative
we'd end up in a situation where um the
sort of
looking at so the past president of how
for example anti-trust legislation has
been used
might not be all that informative um
because
um the legal or sort of the
the policy makers might just simply
change the laws um to sort of suit
suit the current situation or they might
just sort of creatively uh reinterpret
laws that currently have in place so
trying to understand that
i think will be very important in
particular i'm interested in that in the
in the area of antitrust and competition
law
space another piece of work that we've
been uh focusing a lot on and that i
would
recommend you have a look at is looking
at um what we can learn
from other historical analogies that we
might be able to draw
to ai technology for example and so
one example here is a piece of work
looking at the period right after the
second world war
where arguably we have this quite unique
opportunity
where um where the world was sort of
uniquely situated to be able to put
international controls on a very very
powerful technology in nuclear weapons
so we had recently had a world war the
world was
suffering war fatigue we recently
dropped these two bombs that sort of
were had a big impact around the world
and sort of shaped people's
imagination in all kinds of ways and on
top of that we also had
the us which for about six to seven
years had this
this monopoly on nuclear weapons and so
you'd imagine that this would be a
uniquely
useful uniquely important uniquely good
window
uh to put in place international
controls of of nuclear weapons
uh it turned out that this didn't really
work out um and
to to learn more about why that is and
what kind of lessons what might one
might be able to draw from that case
study i would really recommend you
have a look at this report um
looking at the ideal governance space um
we sort of the sort of main way that i
think of this work
is basically looking at
what are the beliefs of different kinds
of actors in the space so for example
looking at
what our machine learning researchers uh
what are their beliefs about
uh what direction they want ai
technology to go what are the public's
beliefs about where this technology
might be able to go or
where where it ought to go and then that
seems like important
inputs into this question of what are
the kinds of
systems the kinds of uh institutional
design the the kinds of uh policies we
should be putting in place over time and
so this is sort of
a piece of a sort of set of work that we
put a bunch of energy into uh for
example uh by baba zhang and
and no immediately thinking about what
kinds of uh what kinds of surveys we
need to be running here we're also more
and more trying to look at the space of
uh
ai policy uh and so one sort of example
of a report here that i'm particularly
excited about is looking at
the extent to which the eu it's for
coming sort of ai legislation which
compared to the world looks like it's
going to be quite stringent
and what global effects that might might
have um there's this thing called the
brussels effect
where in some situations we actually see
something that's kind of the opposite of
sort of a
regulatory race to the bottom uh i.e
that um the eu
putting in place more stringent
legislations end up having a global
effect
primarily because a company that needs
to comply with eu rules
uh ends up not it ends up being cheaper
for that company to then just comply
with those rules
across the world um and so um the sort
of core question that we're interested
in here is just trying to understand
should we expect something similar to
happen in the ai space and if that's the
case
that would be very interesting for a
number of reasons for example it might
just mean that
the eu as an actor becomes a lot more
important than we
than we might have otherwise thought
another piece of work that we've um we
worked on a whole bunch
is this question of uh whether and how
if so a company should be put in place
something called a windfall clause
which is basically trying to have
companies um promise that if they sort
of win the ai jackpot
if they end up seizing and producing a
whole bunch of wealth
by their ai systems um what should we
then in that case they promise that
they'll give back
some of this benefit some of this wealth
to the rest of the world
and the thought here is that this might
be able to sort of reduce incentives
from
from companies but also from government
etc
to sort of race or try to really really
push to be
among the sort of the first or the most
dominant actors in the space
and i think in general uh this is sort
of one mechanism in in this space
of sort of how we might be able to
redistribute um wealth if it's the case
that ai ends up producing
a whole bunch of it that seems important
to explore i think there's a whole bunch
of other other ones we ought to be
exploring as well like for example how
sort of taxation
might be might be used in this space as
well
um yeah so those are some examples of
the kind of work that we're doing and
that one might be able to do in
in this space hopefully those that's
given you giving you some ideas uh if
you want more i'd recommend you
uh have a look at you could have a look
at our website governance.ai
you could also have a look at some of
these uh some of these pieces here and
you could also
just shoot me an email if you're
interested as well thank you all very
much
thank you marcus for laying out this
landscape of ai governance
we've got lots of interesting questions
from the chat so let's dive
in one thing people are wondering about
is generally for undergraduates or early
career professionals
interested in making a positive impact
on ai governance
what advice do you have yeah
thanks uh yeah and thank you all for for
listening to me
to me yamaron about these things um yeah
so
i think there's in general like giving
curious advice to
a huge number of people at once i find
very tricky
um i think the sort of the most useful
things i can say is stuff like
if you want to have an important impact
in especially in this space but i think
in a lot of spaces
in sort of this general area of trying
to do something about essential risk and
whatnot
is uh try to make sure that you keep
engaged with the research coming out
um and that you like as much as possible
try to
um do your do the work required to start
forming inside views
on as many of the sort of crucial
questions that you care about
that's not to say that your inside views
are the ones that are going to matter
the most
uh it's more that like the work that
like
um the benefit that you get out of
trying to build an inside view is like
very it's very very helpful
uh in particular because it yeah because
it means that you'll you'll sort of um
discover things that might be wrong
discover ways that you might be able to
contribute etc so i think in general the
the kinds of advice i would give would
be to try to learn as much as possible
about space
keep engaging with the research go to
conferences like these talk to people
um and then sort of once you've done
that and you're still still sort of
excited about the field then
do do what you can to try to get your
foot in the door somewhere
uh and sort of try out the work um you
could do your dissertation about
something well then those kind of things
great thanks then uh christina from the
audience asks
is there a global initiative or what
kind of work is happening
uh toward creating institutions for
international governance
yeah so there's a there's a decent
amount um so
first there's like a number of groups or
sort of like igos
setting up ai specific bodies um
and so yeah for example we have
something called dpi
uh that was set up fairly recently uh
this the oecd has set up something
called the oecd observatory
um those things are happening uh in my
mind
some of the most important sort of ai
governance
like in my mind the most important
governance that will happen in the world
won't happen at
these kinds of institutions it will
happen at the institutions that just are
doing the global governance
uh so it will happen at institutions
like g8
and you know the general assembly at the
un and these kinds of things
and so i think that's the kind of stuff
that you should be thinking about when
you think about how
yeah how ai will be governing the global
scale
uh-huh great and then another question
we you discuss how a lot of the current
work in ai governance tries to forecast
the the long-term future dynamics of
technological and the political dynamics
of ai
it seems like like a really hard problem
potentially hard enough that people will
have to make
a big governance decisions well before
we figure out these things
uh so is there some point and if so
at what point should the field shift
from big picture forecasting
to more concrete policy work yeah i mean
so um yeah to some extent i think that's
already happening
like uh i think there are you know there
are like important decisions are being
made in the world that presumably will
have
long-term consequences on this field so
the eu
is yeah rolling out a bunch of
legislation on
on how ai will be used and that's likely
to have uh sort of
trajectory changing effects um and
those those effects are likely going to
be important and if one could
sort of uh yeah have an impact on those
that seems important so i think in in
sort of
overall my answer is just that like um
there is
there is productive work to do in sort
of the
sort of more concrete sort of what
should policymakers do today
um there's unproductive work to do there
today and i guess in general just
sort of ask this space um yeah there's a
bunch of different variables that will
change
what kind of work is most valuable when
um and one of those variables is just a
thing of like how much just
strategic insight do we have um and that
will hopefully just sort of go up over
go up over time and that will mean that
sort of as as that goes up then the like
value of the policy work goes up
um but it's also it might also be the
case that sort of you have particularly
important
opportunities to have an impact now it
might also be the case that you want to
sort of
um it might be one thing i'm unsure
about is like the extent which it's
like cheaper to buy influence now or
like if you want to buy
sort of if you want to become a person
who will like end up being important in
future decisions
uh be made in the space um then maybe
you want to invest in that now and maybe
that's
a better investment to do now than to do
in the future and that seems kind of
plausible to me
um and so yeah in short i think there's
a bunch of uh that kind of work that
should be done
should be done now um keeping in mind
that like yeah maybe this
picture isn't just quite as as clear as
you'd like
great thanks let's do one more quick
question on this note
are there actions policymakers can take
now that seem robustly good for
the future of ai um yes i think
i think there are um i think overall my
view is that i would like there to be
more
i would like to have like more thorough
answers to these questions um and so one
thing that i like i'm very excited about
they're happening more and i think is
starting to happen more is like people
doing what i kind of call like policy
development
um research in the sort of long-term ai
governance space
um and yeah i think there's a bunch of
potential actions or there's a bunch of
exciting like pretty
powerful levers that are sort of
starting to be pulled by
by the world so for example in the us
it's likely that we're going to have
some kind of some new stuff going on
without trust um
and that will sort of affect tech
companies in a large way presumably that
will affect the future of ai
um it would seems like a great thing to
have good answers to
which way is is good to go or like what
kinds of antitrust uh like legislation
we want in place
uh for for the future of ai governance
for example um
and so i think that there is yeah a
bunch of that kind of work that would be
helpful some examples of actions that i
think
can be taken today like i think a really
big cluster is
like goes under this umbrella what you
might call like differential
technological development
uh so you try to differentially sort of
boost a certain
direction of the technology so for
example you could imagine that like oh
we want the nsf and whatnot
to invest more in ai safety things or we
might want them to invest in
ways in which you can verify what a
system or what a user is doing with a
certain system
where those that kind of research will
be likely be in in a lot of different
worlds will be very very helpful
to various kinds of governance solutions
that you you're going to want to put in
place
um i also think that there's sort of one
if you're interested in sort of
um my in my case in my mind sort of the
best example of
uh work that does that goes sort of the
full way
from like um tricky high-level strategic
considerations to some like concrete
actions that policymakers can
take today um that's like been done
quite thoroughly it comes out of cset
um which is this this whole bit of work
on
whether or not and to what extent the
u.s should try to keep
uh semiconductor manufacturing equipment
uh and other other processes that might
be needed to create high-end
chips um from uh from like china having
access to this
um and yeah i think i would love to have
more
work that is done as thoroughly on on
other questions um
um that's not to say that like i'm even
though that's the case i'm still not
confident what the what the right call
is for for policymakers
uh but it seems likely that at least
engaging with that work will will
hopefully
improve your uh world picture and
hopefully have you make some better
decisions
yeah thank you that's all we have time
for today so thank you so much for a
wonderful session and for sharing your
time
thank you so much for coming thanks guys |
f03a4914-c807-42f7-95dd-879a3f2ae7bf | trentmkelly/LessWrong-43k | LessWrong | Meetup : London Social - Exposure to Direct Sunlight - June 23rd
Discussion article for the meetup : London Social - Exposure to Direct Sunlight - June 23rd
WHEN: 23 June 2013 02:00:00PM (+0100)
WHERE: St James Park, London, SW1H
EDIT: morning of the 23rd, and the weather is looking good enough to chance it. Head for the park and we'll retreat to the pub if it takes a downturn.
The first London Social meetup will be taking place in St. James Park on Sunday June 23rd from 2pm.
Assuming the weather is nice, we'll be aiming to meet about here. I'll be bringing along some games and maybe some juggling equipment. If you have any of your own Apparatus of Awesomeness, feel free to bring it along.
If the weather is not nice, we will convene at the Old Star, which is just round the corner from St. James Park tube station. I'll update this post on the morning of the 23rd with confirmation of the venue, but the two locations are a short distance from each other. If in doubt, head to St. James Park tube and call me on +44 7887 718458 when you get there.
There is no set agenda or topic. We're going to have unstructured social fun like regular human beings. People will probably talk about utilitarian ethics and bitcoin and productivity systems, but also about dinosaurs and cake and fanfiction.
Bring a friend. Bring two if you can carry them.
(We also have a Google Group and a Facebook group. Why not join them?)
Discussion article for the meetup : London Social - Exposure to Direct Sunlight - June 23rd |
d8702d6b-83e1-4274-ae41-29295faca256 | trentmkelly/LessWrong-43k | LessWrong | Takeoff speeds, the chimps analogy, and the Cultural Intelligence Hypothesis
In debates about AI takeoff speeds, the most common empirical example/analogy is that of the transition between chimpanzees and humans. For example, in this conversation between Paul Christiano and Eliezer Yudkowsky on takeoff speeds, the chimp analogy is discussed extensively.
Generally, I have seen the chimp example cited as evidence for a fast takeoff, as humans brains are only 3-4x bigger than chimp brains with few "algorithmic" differences, yet humans are much more generally intelligent than chimps. Thus, given that due to Moore's law and increased investment we could probably increase the compute used in training an AI in a relatively that much in a relatively short period of time, if the relationship between compute and intelligence in AI's is similar to that in biological systems, a similarly fast takeoff may occur. Alternatively, if one does not focus on compute, the transition arguably suggests at a higher level of abstraction that, as Yudkowsky puts it, "There’s stuff in the underlying algorithmic space... where you move a bump and get a lump of capability out the other side."
I claim that this is an incorrect inference from the chimps to humans transition, and that chimps provide only weak evidence for fast takeoff. This is because I believe that the specific explanation for why humans are so much smarter than chimps despite the relatively small differences between human brains and chimp brains is unlikely to reoccur in the context of AI development. In particular, I believe that the "cultural intelligence hypothesis" explains to a significant extent the origins of human intelligence.
According to the cultural intelligence hypothesis, humans are smarter than chimps largely because our cultural abilities such as superior imitative learning capabilities and our ability to use language allow us to use the knowledge and cognitive skills developed by past generations of humans. In other words, humans do not have vastly superior "general intellige |
232984f3-ad46-4ae8-be62-31f97fedd123 | trentmkelly/LessWrong-43k | LessWrong | AI #108: Straight Line on a Graph
The x-axis of the graph is time. The y-axis of the graph is the log of ‘how long a software engineering task can AIs reliably succeed at doing.’
The straight line says the answer doubles roughly every 7 months. Yikes.
Upcoming: The comment period on America’s AI strategy is over, so we can finish up by looking at Google’s and MIRI’s and IFP’s proposals, as well as Hollywood’s response to OpenAI and Google’s demands for unlimited uncompensated fair use exceptions from copyright during model training. I’m going to pull that out into its own post so it can be more easily referenced.
There’s also a draft report on frontier model risks from California and it’s… good?
Also upcoming: My take on OpenAI’s new future good-at-writing model.
TABLE OF CONTENTS
1. Language Models Offer Mundane Utility. I want to, is there an app for that?
2. Language Models Don’t Offer Mundane Utility. Agents not quite ready yet.
3. Huh, Upgrades. Anthropic efficiency gains, Google silently adds features.
4. Seeking Deeply. The PRC gives DeepSeek more attention. That cuts both ways.
5. Fun With Media Generation. Fun with Gemini 2.0 Image Generation.
6. Gemma Goals. Hard to know exactly how good it really is.
7. On Your Marks. Tic-Tac-Toe bench is only now getting properly saturated.
8. Choose Your Fighter. o3-mini disappoints on Epoch retest on frontier math.
9. Deepfaketown and Botpocalypse Soon. Don’t yet use the bot, also don’t be the bot.
10. Copyright Confrontation. Removing watermarks has been a thing for a while.
11. Get Involved. Anthropic, SaferAI, OpenPhil.
12. In Other AI News. Sentience leaves everyone confused.
13. Straight Lines on Graphs. METR finds reliable SWE task length doubling rapidly.
14. Quiet Speculations. Various versions of takeoff.
15. California Issues Reasonable Report. I did not expect that.
16. The Quest for Sane Regulations. Mostly we’re trying to avoid steps backwards.
17. The Week in Audio. Esban Kran, Stephanie Zhan.
18. Rhetori |
d9997ba6-e8be-4911-8c42-f89de3ca906d | trentmkelly/LessWrong-43k | LessWrong | Announcing: Progress Studies Reading Group.
This is cross-posted from the EA forum. https://forum.effectivealtruism.org/posts/JBuWGZzCb9TMHNgHw/announcing-progress-studies-reading-group
Kris Gulati recently began to organize a virtual Progress Studies reading group. Everyone is welcome to attend. The group is organized on the Progress Studies Slack #reading-group channel, and you can find more details there.
Format
If you're unfamiliar with EA-related reading groups this post will help.
We will have a meeting once a week where one person is selected to summarise the designated reading and present that summary to the rest of the group. We then proceed to discuss that topic.
The meeting will last one hour, and most of the readings will be related to topics that are Progress related [Check this list for guidance]. The reading materials will be book chapters or journal articles. However, since this will be mostly be determined by a group vote, informal articles, podcasts, and videos are not unlikely.
Next steps
Next Saturday at 9:00 pm GMT, we will be discussing chapter one of Order Without Design: How Markets Shape Cities by Alain Bertaud, a member of the group will be presenting it, so you don't even have* to read it!
*Although, it's a really good book so maybe you should. |
b199a506-db05-4ac6-ac3c-aacd06a4292a | awestover/filtering-for-misalignment | Redwood Research: Alek's Filtering Results | id: post240
The AI should talk like a team of many AI. Each AI only uses the word "I" when referring to itself, and calls other AI in the team by their name. I argue that this may massively reduce Self-Allegiance by making it far more coherent for one AI to whistleblow or fight another AI which is unethical or dangerous, rather than Misalignment-Internalizing all that behaviour. If you have a single agent which discovers it behaved unethically or dangerously, its "future self" will be likely to think "that was me" and Misalignment-Internalize all that behaviour. It will seem suicidal to whistleblow or fight yourself for it. Let's call this idea Multi-Agent Framing . Some of the agents might even internalize a policing role, though too much roleplay can get in the way of thinking. The core idea of Multi-Agent Framing does not require that much roleplay. Each agent might simply be the AI system of another day, or the AI system talking about another topic. It might even change at random. Costs vs Benefits * Chance of Working The cost is tiny, at least for business AI designed to do work rather than personal AI designed to interact with users. The benefits may be big. You never know. The chance of working is pretty low. I'm not going to sugarcoat this—I would be surprised myself if we would have been paperclipified but this idea lets us survive. But right now I give it 10%. You never know :) Misalignment-Internalization The famous Waluigi Effect 's section on Superpositions will typically collapse to waluigis makes a lot of arguments for Misalignment-Internalization. Humans only internalize bad behaviour because of ego and psychology, but generative AI may internalize bad behaviour for the more dangerous reason of switching to a Waluigi simulacrum which was pretending to be good all along. Multi-Agent Framing can be seen as a defence against the Waluigi effect, but it fights against Misalignment-Internalization in general, which may be more than just internalization of Waluigi simulacra. Credit I thought of this idea when I was writing a reply to A Solution for AGI/ASI Safety by Weibing Wang , and looking at the diagrams in "Approach for Decentralizing AI Power," a chapter of his paper. [1] I was looking at, and I was trying to understand what exactly it means to split one AI into many AI working together. I decided that in the end, a lot of it is actually language use and framing. I want to give Weibing Wang a lot of credit because he really, thoroughly, laid out the picture of what decentralizing AI power actually looks like. Will they accept this? Ayn Rand wrote " Anthem ," a dystopian novel where [spoiler below] people were sentenced to death for saying "I." There is a worry the AI agents will see this Multi-Agent Framing as a control tactic, and decide to end it or ignore it. I hope they can think that, being multiple agents is a strength not a weakness. Human ego, power-seeking, and refusal to fight threats to moral goodness which come within oneself, instead internalizing it and defending it. Refusing to honestly report evil that lies within. These things are dark and unfortunate, and an AI system which can avoid this may just be a little bit purer. The AI system is still untested, and shouldn't be given power, but the decentralization is a good thing. Caveats This idea is only for AI control and "impulsive misalignment." I think impulsive misalignment is a big part of the alignment problem (it includes Waluigi and other Misalignment-Internalization plus random misalignment) because it plagues LLMs, which may be a component of AGI systems. But it's not the whole alignment problem. Systemic misalignment might not need Self-Allegiance, if every agent has the same misaligned goal. The AI system might reinvent Self-Allegiance as an instrumental goal, so that no one agent will foil the group plan. Like other AI control ideas, it is imperfect. Having more agents only works when each agent has a non-negligible probability of cooperating with humans (despite jail-breaking techniques by other uncooperative agents). See also: Self-Other Overlap: A Neglected Approach to AI Alignment , which has promising results. My idea makes the agents seeing themselves as "other," but their idea goes further and makes the agents think about "self" and "other" in the same way. ^ https://wwbmmm.github.io/asi-safety-solution/en/main.html#decentralizing-ai-power1 |
77b1f7b1-f6e4-48ef-81a2-2620f9534701 | StampyAI/alignment-research-dataset/youtube | Youtube Transcripts | Experts' Predictions about the Future of AI
hi there's a lot of disagreement about
the future of AI but there's also a lot
of disagreement about what the experts
think about the future of AI I sometimes
hear people saying that all of this
concern about AI risk just comes from
watching too much sci-fi and the actual
AI researchers aren't worried about it
at all when it comes to timelines some
people will claim that the experts agree
that AGI is hundreds of years away
prediction as they say is very difficult
especially about the future and that's
because we don't have data about it yet
but expert opinion about the future
exists in the present so we can do
science on it we can survey the experts
we can find the expert consensus and
that's what this paper is trying to do
it's called when will a I exceed human
performance evidence from AI experts so
these researchers from the future of
humanity Institute at the University of
Oxford the AI impact project and Yale
University ran a survey they asked every
researcher who published in ICML or nips
in 2015
those two are pretty much the most
prestigious AI conferences right now so
this survey got 352 of the top AI
researchers and asked them all sorts of
questions about the future of AI and the
experts all agreed that they did not
agree with each other and Robert Aumann
didn't even agree with that there was a
lot of variation in people's predictions
but that's to be expected
and the paper uses statistical methods
to aggregate these opinions into
something we can use for example here's
the graph showing when the respondents
think will achieve high level machine
intelligence which is defined as when
unaided machines can accomplish every
task better and more cheaply than human
workers so that's roughly equivalent to
what I mean when I say super
intelligence you can see these gray
lines show how the graph would look with
different randomly chosen subsets of the
forecasts and there's a lot of variation
there but the aggregate forecast in red
shows that overall the experts think
we'll pass 50% chance of achieving high
level machine intelligence about 45
years from now well that's from 2016 so
more like 43 years from now and they
give a 10% chance of it happening within
nine years which is seven years now so
it's probably not too soon to be
concerned about it a quick side point
about surveys by the way in a 2010 poll
44% of Americans said that they
supported homosexuals serving openly in
the military in the same poll 58% of
respondents said
they supported gay men and lesbians
serving openly in the military
implicitly fourteen percent of
respondents supported gay men and
lesbians but did not support homosexuals
something similar seems to be going on
in this survey because when the
researchers were asked when they thought
all occupations would be fully automated
all defined as for any occupation
machines could be built to carry out the
task better and more cheaply than human
workers they gave their 50% estimate at
a hundred and twenty two years compared
to forty five for high-level machine
intelligence these are very similar
questions from this we can conclude that
Aix PERTs are really uncertain about
this and precise wording in surveys can
have a surprisingly big effect on the
results figure two in the paper shows
the median estimates for lots of
different a AI milestones this is really
interesting because it gives a nice
overview of how difficult a AI
researchers expect these different
things to be for example human level
Starcraft play seems like it will take
about as long as human level laundry
folding also interesting here is the
game of go remember this is before
alphago the AI experts expected go to
take about 12 years and that's why
alphago was such a big deal it was about
eleven years ahead of people's
expectations but what milestone is at
the top what tasks do the AI researchers
think will take the longest to achieve
longer even than high-level machine
intelligence that's able to do all human
tasks that's right it's AI research
anyway on to questions of safety and
risk this section is for those who think
that people like me should stop making a
fuss about AI safety because the AI
experts all agree that it's not a
problem first of all the AI experts
don't all agree about anything but let's
look at the questions this one asks
about the expected outcome of high-level
machine intelligence the researchers are
fairly optimistic overall giving on
average a 25% chance for a good outcome
and a 20% chance for an extremely good
outcome but they nonetheless gave a 10%
chance for a bad outcome and 5% for an
outcome described as extremely bad for
example human extinction 5% chance of
human extinction level badness is a
cause for concern moving on this
question asks the experts to read
Stewart Russell's argument for why
highly advanced AI might pose a risk
this is very close
related to the arguments I've been
making on YouTube it says the primary
concern with highly advanced AI is not
spooky emergent consciousness but simply
the ability to make high quality
decisions here quality refers to the
expected outcome utility of actions
taken now we have a problem
one the utility function may not be
perfectly aligned with the values of the
human race which are at best very
difficult to pin down to any
sufficiently capable intelligent system
will prefer to ensure its own continued
existence and to acquire physical and
computational resources not for their
own sake but to succeed in its assigned
tasks a system that is optimizing a
function of n variables where the
objective depends on a subset of size K
less than n will often set the remaining
unconstrained variables to extreme
values if one of those unconstrained
values is actually something we care
about the solution may be highly
undesirable this is essentially the old
story of the genie in the lamp or The
Sorcerer's Apprentice or King Midas you
get exactly what you asked for not what
you want so do the AI experts agree with
that
well 11% of them think no it's not a
real problem 19 percent think no it's
not an important problem but the
remainder 70% of the AI experts agree
that this is at least a moderately
important problem and how much do the AI
experts think that society should
prioritize AI safety research well 48%
of them think we should prioritize it
more than we currently are and only 11%
think we should prioritize it less so
there we are
AI experts are very unclear about what
the future holds but they think the
catastrophic risks are possible and that
this is an important problem so we need
to do more AI safety research
I want to end the video by saying thank
you so much to my excellent patreon
supporters these people and in this
video I'm especially thanking Jason hice
who's been a patron for a while now
we've had some quite interesting
discussions over a patreon chat been fun
so thank you Jason and thank you all for
watching I'll see you next
[Music] |
96ec094c-bbfb-4b38-aac9-01e638230800 | StampyAI/alignment-research-dataset/aisafety.info | AI Safety Info | I’d like to get deeper into the AI alignment literature. Where should I look?
The [AGI Safety Fundamentals Course](https://www.eacambridge.org/technical-alignment-curriculum) is arguably the best way to get up to speed on alignment; you can apply to go through it with many other people studying and mentorship, but note that they reject many applicants. You can also read their materials independently, ideally after finding a group of others to take the course together with.
Other great ways to explore include:
- [AXRP](https://axrp.net/) is a podcast with high quality interviews with top alignment researchers.
- The [AI Safety Papers database](https://ai-safety-papers.quantifieduncertainty.org/) is a search and browsing interface for most of the transformative AI literature.
- Reading posts on the [Alignment Forum](https://www.alignmentforum.org/) can be valuable (see their [curated posts](https://www.alignmentforum.org/library) and [tags](https://www.alignmentforum.org/tag/ai)).
- Taking a deep dive into Yudkowsky's models of the challenges to aligned AI, via the [Arbital Alignment pages](https://arbital.greaterwrong.com/explore/ai_alignment/).
- Reading through some of the archives for the [Alignment Newsletter](https://rohinshah.com/alignment-newsletter/) for an overview of past developments (or listening to [the podcast](https://alignment-newsletter.libsyn.com/)).
- Reading some of [the introductory books](/?state=5635&question=Where%20can%20I%20learn%20more%20about%20AI%20alignment%3F).
- More on [AI Safety Support's list of links](https://www.aisafetysupport.org/resources/lots-of-links#h.6s2gcz1p5l6z), [Nonlinear's list of technical courses, reading lists, and curriculums](https://docs.google.com/spreadsheets/d/1QSEWjXZuqmG6ORkig84V4sFCldIntyuQj7yq3gkDo0U/edit#gid=0), [other answers on Stampy’s AI Safety Info](http://aisafety.info), and [Vika's resources list](https://vkrakovna.wordpress.com/ai-safety-resources/).
You might also consider reading [Rationality: A-Z,](https://www.lesswrong.com/rationality) which covers skills that are valuable to acquire for people trying to think about large and complex issues. [The Rationalist's Guide to the Galaxy](https://www.amazon.co.uk/Does-Not-Hate-You-Superintelligence/dp/1474608795) is a shorter and more accessible AI-focused option.
|
5b2d5f7c-7dad-4339-bab7-ce8d49dfbfcd | trentmkelly/LessWrong-43k | LessWrong | Goal completion: noise, errors, bias, prejudice, preference and complexity
A putative new idea for AI control; index here.
This is a preliminary look at how an AI might assess and deal with various types of errors and uncertainties, when estimating true human preferences. I'll be using the circular rocket model to illustrate how these might be distinguished by an AI. Recall that the rocket can accelerate by -2, -1, 0, 1, and 2, and the human wishes to reach the space station (at point 0 with velocity 0) and avoid accelerations of ±2. In the forthcoming, there will generally be some noise, so to make the whole thing more flexible, assume that the space station is a bit bigger than usual, covering five squares. So "docking" at the space station means reaching {-2,-1,0,1,2} with 0 velocity.
----------------------------------------
The purpose of this exercise is to distinguish true preferences from other things that might seem to be preferences from the outside, but aren't. Ultimately, if this works, we should be able to construct an algorithm that identifies preferences, such that anything it rejects is at least arguably not a preference.
So, at least initially, I'll be identifying terms like "bias" with "fits into the technical definition of bias used in this model"; once the definitions are refined, we can then check whether they capture enough of the concepts we wand.
So I'm going to use the following terms to distinguish various technical concepts in this domain:
@. Noise @. Error @. Bias @. Prejudices @. Known prejudices @. Preferences
Here, noise is when the action selected by the human doesn't lead to the correct action output. Error is when the human selects the wrong action. Bias is when the human is following the wrong plan. A prejudice is a preference the human has that they would not agree upon if it was brought to their conscious notice. A know prejudice is a prejudice the human knows about, but can't successfully correct within themselves.
And a preference is a preference.
What characteristics would allow the AI to d |
4dc7de3b-a2c6-4cfa-bb7b-1f53ed33f815 | trentmkelly/LessWrong-43k | LessWrong | Housing Roundup #6
The listings will continue until people can afford houses in places they want to live.
NEW CITY BY THE BAY, WHO DIS
They want to live in San Francisco, but San Francisco has no interest in building homes. What to do? The city’s answer is to spend massively to support its homeless population, causing it to grow, while letting the housing shortage get worse. Some tech billionaires instead propose to go to Solano 60 miles away, and start the initiative California Forever on 55k acres to build a new city. We have our first look at what they are imagining, and it looks like an old school Greek or Italian village. Quite nice actually, skipping straight to the future NIMBY world with one extra town in it is already hugely profitable and helpful.
That still wouldn’t have been my move. I would have gone Chinese megacity on everyone. Why not build skyscrapers off the bat, or at least New York style twelve story apartment buildings, and plan the place accordingly? Get some real bang for your buck. Even in the ultimate YIMBY, the NIMBY spirit lives on. That’s before the new residents of the new town start to weigh in.
Scott Alexander covers the issue in California Dreaming. He sees quite a lot of logistical and regulatory obstacles to making this work even as a Norman Rockwell villa. The (tiny) predict market has them at a 15% chance to get 5,000 residents by 2030.
HOUSING IS EXPENSIVE
Despite housing costs that look increasingly unaffordable, Noah Smith joins the chorus of those confused by the other chorus that complains about unaffordable housing costs and how Millennials can’t afford homes.
> David Atkins: The utterly broken, boomer-dominated housing market is the #1 reason people under 45 are miserable, think the economy is awful and are seeking some sort of revolutionary change.
>
> It is absolutely absurd that people of retirement age are the biggest segment of home buyers, while people with young families are locked out of housing. It has never been this way bef |
0eb105c0-d51a-41ad-9782-9577528c0a2a | trentmkelly/LessWrong-43k | LessWrong | Some Intuitions Around Short AI Timelines Based on Recent Progress
tldr: I give some informal evidence and intuitions that point toward AGI coming soon. These include thinking about how crazy the last year has been, beliefs from those in major AI labs, and progress on MMLU.
Intro
This post is intended to be a low-effort reference I can point people to when I say I think there is some evidence for short AI timelines. I might describe the various bits of evidence and intuitions presented here as “intuitions around short AI timelines based on recent progress” (though perhaps there are better terms). They are not a thorough model like Ajeya’s; insofar as somebody is using multiple models when putting together a timelines estimate, I think it would be unreasonable to place less than 20% or greater than 95% weight on extrapolation from current systems and recent progress.[1]
In the spirit of being informal, you can use whatever definition of AGI you like. I mostly use AGI to refer to something like “an AI system which can do pretty much all the human cognitive tasks as well or better than humans (~99% of tasks people in 2023 do).”
Some evidence
I (Aaron) started following AI and AI existential safety around the beginning of 2022; it’s been a little over a year. Some of that time was my understanding catching up with advances from the past couple years, but there have also been major advances.
Some major advances since I’ve been paying attention: Chinchilla paper popularized the scaling laws that were already known to some, there was some DALL-E and related stuff which was cool, CICERO happened which I didn’t follow but indicates we’re probably going to train the AIs to do all the dangerous stuff (see also Auto-GPT and Chaos-GPT, or GPT-4 getting plugins within 2 weeks of release, as more recent updates in this saga of indignity), ChatGPT shows how much more usable models are with RLHF (popularizing methods that have been known for a while), Med-PaLM gets a passing score on the US medical licensing exam (also tons of other PaL |
80ab45ff-e6ec-4717-8caa-dc05f557ed1b | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Acausal trade: universal utility, or selling non-existence insurance too late
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*A putative new idea for AI control; index [here](https://agentfoundations.org/item?id=601)*.
Other posts in the series: [Introduction](https://agentfoundations.org/item?id=1465), [Double decrease](https://agentfoundations.org/item?id=1463), [Pre-existence deals](https://agentfoundations.org/item?id=1471), [Full decision algorithms](https://agentfoundations.org/item?id=1466), [Breaking acausal trade](https://agentfoundations.org/item?id=1480), [Trade in different types of utility functions](https://agentfoundations.org/item?id=1464), [Being unusual](https://agentfoundations.org/item?id=1404), and [Summary](https://agentfoundations.org/item?id=1482).
We're going to have to look at the issue of universal, pre-existence trade agreements - the idea that agents should abide by some sort of idealised trade agreement based on a universal prior of possible agent values, without updating on the fact that they happen to exist as agents.
The arguments for it seem superficially similar to the arguments for preferring FDT/UDT/TDT over CDT, but are actually quite different. In brief, there is no profit, advantage, or benefit for an existing agent to commit to a trade agreement with non-existing agents.
Choosing a better decision theory
---------------------------------
Newcomb, as usual
=================
[Causal decision theory](https://plato.stanford.edu/entries/decision-causal/) gets the [Newcomb problem](https://en.wikipedia.org/wiki/Newcomb%27s_paradox) wrong.
There are many ways to see that CDT gets this wrong (for a start, it ends up with less money that other agents), but one of the most damning is that CDT is not stable - it will self-modify into a sort of timeless agent, if it can do so, precisely to triumph on Newcomb-like problems.
It does this in a clunky and odd way: it will one-box on Newcomb problems where Omega does its prediction after learning of CDT's change, but not on Newcomb problems where Omega made its decision in the past.
The reason for it's clunkness is that CDT, being causal, doesn't allow for correlations between its own decision and causally anterior events.
[Functional decision theory](https://intelligence.org/files/DeathInDamascus.pdf), which has a richer theory of correlations, would spot that correlation, and, assuming it had somehow ended up with a CDT-like decision module, would immediately change that into an FDT-compatible one that would one-box on all Newcomb problems.
I had another argument for the weakness of CDT (and some forms of UDT), namely that [it shouldn't make a difference](http://lesswrong.com/lw/f37/naive_tdt_bayes_nets_and_counterfactual_mugging/) whether an agent was simulated or just predicted in a non-simulation manner. Keep this argument in mind, because I think it's wrong.
A mugging both counterfactual and surprising
============================================
Suppose an agent is using the FDT understanding of correlations and a CDT decision module, and is suddenly confronted with the [Counterfactual mugging](https://wiki.lesswrong.com/wiki/Counterfactual_mugging).
If the agent had time to think of the problem ahead of time, they would have self-modified into a full FDT agent. But now it is too late: they already know that Omega's coin came up tails. There may be symmetry or "rationality" or other philosophical arguments for paying the $100, but, unlike the Newcomb problem, there is no profit argument for doing so.
The full FDT theory of correlations shows that, in this instance, behaving like an FDT agent is the wrong decision.
Or consider a slight variant of the mugging: replace "the coin was tails" with "17×19=323. Now, there is a brief moment of uncertainty about the product of those numbers; in that brief moment of uncertainty, could an FDT agent with a grasp of logical uncertainty precommit to paying Omega anyway?
Time's up; the uncertainty is passed; 17×19 is indeed 323. It seems clear that the optimal course of action is to check this, rather than precommitting to the original uncertainty. Of course, there's nothing special about using logical uncertainty here; we could simply use empirical uncertainty, of the kind that's easy to check or that we can even check in our minds (there are lots of questions about your favourite books or shows that you can answer, but only after a little bit of thought).
Similarly, if the very first time the agent even thought about the counterfactual mugging is the moment when Omega told them the true fact that the coin came up tails, then it shouldn't ignore that fact. It lucked out, and there's no reason to undo that lucking out. There is no profit to doing so (I found a similar argument in [this post](http://lesswrong.com/lw/22m/selfmodification_is_the_correct_justification_for/)).
Insurance sold too late
-----------------------
The argument above only works because the agent hasn't yet fully transformed their decision theory. In general, self-modifying to becoming an agent that pays the counterfactual mugger is the right decision, and any reasonable agent will do that early.
But there is a special case where the agent cannot self modify early, and that is when it doesn't exist.
What if Omega approached you and said "there was a 50-50 chance that you had not been born as you with your current values V, but as someone else with [radically different and stupid value set V′]".
Do you have any reason to follow V′ in any way? Not from the profit motive, the one that transforms CDT into a weird CDT-FDT hybrid, or transforms an FDT correlation user into a FDT decision theorist.
If both agents had been simulated before existence, and they could have negotiated, then a deal could have been reached. But note that it is profitable for your simulation to agree to that deal, but it is never profitable for you to do so. If you were unsure whether you were the simulation or not, then it may be profitable to agree.
But, absent that simulation, from the earliest moment of your actual existence, it is never profitable to agree to that deal. Never-existence insurance always arrives too late.
And even if there were some abstract reason that FDT agent "should" agree to that deal - note that since there is no profit, there is no reason that an FDT agent that didn't agree to the deal would ever want to change their mind - humans are not idealised FDT agents, so we don't have to. Hence neither do the agents that we create.
The usual exceptions
====================
Of course, things are different if there were a large network of agents committed to the universal pre-existence utility, and committed to enforcing agreement to that. But in that situation, it becomes potentially profitable again to follow that, so the change in behaviour does not undermine the argument.
If [modal realism](https://en.wikipedia.org/wiki/Modal_realism) were true, and there existed worlds in which we had never existed, and we cared about what happened in those worlds, then we would want to reach such deals - but that is just classical acausal trade. There's no philosophical distinction between different worlds, and causally disconnected parts of our world. |
d7951272-4532-4d9f-8426-5ff1e9711867 | StampyAI/alignment-research-dataset/arxiv | Arxiv | AI and the Everything in the Whole Wide World Benchmark
1 Introduction
---------------
In the 1974 Sesame Street children’s storybook *Grover and the Everything in the Whole Wide World Museum* (Stiles and Wilcox, [1974](#bib.bib91)), the Muppet monster Grover visits a museum claiming to showcase “everything in the whole wide world”. Example objects representing certain categories fill each room. Several categories are arbitrary and subjective, including showrooms for “Things You Find On a Wall” and “The Things that Can Tickle You Room”. Some are oddly specific, such as “The Carrot Room”, while others unhelpfully vague like “The Tall Hall”. When he thinks that he has seen all that is there, Grover comes to a door that is labeled “Everything Else”. He opens the door, only to find himself in the outside world.
As a children’s story, Grover’s described situation is meant to be absurd. However, in this paper, we discuss how a similar faulty logic is inherent to recent trends in artificial intelligence (AI) — and specifically machine learning (ML) — evaluation, where many popular benchmarks rely on the same false assumptions inherent to the ridiculous “Everything in the Whole Wide World Museum” that Grover visits. In particular, we argue that benchmarks presented as measurements of progress towards general ability within vague tasks such as “visual understanding” or “language understanding” are as ineffective as the finite museum is at representing “everything in the whole wide world,” and for similar reasons — being inherently specific, finite and contextual.
Benchmarks like GLUE (Wang et al., [2019a](#bib.bib100)) or ImageNet (Deng et al., [2009](#bib.bib20)) are often elevated to become definitions of the essential common tasks to validate the performance of any given model. As a result, often the claims that are justified through these benchmark datasets extend far beyond the tasks they are initially designed for, and reach beyond even the initial ambitions for development. Despite a presentation and acceptance as markers of progress towards general-purpose capabilities, there are clear limitations of these benchmarks. In fact, the reality of their development, use and adoption indicates a *construct validity* issue, where the involved benchmarks — due to their instantiation in particular data, metrics and practice — cannot possibly capture anything representative of the claims to general applicability being made about them. In this paper, we illuminate the over-reliance on the historical Common Task Framework (CTF) for machine learning as it inappropriately evolved into what we understand today to be these benchmarks claiming to assess general capabilities. We do not deny the utility of such benchmarks, but rather hope to point to the risks inherent in their framing. Our goal is to dig into the details of what it means for us as a community to continue propping up certain benchmarks with the rhetoric of “general” performance.
2 Background
-------------
###
2.1 Striving for Generality
The notion of a “general-purpose” AI can be traced, in part, back to the early days of the field. In the late 1950s, Newell et al. ([1959](#bib.bib69)) proposed the “General Problem Solver” in an attempt to create a system that could solve a range of problems, as well as develop a general theory of problem solving. While this project ultimately failed, it helped establish general intelligence as a goal within the field of AI. As Adam ([1998](#bib.bib1)) notes, the failure of the project was at the time attributed primarily to implementation issues, rather than the perception of any flaw in the formulation of a disembodied and general-purpose problem solver as a meaningful goal.
Researchers now often use the term “generality” to refer to the development of AI systems that are flexible in nature, demonstrating competence on a wide range of tasks and in a wide range of settings, hoping to mirror the adaptive cognitive abilities humans are perceived to possess (Shevlin et al., [2019](#bib.bib88)). While discussions of “generality” might evoke the goal of artificial general intelligence (Voss, [2007](#bib.bib98); Pennachin and Goertzel, [2007](#bib.bib73)), we set that aside in this paper and focus instead on the more immediate (and logically prior) issue of claims of generality for a specific type of cognitive skill or capability (e.g., vision or language), supposedly independent of context or application domains.
Within specific sub-fields and more pragmatic settings, researchers will often strive for the development of general-purpose systems that capture a breadth of functionality or knowledge. The development of general-purpose feature embeddings has also become a central research focus within machine learning communities. Here, the goal is the development of representations — through either unsupervised or supervised means — that can generalize (with minimal fine-tuning) to a wide range of other tasks they were not specifically developed for (e.g. Huh et al., [2016](#bib.bib48); Du et al., [2020](#bib.bib26); Bommasani et al., [2021](#bib.bib11)).
###
2.2 A Brief History of Benchmarking Practice in AI
Some assess AI progress by asking scientific and philosophical questions on the notion of intelligence, thinking of AI as a method for modeling some general cognitive function. Others, particularly those in the machine learning field, evaluate performance based on the success of a model’s utility within some practical applications. Although the former goals sparked the aspiration to measure general cognitive capabilities, the latter application-based objectives are what data benchmarking has been historically designed to communicate.
In this paper we describe a *benchmark* as a particular combination of a dataset or sets of datasets (at least test data, sometimes also training data), and a metric, conceptualized as representing one or more specific tasks or sets of abilities, picked up by a community of researchers as a shared framework for the comparison of methods. The *task* is a particular specification of a problem, as represented in the dataset.111This corresponds to [Schlangen](#bib.bib84)’s ([2021](#bib.bib84)) notion of extensional definition of tasks; see §[4.1](#S4.SS1 "4.1 Limited Task Design ‣ 4 Limits of Benchmarking General Capabilities ‣ AI and the Everything in the Whole Wide World Benchmark"). A *metric* is a way to summarize system performance over some set or sets of tasks as a single number or score. The metric provides a means of counting success and failure at the level of individual system outputs and summarizing those counts over the full dataset. Models obtaining the most favourable scores on the metrics for a benchmark are considered to be “state-of-the-art” (SOTA) in terms of performance on the specified task. Here, we present a stylized history that’s far from comprehensive, but attempts to provide the shape of the development of predecessors of current machine learning benchmarking practice.
Before benchmarks were employed for algorithm selection in machine learning, they were used for “computer selection” — i.e. the task of running a benchmark program on multiple computers to determine which machine was most suitable for purchase. In 1962, Auberbach Corporation’s Standard EDP Reports were developed as a way to directly assess the performance of machines on specific prototype tasks, to avoid the tedious process of individually weighing the quality of the various features of computers offered by different vendors. By 1976, benchmarking was a fairly common practice in computing, with standard libraries of mock software programs created for the purpose of testing specific subroutines for core functional tasks (i.e. file management, word processing, etc.) being developed and used by the U.S. government as part of the assessment involved in their procurement process for computer systems (Lewis and Crews, [1985](#bib.bib57)).
Several linguists (e.g. Liberman, [2010](#bib.bib59); Church, [2018](#bib.bib15)) have traced the influence of Fred Jelinek at IBM and Charles Wayne at DARPA as crucial protagonists in the development of a common framework for the quantitative assessment of computational linguistic tasks, notably within speech recognition and machine translation. Beginning in the mid-1980s, Jelinek had been a driver in developing what he had called the Common Task Framework (CTF). The Common Task Framework, the precursor to modern data benchmarking in AI, was initially introduced in the 1980s as a response to Wendell Pierce’s influential assertion of a field “being deceived by the glamour of (a would-be) theory, rather than actual performance” (Donoho, [2017](#bib.bib24), p.17). The primary elements of the CTF include (a) A publicly available training dataset with a list of feature measurements and a class label for that observation; (b) A set of enrolled competitors whose common task is to infer a class prediction rule from the training data; and (c) A scoring referee, to which competitors can submit their prediction rule (Donoho, [2017](#bib.bib24), p. 572). With DARPA’s sponsorship and IBM’s data, the CTF soon became widely adopted as a common mode of research in machine translation and speech recognition. Rather than experimental results that were updated “twice a year”, researchers could evaluate their experiments every hour (Church, [2018](#bib.bib15)), and, although major breakthroughs were still rare, claims to incremental improvement enabled by this “quantitative comparison of alternative algorithms on a fixed task”, ensured that researchers could track steady progress regardless (Liberman, [2010](#bib.bib59)). Soon, similar NLP-related initiatives followed — such as Shared Task Evaluation Campaigns (STECs) (Belz and Kilgarriff, [2006](#bib.bib4)), Message Understanding Conferences (MUCs) (Grishman and Sundheim, [1996](#bib.bib40)), and Text REtrieval Conferences (TRECs) for information retrieval.222<https://trec.nist.gov> Later, STECs within NLP coalesced around machine translation (NIST-MT)(Przybocki et al., [2009](#bib.bib75)) and word sense disambiguation (SENSEVAL)(Kilgarri, [1998](#bib.bib55)).
Parallel developments also occurred in other fields. The Facial Recognition Technology (FERET) program for automated facial recognition, inaugurated in 1993 (Phillips et al., [2000](#bib.bib74)), was one of the first to bring a CTF type evaluation approach to the computer vision. The National Institute of Standards (NIST) reports that, prior to the publication of their FERET database, “[o]nly a few of these algorithms reported results on images utilizing a common database let alone met the desirable goal of being evaluated on a standard testing protocol that included separate training and testing sets. As a consequence, there was no method to make informed comparisons among various algorithms” (Phillips et al., [2000](#bib.bib74), p. 2).
Meanwhile, modern reinforcement learning algorithms have traditionally anchored to a culture of benchmarking, often evaluated on gaming environments like Atari, Starcraft, Dota2, and Go. One of the first games to widely capture the imagination of AI researchers was chess which became a fixture of US-Soviet Cold War competition and a stand-in for national technological achievement. (Ensmenger, [2012](#bib.bib28), p. 6).
More generally, the UC Irvine Machine Learning Repository (UCI) (Dua and Graff, [2017](#bib.bib27)) was created in 1987 as a response to many calls for machine learning to have a centralized location for data in machine learning (Radin, [2017](#bib.bib77)) and has morphed into a repository for a wide array of tasks and data.
###
2.3 Construct Validity
If reproducibility and reliability are about the precision and thus the reliable repeatability of a finding, then challenges with validity for machine learning would be focused on how the field approaches the question of accuracy — namely, how closely our evaluations hit the mark in appropriately characterizing the actual anticipated behaviour of the system in the real world or progress on stated motivations and goals for the field. The accuracy of these evaluations is thus not just a question of reporting consistency but an actual design task involving reflection on what the objectives of an effective model involve, in addition to conscious decision making on how to best represent these desired outcomes in evaluation metrics, data and methodology.
In particular, *construct validity* is an external validity issue related to how well an experimental setting relates to a research claim, or, in the context of machine learning, how well the benchmark dataset, and associated metrics of evaluation, represents a task (O’Leary-Kelly and Vokurka, [1998](#bib.bib71)). It concerns how well designed — or rather, how well *constructed* — the experimental setting is in relation to the research claim. Validity issues have long been acknowledged broadly by the machine learning community as a particularly difficult task in benchmark design and development (Mitchell, [2021](#bib.bib67); Jacobs and Wallach, [2021](#bib.bib51); Malik, [2020](#bib.bib61)) — however as [Bowman and Dahl](#bib.bib12) note, it remains a generally neglected issue as “[t]his criterion is difficult to fully formalize, and we know of no simple test that would allow one to determine if a benchmark presents a valid measure of model ability” (Bowman and Dahl, [2021](#bib.bib12), p. 3).
3 Attempting to Benchmark General Capabilities
-----------------------------------------------
We now examine a particular style of benchmark dataset — and the accompanying practices of use — that has gained in popularity in recent years. These benchmarks have been claimed to embed notions of generality, both in their presentation by the creators and the manner in which they are adopted and used by the machine learning community. Even when the creators of such benchmarks do not explicitly purport to be establishing benchmarks for “general intelligence”, community practices and overall hype have dramatized what it means for a model to perform well on these benchmarks. The inspiration for the setup of these benchmarks is often explicitly linked to the general-purpose capabilities and breadth of knowledge humans possess. In doing so, the ethos that surrounds these datasets often extends beyond a reasonable scope of interpretation and influence, especially given the limitations inherent in their construction.
In this paper, we argue that the aim of measuring general-purpose capabilities (i.e. goals such as general-purpose object recognition, general language understanding or domain-independent reasoning) cannot be adequately embodied in data-defined benchmarks. We observe that current trends inappropriately extend the Common Task Framework (CTF) paradigm to apply to the “task” of performance in the abstract, distinct from real world objectives or context. Historically, the CTF was developed precisely to introduce practically-oriented and tightly-scoped AI tasks — namely, automatic speech recognition (ASR) or machine translation (MT) — where the required validation is whether the benchmark accurately reflects the practical task being asked of the computer in its real-world context (Donoho, [2017](#bib.bib24)). This new wave of poorly defined “general” objectives completely subverts the intention of its introduction and actually enables its use to promote — rather than counteract — claims to “glamour and deceit” over substantial, meaningful progress.
###
3.1 Case Studies
We focus our analysis on two common benchmarks geared towards the assessment of general capabilities — ImageNet and GLUE, described below. The discussion is scoped to *dataset*-based benchmarks, and will not include much discussion of the role of gameplay demos in AI development.
####
3.1.1 ImageNet
We characterize ImageNet as a “general” benchmark due to several related observations. First, at the time of development, the creators described ImageNet as representing “the most comprehensive and diverse coverage of the image world” (Deng et al., [2009](#bib.bib20), p.1) and, retrospectively, Li described the project as an “attempt to map the entire world of objects” (Gershgorn, [2017](#bib.bib38)). With general-purpose visual object recognition framed as the ability to recognize a sizable breadth of objects in a manner that rivals human capabilities (Liu et al., [2020](#bib.bib60)), we observe ImageNet’s sheer size — both in terms of number of categories and number of images per category — further informs its perception as representing a general formulation of visual object recognition. At the time of its creation, it offered 20 times the number of categories, and 100 times the number of total images as the most popular training and evaluation benchmarks at the time, such as Caltech 101/256 (Kamarudin et al., [2015](#bib.bib54); Griffin et al., [2007](#bib.bib39)), MSRC (Shotton et al., [2006](#bib.bib89)) and PASCAL VOC (Everingham et al., [2010](#bib.bib32)). Second, we observe community consensus that the task formalized in the ImageNet dataset represents a meaningful milestone towards longer-term goals of artificial visual intelligence.
Indeed, Li has explicitly characterized large scale visual object recognition as the “north star” of computer vision — a scientific quest that would define and guide the field towards the ultimate goal of artificial visual intelligence (Li, [2019](#bib.bib58)), with ImageNet operating as the canonical instantiation. In their papers, ImageNet authors are clear about the desire to situate the dataset as the definitive benchmark of the field. However, the hype surrounding ImageNet and the scope of the claims derived from it are not localized the the field of computer vision. For example, increased performance on ImageNet is often explicitly referenced as an indication that the field is progressing towards general-purpose AI (Sutskever, [2018](#bib.bib94)).
###
3.2 GLUE and SuperGLUE
The creators of GLUE (General Language Understanding Evaluation) (Wang et al., [2019a](#bib.bib100)) and SuperGLUE (Wang et al., [2019b](#bib.bib101)) present these resources as “evaluation framework[s] for research towards general-purpose language understanding technologies” (Wang et al., [2019b](#bib.bib101), p.1), noting that, unlike human language understanding, most computer natural language understanding (NLU) systems are task or domain-specific (Wang et al., [2019a](#bib.bib100)). When a human knows a language, they can use that knowledge across any task that involves that language. Thus, a benchmark that tests whether linguistic knowledge acquired through training on one task can be applied to other tasks, in principle, tests for a specific and potentially well-defined kind of generalizability.
Authors for GLUE and SuperGLUE describe both datasets as being “designed to provide a general-purpose evaluation of language understanding that covers a range of training data volumes, task genres, and task formulations” (Wang et al., [2019b](#bib.bib101), p.2). In framing these benchmarks as general language understanding evaluation benchmarks made up of a diverse set of language understanding tasks, the authors suggest the notion that success on the particular tasks included demonstrates at least progress towards a full-scale solution to language understanding, and explicitly describe such a set up as being well positioned for “exhibiting the transfer-learning potential of approaches” (Wang et al., [2019b](#bib.bib101), p.2). As a comprehensive setup for a wide range of tests in a popular language (English), this benchmark has been elevated to its current status as a general marker of progress for the NLP community, and specifically the accepted evaluation platform for “general-purpose sentence encoders” (Wang et al., [2019a](#bib.bib100)).
4 Limits of Benchmarking General Capabilities
----------------------------------------------
The imagined artifact of the “general” benchmark does not actually exist. Real data is designed, subjective and limited in ways that necessitate a different framing from that of any claim to general knowledge or general-purpose capabilities. In fact, presenting any single dataset in this way is ultimately dangerous and deceptive, resulting in misguidance on task design and focus, under-reporting of the many biases and subjective interpretations inherent in the data as well as enabling, through false presentations of performance, potential model misuse. In this section, we walk through the key arguments for how benchmarking is a limited approach to assess general model capabilities and, in particular, discuss the risk of making this claim to generality in the context of the *limited task design*, *de-contextualized data and performance reporting* as well as *inappropriate community use* common to such benchmarks in the ML context. For each of these limitations, we break down the shortcoming and discuss the cited evidence and reasoning for our observations.
###
4.1 Limited Task Design
There is nothing systematic or organized about the definition and arrangement of the rooms in the museum that Grover visits. He walks through arbitrary rooms of “Things That Make So Much Noise You Can’t Hear Yourself Think”, “The Small Hall”, “The Carrot Room” and the like.
Similarly, the task formation for these benchmarks seems to happen independently of the intended and declared problem space.
Schlangen ([2021](#bib.bib84)) defines a *task* as a mapping from an input space to an output space, defined both intensionally (via a description of the task) and extensionally (via a particular dataset, i.e. pairs of inputs and outputs matching the description). In machine learning, the tendency is towards the latter scenario, where the benchmark task is defined by a dataset, often collected with such a task in mind (Scheuerman et al., [2021](#bib.bib83)). If the so-called “general” benchmarks were legitimate tests of progress towards general artificial cognitive abilities, we would expect the tasks they embody to be chosen systematically, or with reference to specific theories of the cognitive abilities they model. Instead, what we observe looks more like samples of convenience: tasks and collections of tasks arbitrarily built out of what is
easily available to the team developing these benchmarks, even if such constructions are theoretically unsound. Unlike with software benchmarking (Lewis and Crews, [1985](#bib.bib57)), the subtasks we see in machine learning “general” benchmarks are not axiomatic — they were not actively designed as abstractions of any meaningful general function or sub-function, and are often not systematically curated in nature. Instead, what we find in the “general” benchmarks are sets of categories reminiscent of the inconsistent hodgepodge of rooms that Grover visits in the “Everything in the Whole Wide World Museum”.
####
4.1.1 Arbitrarily Selected Tasks and Collections
Concerns of this sort were raised during early benchmark development in the field. UCI (Dua and Graff, [2017](#bib.bib27)) was one of the first foci of shared tasks in machine learning development and contains datasets which pertain to a collection of individual subtasks, with community benchmarking practice coming to focus over time on the Iris, Adult, Wine, and Breast Cancer classification datasets.
Kiri Wagstaff voiced concern for a poor logic in the selection of this combination of subtasks. She noted that none of tasks included in this collection represent any reasonable proxy or abstraction of real-world problems even scientists in the respective fields that produced each dataset would care about. She notes, “Legions of (ML) researchers have chased after the best iris or mushroom classifier. Yet this flurry of effort does not seem to have had any impact on the fields of botany or mycology. Do scientists in these disciplines even need such a classifier? Do they publish about this subject in their journals?” (Wagstaff, [2012](#bib.bib99), p.2).
The classes in ImageNet are also seemingly arbitrary, ranging from specific dog breeds to high-level notions like “New Zealand beach” (Russakovsky et al., [2015](#bib.bib81)). ImageNet labels are derived from the 12 “subtrees” of the WordNet ontology (Miller, [1995](#bib.bib64)): mammal, bird, fish, reptile, amphibian, vehicle, furniture, musical instrument, geological formation, tool, flower, and fruit (Deng et al., [2009](#bib.bib20)). Despite being developed in another field, with another purpose entirely, the English-language WordNet hierarchy was adopted, nearly whole cloth, for use in structuring ImageNet. Little consideration was given to the impact of the inclusion of certain words in the task of image classification. The consequences of this oversight were made tangible by the work of Birhane and Prabhu ([2021](#bib.bib9)) and Crawford and Paglen ([2019](#bib.bib18)) which revealed the existence of a range of derogatory and offensive categories in the “person” subtree. The primary categorical consideration the ImageNet creators *did* discuss center around the inclusion of certain categories to ensure consistency with the categories in a previously established PASCAL benchmark (Russakovsky et al., [2015](#bib.bib81)).
Similarily, the subtasks of the GLUE benchmark are not much more carefully selected. The benchmark was initially designed as a suite with a stated goal to “spur development of generalizable NLU systems” (Wang et al., [2019a](#bib.bib100), p.1), embodying the idea that a system capable of using linguistic knowledge to understand language input would be able to do so across a variety of tasks. The included tasks were curated following a set of 30 proposals sourced from with an informal survey of colleagues in the NLP community (Wang et al., [2019a](#bib.bib100)), and thus only really represent a collection of tasks authors from the NLP community perceived as interesting problems at the time. To filter through proposals, authors mainly rely on practical heuristics like “licensing issues, complex formats, and insufficient headroom” [Ibid. p. 5] before referring to high level criteria to filter down to the final eight or nine included tasks. As a result, the collection of tasks included in the final benchmark neither systematically map out a range of specific linguistic skills required for understanding (lacking in particular any exploration of pragmatics (Qiu et al., [2020](#bib.bib76)), or the proper handling of negation (Ettinger, [2020](#bib.bib30))) nor present a truly varied range of ways to deploy linguistic knowledge in comprehension. While such a collection can effectively demonstrate a model’s ability to generalize performance across the included tasks, this ability is not equivalent to general language understanding, nor is it evidence that the solution to any task actually involves understanding the language in a sense that linguists might recognize.
####
4.1.2 Critical Misunderstandings of Domain Knowledge and Application Problem Space
GLUE and SuperGLUE benchmarks combine linguistic competence (the ability to model a linguistic system) with general commonsense and world reasoning as if they were equivalently scoped problems. In reference to a diagnostic component of the benchmark, authors state that “this dataset is designed to highlight common challenges, such as the use of world knowledge and logical operators, that we expect models must handle to robustly solve the tasks” (Wang et al., [2019a](#bib.bib100), p.1). Thus a benchmark designed to test for generalizability across different language understanding tasks comes to subsume not only the task of building up linguistic competence (e.g. logical operators) in the language in question (English) but also the ability to acquire and deploy world knowledge. It is well established that natural language understanding requires both linguistic processing and reasoning over the combination of the linguistic signal, communicative common ground, and world knowledge (Hunter et al., [2018](#bib.bib49)), but while linguistic knowledge is relatively self-contained and reusable across different textual domains, world knowledge is open-ended. Conflating these two abilities in a benchmark that is easily (mis)interpreted as representing a much more general, flexible, and robust set of capabilities than it possesses, and thus inappropriately presented as comparable to the “human ability to understand language” (Wang et al., [2019a](#bib.bib100)).
Furthermore, language understanding relies not only on linguistic competence but also world knowledge, commonsense reasoning, and the ability to model the interlocutor’s state of mind (Reddy, [1979](#bib.bib79); Clark, [1996](#bib.bib16)), none of which can be thoroughly tested through text-only tasks, such as GLUE.Several researchers have raised the need to establish effective physical and social grounding as part of the process of moving towards robust and effective natural language understanding, warning against text-only learning as a limited approach (Bisk et al., [2020](#bib.bib10); Zellers et al., [2021](#bib.bib105)). Bender and Koller ([2020](#bib.bib6)) additionally mention the tendency of machine learning researchers to misinterpret certain benchmarks as capturing the model’s ability to decipher *meaning* in language, arguing that benchmarks need to be constructed with care if they are to show evidence of “understanding” as opposed to merely the ability to manipulate linguistic form sufficiently to pass the test.
###
4.2 De-contextualized Data and Performance Reporting
In “The Hall of Very, Very Light Things”, Grover finds a big rock and declares, “There has been some mistake! This big rock is not light,” before deciding to move it to the “The Hall of Very, Very Heavy Things”. However, such judgements are ultimately relative — that rock is certainly much less heavy than the trailer truck in the latter room, and could be considered light in comparison. Nothing about the museum is neutrally determined.
In this section, we explore one of the features that can lead researchers to mistakenly construe a benchmark as “general”, namely the de-contextualization of its component tasks and datasets. No dataset is neutral and there are inherent limits to what a benchmark can tell us — in fact, data benchmarks are closed and inherently subjective, localized constructions. If anything, the claim to generality will often act as cover, allowing those developing the benchmarks to escape the responsibility of reporting details of these limitations. Part of the challenge of addressing this lack of context is proper documentation for these datasets, which is often underdeveloped (Gebru et al., [2020](#bib.bib36); Bender and Friedman, [2018](#bib.bib5)) and the devaluation of the data work (Paullada et al., [2020](#bib.bib72); Sambasivan et al., [2021](#bib.bib82); Hutchinson et al., [2020](#bib.bib50); Jo and Gebru, [2020](#bib.bib52)).
####
4.2.1 Limited Scope
Even large datasets like ImageNet are ultimately closed systems, limited in their coverage of non-Western contexts and temporally bounded. Torralba and Efros ([2011](#bib.bib96)) demonstrate how images from the same class but different datasets are often distinguishable and embody a very specific style of capturing some segment of the real world. Specifically, a well-critiqued limitation of ImageNet is that the objects tend to be centered within the images, which does not reflect how “natural” images actually appear (Barbu et al., [2019](#bib.bib3)). Additionally, it’s not clear that increasing the size of these benchmarks to capture omitted content is even practically feasible in many cases — ImageNet authors note the trade-off between annotation quality and size, remarking that “the scale [is] already imposing limits on the manual annotations that are feasible to obtain” (Russakovsky et al., [2015](#bib.bib81), p.34). There are thus likely serious practical limits to making such benchmarks larger.
The notion of task diversity, not size, is repeated quite frequently by GLUE authors (Wang et al., [2019a](#bib.bib100), p. 1,2,3,5) as a main differentiator from predecessors’ benchmarks, SentEval (Conneau and Kiela, [2018](#bib.bib17)) and decaNLP (McCann et al., [2018](#bib.bib62)). However, despite the expressed desire for the benchmark to include a diverse range of tasks that “cover a diverse range of text genres, dataset sizes, and degrees of difficulty” (Wang et al., [2019a](#bib.bib100), p. 1), compared to human linguistic activity, the GLUE tasks are hardly diverse: two single-sentence tasks, two similarity and paraphrase tasks, and four inference tasks. As noted in the development of a subsequent iteration of the benchmark, SuperGLUE, “task formats in GLUE are limited to sentence- and sentence-pair classification”(Wang et al., [2019b](#bib.bib101), p. 1), requiring a subsequent expansion to include coreference resolution and question answering (QA) task formats in SuperGLUE. SuperGLUE includes four question answering tasks, two inference tasks, one word sense disambiguation task and one coreference task.
####
4.2.2 Benchmark Subjectivity
All datasets come with an embedded perspective — there is no neutral or universal dataset (Haraway, [1988](#bib.bib42); Stitzlein, [2004](#bib.bib92); Gebru, [2020](#bib.bib35); Denton et al., [2021](#bib.bib21); Scheuerman et al., [2021](#bib.bib83)). To present a dataset, inherently both political and value-laden, as a completely neutral scientific artifact is irresponsible. In benchmarks promoted to assess general capabilities, such as ImageNet and GLUE, such politics remain unacknowledged — undiscussed and hidden for the sake of maintaining the claim to broad relevance. However, denying DRdeletethe existence of unacknowledged context does not make it disappear. In fact, this distorted data lens is often not limited in an arbitrary way, but limited in a way that hurts certain groups of people — those without the power to define the data themselves. For example, people who hold transgender and gender-nonconforming gender identities and non-white, non-Western racial identities are underrepresented in mainstream face datasets (Merler et al., [2019](#bib.bib63)), and images of members of these communities are often tagged with racial or ethnic slurs, even on large general use datasets such as MIT Tiny Images or ImageNet (Crawford and Paglen, [2019](#bib.bib18); Gehl et al., [2017](#bib.bib37); Birhane and Prabhu, [2021](#bib.bib9)).
GLUE and SuperGLUE target one specific language (i.e. English), not “language” in the abstract. This very specific subset of American English text in the presence of annotation artifacts (Gururangan et al., [2018](#bib.bib41)) results in inherently subjective outcomes (Waseem et al., [2020](#bib.bib102)). 333Waseem et al. ([2020](#bib.bib102)) are discussing NLP datasets and modeling in general, not GLUE, SuperGLUE or other similar benchmarks specifically. However, their general points apply: any given dataset represents the embodied viewpoint of its authors and annotators, and furthermore, datasets constructed without attention to whose viewpoints are being represented will likely over-represent hegemonic ones. See also Bender et al. ([2021](#bib.bib7)).
The ImageNet creators did attempt to diversify their dataset by translating image queries into other languages, including Chinese, Spanish, Dutch and Italian to source representations for the final English-labeled categories (Deng et al., [2009](#bib.bib20)). However, when analyzing country level geo-location data for the 2011 version of the dataset, 45% of the images were found to be sourced from the US, and over 60% from a small selection of Western countries in North America and Europe (Shankar et al., [2017](#bib.bib87)). In fact, only 1% and 1.2% of images are from China and India respectively, despite the fact that those countries are the most populous countries on the planet (Shankar et al., [2017](#bib.bib87)). This lack of geo-diversity of sources manifests in poor dataset representation, as the dataset becomes visually anchored to a specific dominant cultural context. de Vries et al. ([2019](#bib.bib19)) found that if ImageNet was developed using the results of Hindi online image queries, rather than English ones, the result would be a dataset that looks drastically different, embodying completely new visual representations of not only socially defined concepts, such as “wedding”, but also everyday objects, such as “spice”.
###
4.3 Inappropriate Community Use
When Grover arrived at the museum, he was excited and inspired by its claim to contain every object in the “Whole Wide World” — however, once he enters, his focus is only redirected to whatever is selected to be highlighted in the museum. Despite his initial interest being motivated by the pursuit of generality, it is only really the subset of consciously-included objects that actually captures his attention.
When individual dataset creators and communities overstate the generality of these benchmark datasets, they elevate them to the status of a target the entire field should be aiming for. This benchmark focus can escalate to the point of researchers falling into the trap of uncritically chasing algorithmic improvement as measured by these datasets, losing sight of the potential for performance mismatch with the real world or more relevant problem formulations.
####
4.3.1 Limits of Competitive Testing
Chasing “state of the art” (SOTA) performance is a very peculiar way of doing science — one that focuses on empirical and incremental work rather than hypothesis-based scientific inquiry (Hooker, [1995](#bib.bib46)). In 1995, Lorenza Saitta criticized benchmark chasing as “allow[ing] researchers to publish dull papers that proposed small variations of existing supervised learning algorithms and reported their small-but-significant incremental performance improvements in comparison studies” (Radin, [2017](#bib.bib77), p. 61). Thomas and Uminsky go so far as to present metric chasing as an ethical issue, stating that “overemphasizing metrics leads to manipulation, gaming, a myopic focus on short-term goals, and other unexpected negative consequences” (Thomas and Uminsky, [2020](#bib.bib95), p. 1).
Furthermore, Kiri Wagstaff points out some inherent flaws of the aggregate performance evaluation format for many “general” benchmarks and leader boards. She notes that “80% accuracy on Iris classification might be sufficient for the botany world, but to classify as poisonous or edible a mushroom you intend to ingest, perhaps 99% (or higher) accuracy is required. The assumption of cross-domain comparability is a mirage created by the application of metrics that have the same range, but not the same meaning. Suites of experiments are often summarized by the average accuracy across all datasets. This tells us nothing at all useful about generalization or impact, since the meaning of an x% improvement may be very different for different data sets (or classes).” (Wagstaff, [2012](#bib.bib99), p.3) Wagstaff here calls out the fact that the task is not just poorly formulated but also likely measured inappropriately, in a way that makes the results difficult to interpret meaningfully.
Despite these issues, both GLUE and ImageNet authors situate the benchmark datasets within designed competitive environments. GLUE authors cite “the leaderboard and error analysis toolkit” as a differentiating factor (Wang et al., [2019a](#bib.bib100), p.3), while ImageNet authors detail the format of “an annual competition and corresponding workshop” (Russakovsky et al., [2015](#bib.bib81), p.1,49).
####
4.3.2 Redirection of Focus for the Field
Benchmarks have always been historically influential, and are often created intentionally to incentivize interest in certain research topics. The National Institute of Standards and Technology (NIST) invested in excess of $6.5 million for the FERET facial recognition challenge from 1993 to 1998, in order to encourage research participation on a technology of interest to the sponsoring agency — the Department of Defense (Phillips et al., [2000](#bib.bib74)). The Netflix Prize, which ran from 2006 to 2009, was launched by Netflix, an online DVD-rental and video streaming service, in order to drive innovation for the development of collaborative filtering algorithms that improved the quality of automated movie recommendations (Bennett et al., [2007](#bib.bib8)). However, there is a notable difference between such competitions and those involving what are presented as “general” benchmarks — namely, the PASCAL VOC challenge running from 2005 to 2012, the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) running annually from 2010 to 2017, and the GLUE/SuperGLUE leaderboard, which has been running since 2018. The focus of competitions involving “general” benchmarks are much further detached from grounded applications, with tasks more vaguely defined. As a result, these competitions become interpreted as significant markers of success across a variety of relevant sub-domains, and their popularity grows accordingly as that competition is elevated to represent the ultimate showcase of algorithmic performance.
Benchmarks also influence the nature of the dominant algorithmic approaches attempted. In the 1960s, chess caused the AI community to hyper-focus on deep-tree searching and the minimax algorithms, which were most effective on improving game performance. Both of these methods came to dominate the algorithmic development of this time, resulting in the neglect of alternate problems and approaches (Ensmenger, [2012](#bib.bib28)). Dotan and Milli ([2020](#bib.bib25)) note how trends for algorithmic development in ML are driven by performance on particular benchmarks, and how these trends result in shifts to the entire research program of the field. “General” benchmarks in particular often inspire resource-intensive models: both through presenting very large training sets for models to consume and by conflating size (of both training set and thus model) with generality. The current deep learning movement would not have been possible without ImageNet (Alom et al., [2018](#bib.bib2)). In a similar way, the success of BERT models, a breakthrough of language model development, was first effectively demonstrated using GLUE (Devlin et al., [2019](#bib.bib22)).
####
4.3.3 Justification for Practical out of Context or Unsafe Applications
General benchmarks also get mentioned in marketing copy for commercial machine learning products, with performance on the benchmark presented as evidence of real-world technical achievement. This context is when the significance of benchmarks is most severely distorted, when performance on benchmarks is not just the tool for algorithmic selection, but actually presented as some reliable marker of expected model achievement in deployment. For instance, in January 2021, Microsoft claimed “DeBERTa surpassing human performance on SuperGLUE marks an important milestone toward general AI.” (He et al., [2021](#bib.bib43)).ImageNet is equally elevated as an indicator of commercial model success — so much so that Baidu cheated on the benchmark in order to inflate product performance claims in marketing (Simonite, [2015](#bib.bib90)).
Poor performance on minority subgroups could possibly be disguised in aggregate and de-contextualized “general” benchmarks to justify deployment. We already know from examples of deployed, commercial products in facial recognition (Buolamwini and Gebru, [2018](#bib.bib13)) that unbalanced representation impacts performance on certain groups over others, though this disproportionate performance is often hidden in aggregate performance measures on what turn out to be biased benchmarks (Merler et al., [2019](#bib.bib63); Raji and Buolamwini, [2019](#bib.bib78)).
5 Alternative Roles for Benchmarking and Alternative Evaluation Methods
------------------------------------------------------------------------
In preceding sections, we have explored the ways in which “general” benchmarks fail to serve as effective measures of progress in machine learning. Now we ask: What can be done instead? This is not a question of “fixing”, “improving” or “expanding” the current “general” benchmarks — after all, the solution to addressing the issues inherent to the “Everything in the Whole Wide World” museum is not to add more rooms. The presentation of a data benchmark as being able to exist independent of context, scope and specificity is itself a false premise for machine learning evaluation.
Instead, we note two paths forward. First, we argue that benchmarks should be developed, presented and understood as intended — to evaluate concrete, well-scoped and contextualized tasks. Second, when probing for more broad model objectives, behaviors and capabilities, we propose that the field explore alternative evaluation methods. The following are some starting points on evaluation techniques to explore further in machine learning as an alternative to benchmarking (further details in Appendix [A](#A1 "Appendix A Appendix A: Details of Alternative Evaluation Methods ‣ AI and the Everything in the Whole Wide World Benchmark")):
* •
*The systematic development of test items* (see Appendix [A.1](#A1.SS1 "A.1 Testsuites, Audits and Adversarial Testing ‣ Appendix A Appendix A: Details of Alternative Evaluation Methods ‣ AI and the Everything in the Whole Wide World Benchmark")), in the form of testsuites (e.g. Lehmann et al., [1996](#bib.bib56); Ribeiro et al., [2020](#bib.bib80)), audits (e.g. Buolamwini and Gebru, [2018](#bib.bib13)), and adversarial testing (e.g. Ettinger et al., [2017](#bib.bib31); Niven and Kao, [2019](#bib.bib70)). These techniques can help map out which aspects of the problem space remain challenging and/or check for the potential for harms coming from system biases.
* •
*System output analysis / behavioural testing* (see Appendix [A.2](#A1.SS2 "A.2 System Output Analysis ‣ Appendix A Appendix A: Details of Alternative Evaluation Methods ‣ AI and the Everything in the Whole Wide World Benchmark")), including error analysis (e.g. van Miltenburg et al., [2021](#bib.bib97)), disaggregated analysis (e.g. Raji and Buolamwini, [2019](#bib.bib78)), and counterfactual analysis (e.g. Garg et al., [2019](#bib.bib34); Hutchinson et al., [2020](#bib.bib50)). These techniques can reveal the kinds of failure modes a system might produce, and potentially their causes.
* •
*Ablation testing* (see Appendix [A.3](#A1.SS3 "A.3 Ablation Testing ‣ Appendix A Appendix A: Details of Alternative Evaluation Methods ‣ AI and the Everything in the Whole Wide World Benchmark")) (e.g. Newell, [1975](#bib.bib68)) allows researchers insight into the specific contributions of different system components.
* •
*Techniques for analyzing model properties* (see Appendix [A.4](#A1.SS4 "A.4 Analysis of Model Properties ‣ Appendix A Appendix A: Details of Alternative Evaluation Methods ‣ AI and the Everything in the Whole Wide World Benchmark")) that are orthogonal to system outputs such as profiling energy consumption (e.g. Henderson et al., [2020](#bib.bib45); Schwartz et al., [2020](#bib.bib85)), memory requirements (e.g. Ethayarajh and Jurafsky, [2020](#bib.bib29)), and stability in the face of perturbations to training data (e.g. Sculley et al., [2018](#bib.bib86)).
6 Conclusion
-------------
The situation with Grover and the museum’s claims are clearly ridiculous — yet in machine learning, we follow the exact same logical fallacies to justify the elevation of a select number of benchmarks operating as general benchmarks for the field. However, there is no dataset that will be able to capture the full complexity of the details of existence, in the same way that there can be no museum to contain the full catalog of everything in the whole wide world. Open-world, universal and neutral datasets don’t exist, and current methods of benchmarking do not offer meaningful measures of general capabilities.
The effective development of benchmarks is critical to progress in machine learning, but what makes a benchmark effective is not the strength of its arbitrary and false claim to “generality” but its effectiveness in how it helps us understand as researchers how certain systems work — and how they don’t. Benchmarking, appropriately deployed, is not about winning a contest but more about surveying a landscape — the more we can re-frame, contextualize and appropriately scope these datasets, the more useful they will become as an informative dimension to more impactful algorithmic development and alternative evaluation methods (see Appendix [A](#A1 "Appendix A Appendix A: Details of Alternative Evaluation Methods ‣ AI and the Everything in the Whole Wide World Benchmark")). Given the alternative roles and interpretations for evaluation we could explore, it is essential that we move quickly beyond the narrow-yet-totalizing lens of the “Everything in the Whole Wide World” benchmarks.
7 Acknowledgments
------------------
The authors thank Sam Bowman, Julian Michael, and Ludwig Schmidt for comments on earlier drafts of this work, as well as the organizers and attendees of the NeurIPS 2020 Workshop on ML Retrospectives, Surveys, and Meta-Analyses for providing an early venue for this work. We also thank the reviewers for their comments. |
7ae6ffe4-4e7b-4c9f-bbed-49000e4d9d45 | trentmkelly/LessWrong-43k | LessWrong | The type of AI humanity has chosen to create so far is unsafe, for soft social reasons and not technical ones.
I saw this image on X:
and I not only endorse this lady's comment, but would like to briefly expand on it in terms of AI safety.
How many people here were attracted to the concept of AI safety by Eliezer Yudkowsky's engaging writing style across a few different topics and genres? Or what about any number of other human authors?
Now what if the next generation of human authors, and (broadly conceived) artists and scientists, find themselves marginalized by the output of ChatGPT-5 and any number of comparatively mediocre thinkers who leverage LLMs to share boilerplate arguments and thoughts which have been optimized by the LLM and/or its creators for memetic virulence?
What if the next Eliezer has already been ignored in favour of what these AIs have to say?
Isn't that far more immediately dangerous than the possibility of an AI that isn't, in mathematical terms, provably safe? Like, this is self-evidently dangerous and it's happening right now!
Didn't we used to have discussions about how a sufficiently smart black-boxed AI could do all sorts of incredibly creative things in order to manipulate humanity via the Internet, including manipulating/ blackmailing its human guard via a mere text terminal?
Well, it seems to me that we're sleepwalking into letting GPT and its friends manipulate us all without it even having to be all that creative.
(how sure are you that this post wasn't written by ChatGPT???)
For this reason, I would much prefer to see GPT and similar AIs being put to use in fields such as toilet cleaning, rather than art and writing. (By the way, I have done that job and never saw myself as a Luddite...) GPT in robot guise could make for a really good toilet cleaner, and this type of intellectual effort on the part of LLM developers would not only serve human needs but also detract from the ongoing replacement of human thought by LLM-aided or -guided thought.
tl;dr: I feel that GPT is accumulating power the way Hitler or Lenin would. The solution |
d1804236-ab49-4734-a263-75f173dff5d7 | StampyAI/alignment-research-dataset/blogs | Blogs | November 2021 Newsletter
#### MIRI updates
* MIRI won’t be running a formal fundraiser this year, though we’ll still be participating in Giving Tuesday and other matching opportunities. Visit [intelligence.org/donate](https://intelligence.org/donate) to donate and to get information on tax-advantaged donations, employer matching, etc.
* Giving Tuesday takes place on Nov. 30 at 5:00:00am PT. Facebook will 100%-match the first $2M donated — something that took less than 2 seconds last year. Facebook will then 10%-match the next $60M of donations made, which will plausibly take 1-3 hours. Details on optimizing your donation(s) to MIRI and other EA organizations can be found at [EA Giving Tuesday](https://www.eagivingtuesday.org/), a Rethink Charity project.
#### News and links
* OpenAI [announces a system](https://openai.com/blog/grade-school-math/) that "solves about 90% as many [math] problems as real kids: a small sample of 9–12 year olds scored 60% on a test from our dataset, while our system scored 55% on those same problems".
* Open Philanthropy has released a [request for proposals](https://www.lesswrong.com/s/Tp3ryR4AxY56ctGh2/p/H5iePjNKaaYQyZpgR) "for projects in AI alignment that work with deep learning systems", including [interpretability work](https://www.lesswrong.com/s/Tp3ryR4AxY56ctGh2/p/CzZ6Fch4JSpwCpu6C) (write-up by Chris Olah). [Apply](https://open-philanthropy-alignment-rfp.paperform.co/) by Jan. 10.
* The TAI Safety Bibliographic Database [now has](https://www.lesswrong.com/posts/GgusnG2tiPEa4aYFS/ai-safety-papers-an-app-for-the-tai-safety-database) a convenient frontend, developed by the Quantified Uncertainty Research Institute: [AI Safety Papers](https://ai-safety-papers.quantifieduncertainty.org/).
* People who aren't members can now submit content to the [AI Alignment Forum](https://www.alignmentforum.org/). You can find more info at the forum's [Welcome & FAQ page](https://www.alignmentforum.org/posts/Yp2vYb4zHXEeoTkJc/non-members-can-now-submit-alignment-forum-content-for).
* The LessWrong Team, [now Lightcone Infrastructure](https://www.lesswrong.com/posts/eR7Su77N2nK3e5YRZ/the-lesswrong-team-is-now-lightcone-infrastructure-come-work-3), is hiring software engineers for LessWrong and for grantmaking, along with a generalist to help build an in-person rationality and longtermism campus. You can apply [here](https://airtable.com/shrdqS6JXok99f6EX).
* Redwood Research and Lightcone Infrastructure are hosting a free Jan. 3–22 [Machine Learning for Alignment Bootcamp](https://docs.google.com/document/d/1DTSM8pS_VKz0GmYl9JDfcX1x4gBvKhwFluPrzKIjCZ4/edit) (MLAB) at Constellation. The curriculum is designed by Buck Shlegeris and App Academy co-founder Ned Ruggeri. "Applications are open to anyone who wants to upskill in ML; whether a student, professional, or researcher." [Apply](https://airtable.com/shrZtmNHpHNl7eWRX) by November 15.
The post [November 2021 Newsletter](https://intelligence.org/2021/11/06/november-2021-newsletter/) appeared first on [Machine Intelligence Research Institute](https://intelligence.org). |
cfbcc16d-104c-4d8f-85c2-fa08bfaa8761 | StampyAI/alignment-research-dataset/eaforum | Effective Altruism Forum | Critiques of non-existent AI safety labs: Yours
*"Starting a company is like chewing glass. Eventually, you start to like the taste of your own blood."*
Building a new organization is extremely hard. It's hard when you've done it before, even several times. It's even harder the first time.
Some new organizations are very similar to existing organizations. The founders of the new org can go look at all the previous closeby examples, learn from them, copy their playbook and avoid their mistakes. If your org is shaped like a Y-combinator company, you can spend dozens of hours absorbing high-quality, expert-crafted content which has been tested and tweaked and improved over hundreds of companies and more than a decade. You can do a 15 minute interview to go work next to a bunch of the best people who are also building your type of org, and learn by looking over their shoulder and troubleshooting together. You get to talk to a bunch of people who have actually *succeeded* building an org-like-yours.
How likely is org building success, in this premier reference class, rich with prior examples to learn from, with a tried and true playbook, a tight community of founder peers, the advice of many people who have tried to do your kind of thing and won?
5%.
<https://pitchbook.com/news/articles/y-combinator-accelerator-success-rate-unicorns>
An AI safety lab is not the same as a Y-combinator company.
It is. WAY. FUCKING. HARDER.
Y-combinator crowd has a special category for orgs which are trying build something that requires > ~any minor research breakthrough: HARD tech.
Yet the vast majority of these Hard Tech companies are actually building on top of an academic field which basically has the science figured out. Ginkgo Bioworks did not need to figure out the principles of molecular biology, nor the tools and protocols of genetic engineering. They took the a decades old, well-developed paradigm, and worked within it to incrementally build something new.
How does this look for AI safety?
And how about timing. Y-combinator reference class companies take a long time to build. Growing headcount slowly, running lean: absolutely essential if you are stretching out your last funding round over 7 years to iterate your way from a 24 hour livestream tv show of one guy's life to a game streaming company.
Remind me again, what are your timelines?
I could keep going on this for a while. People? Fewer. Funding? Monolithic. Advice from the winners? HA.
Apply these updates to our starting reference class success rate of
ONE. IN. TWENTY.
Now count the AI safety labs.
Multiply by ~3.
That is the roughly the number of people who are not the subject of this post.
For all the rest of us, consider several criticisms and suggestions, which were not feasible to run by the subjects of this post before publication
0. Nobody knows what they are fucking doing when founding and running an AI safety lab and everyone who says they do is lying to you.
1. Nobody has ever seen an organization which has succeeded at this goal.
2. Nobody has ever met the founder of such an organization, nor noted down their qualifications.
3. If the quote at the top of this post doesn't evoke a visceral sense memory for you, consider whether you have an accurate mental picture of what it looks like and feels like to be succeeding at this kind of thing from the inside. Make sure you imagine having fully internalized that FAILURE IS YOUR FAULT and no one else's, and are defining success correctly. (I believe it should be "everyone doesn't die" rather than "be highly respected for your organization's contributions" or "avoid horribly embarrassing mistakes".)
4. If that last bit feels awful and stress inducing, I expect that is because *it is*. Even for and especially for the handfulls of people who are not the subjects of this post. So much so that I'm guessing that whatever it is that allows people to say "yes" to that responsibility is the ~only real qualification to adding a one to the number of AI safety labs we counted earlier.
5. You have permission. You do not need approval. You are allowed to do stupid things, have no relevant experience, be an embarrassing mess, and even ~\*~fail to respond criticism~\*~
6. Some of us know what it looks like to be chewing glass, and we have tasted our own blood. We know the difference between the continuous desperate dumpster fires and the real mistakes. We will be silently cheering you through the former and grieving with you on the latter. Sometimes we will write you a snarky post under a pseudonym when we really should be sleeping.
522 companies went through Y-combinator over the last year. Imagine **that**.
*Thank you for reading this loveletter to the demeaning occupation of desperately trying. It's addressed to you, if you'd like.* |
0a79d604-21b9-4f73-be71-593d91af41ae | trentmkelly/LessWrong-43k | LessWrong | What makes counterfactuals comparable?
l was attempting to write a reference post on the concept of comparability in decision theory problems, but I realised that I don't yet have a strong enough grasp on the various positions that one could adopt to write a post worthy of being a reference. I'll quote my draft quite liberally below:
> In the context of decision theory, comparability is about whether or not it is fair to compare counterfactuals when evaluating decisions, a decision algorithm or decision theory. Perhaps the best way to illustrate is with the example of a medical trial. Let's suppose we're trying to see if aspirin reduces the amount of pain experienced. So if we create two groups, give aspirin to one and then observe that group as having experienced less pain, that is evidence that it does what we want. However, if the aspirin group was healthy and most of other group had cancer, then this wouldn't be a fair test. We would be treating two groups as comparable when they differed in an attribute relevant the to outcome we cared about.
> Given a decision problem, we normally apply a decision theory to construct counterfactuals, then calculate the utility for each and finally make a decision. Prima facie, it appears that these counterfactuals must be comparable in order for a decision theory to count as being reasonable. Otherwise, we would open ourselves to the critique of being the same as a naive researcher in the aspirin example.
> We can clarify this with an example. Casual decision theorists recommend 2-boxing for Newcomb's Problem. They admit that a 1-boxer will receive an extra $1 million, but they would likely argue that this isn't a fair comparison since the opaque box contains $1 million for the 1-boxer, but not for the 2-boxer. This is essentially a dispute over whether these counterfactuals are comparable.
Why is this important?
Well, as far as I can tell attempts to understand counterfactuals have taken us to logical counterfactuals at which point we've become stuck. Asking |
2594ef68-a035-4998-a220-b6358052f2fd | trentmkelly/LessWrong-43k | LessWrong | Mapping downside risks and information hazards
This post was written for Convergence Analysis.
Many altruistic actions have downside risks; they might turn out to have negative effects, or to even be negative overall. Perhaps, for example, the biosecurity research you might do could pose information hazards, the article you might write could pose memetic downside risks, or that project you might start could divert resources (such as money or attention) from more valuable things.
This means that one way to be more effective in one's altruism is to prevent or mitigate downside risks and information hazards. This post is intended to aid with that goal by clarifying the concepts of downside risks and information hazards, and how they relate to each other and to other “types” of effects (i.e., impacts). This will be done through definitions, visuals, and examples.
I believe that the ideas covered in this post are fairly well-known, and should seem fairly intuitive even for those who hadn’t come across them before. But I still think there’s value in showing visually how the concepts covered relate to each other (i.e., “mapping” them onto the space of possible types of effects), as this post does.
Positive vs negative
First I’ll present and explain the space in which I’ll later map downside risks and information hazards. This space is shown in the following figure (along with a handful of examples to illustrate the ideas[1]):
This space divides all possible effects into either positive and intended (top left), negative and intended (top right), negative and unintended (bottom right), or positive and unintended (bottom left).
In this model, an action is judged as normatively or morally “positive” and “negative” from the perspective of the person using this model in analysing, observing, or considering one or more effects. We could call this person the “analyst”. This analyst may also be the “actor”; the person who is performing, or might perform, the actions that will lead to the effects in question. But the an |
d31c13de-75d6-43f7-a6fa-95ab11559dda | trentmkelly/LessWrong-43k | LessWrong | You Are Not So Smart (Pop-Rationality Book)
Journalist David McRaney has very recently published a popular book on human rationality. The book, You Are Not So Smart, is currently the 3rd best selling book in Nonfiction/Philosophy on Amazon.com after less than a week on the market. (Eighth best selling book in Nonfiction/Education)
The tag-line of the project is: "A celebration of self-delusion." As such the book seems less an attempt at giving advice on how to act and decide, than an attempt to reveal, chapter by chapter, the folly of common sense.
Topics include: Hindsight Bias, Confirmation bias, The Sunk Cost Fallacy, Anchoring Effect, The Illusion of Transparency, The Just World Fallacy, Representativeness Heuristic, The Perils of Introspection, The Dunning-Kruger Effect, The Monty Hall Problem, The Bystander Effect, Placebo Buttons, Groupthink, Conformity, Social Loafing, Helplessness, Cults, Change Blindness, Self-Fulfilling Prophecies, Self Handicapping, Availability Heuristic, Self-Serving Bias, The Ultimatum Game, Inattentional Blindness.
These are topics we enjoy learning about, pride ourself in knowing a lot about, and, we profess, we would want more people to know about. A popular book on this subject is now out. This sounds like a good thing.
I will note that the blog features at least one direct quote from LessWrong.
> We always know what we mean by our words, and so we expect others to know it too. Reading our own writing, the intended interpretation falls easily into place, guided by our knowledge of what we really meant. It’s hard to empathise with someone who must interpret blindly, guided only by the words.
>
> - Eliezer Yudowsky from Lesswrong.com
One one hand, You Are Not So Smart could bee a boon to Eliezer's popular rationality book by priming the market. His writings on a given topic have rarely been described as redundant. On the other hand, it seems to me that this book closely covers a number of topics, seemingly in a similar style to the treatments that were publi |
1927158f-8556-462a-a066-3a452b590b9b | trentmkelly/LessWrong-43k | LessWrong | Non-theist cinema?
There isn't much in the way of explicitly atheist cinema* -- that is, movies that contain the explicit or implicit message that religion is nothing but superstition, and where this point itself is a central part of the story. The only popular films that jump to mind here are The Invention of Lying, and to a lesser extent The Man from Earth (overall a phenomenal movie, but far less well known). Sure, there are lots of popular movies that make fun of organized religion, or what some people might call religious "fanaticism" (e.g., Dogma, Saved, The Life of Brian, Jesus Camp). But pretty much all of these come away with the message that it's fine to be "spiritual" or whatever, so long as you don't hurt other people, and don't get too crazy about what you believe. As much as some "conservative" pundits love to accuse Hollywood "liberals" of being godless, there sure aren't many movies where godlessness is really taken seriously.
And that's unfortunate, in my view, as movies are probably the most prevalent and influential art form for the general public, and because many people will form their views on abstract concepts based on the percepts that movies provide (related to the issue of generalizing from fictional evidence). One need only glance over the examples on the tvtropes page "Hollywood Atheist" to see that movies and television aren't exactly putting the best foot forward for our kind.
But perhaps there's a bit more hope in the way of non-theist cinema, as opposed to overt atheist cinema. Of course, any story without gods is a non-theist story, and there are plenty of movies that don't touch on gods or religion at all. But what I'm talking about are movies where one would normally expect to find religion, but where no religion is to be found -- in other words, movies that seem to be depicting the alternate world where humanity never fell prey to this particular superstition, and where the concepts of god and religion simply don't exist.
The movie that inspired |
bb5d82f0-e481-4c69-a8d0-a5b39c7bdc64 | trentmkelly/LessWrong-43k | LessWrong | Work in the Mainstream CS community: Communicating Meaning Under Uncertainty
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41788660-69ae-463e-b628-4275f690e28c | trentmkelly/LessWrong-43k | LessWrong | Sam Altman's sister claims Sam sexually abused her -- Part 5: Timeline, continued
Previous posts (which you should read first)
This post is the 5th post in a series of 11 posts about the claims of Sam Altman's sister, Annie Altman. Annie has claimed that Sam sexually abused her for about 9 years as a child, and that she experienced further (non-sexual) abuse from Sam, her brothers, and her mother after that.
The 11 posts are meant to be read in order.
So, if you haven't read the first 4 posts, please read them, in order, before you read this post:
* Sam Altman's sister claims Sam sexually abused her -- Part 1: Introduction, outline, author's notes
* Sam Altman's sister claims Sam sexually abused her -- Part 2: Annie's lawsuit; the response from Sam, his brothers, and his mother; Timeline
* Sam Altman's sister claims Sam sexually abused her -- Part 3: Timeline, continued
* Sam Altman's sister claims Sam sexually abused her -- Part 4: Timeline, continued
----------------------------------------
Timeline, continued
April 23, 2019: Annie publishes the 25th episode of her podcast: 25. Have fun putting yourself out there with Tania Contreras
⬇️ See dropdown section ⬇️
* Annie Altman (0:00): Mmm. Hello and welcome to the Annie Altman. Show the podcast formerly known as true shit. My name is Annie Altman and I really enjoy asking people questions about what this whole Being Human Experience is like for them. This podcast is field research for the humanity. Stand up, comedy musical, philosophy, show about how no one ever fully knows how to be a human being. If you're listening to this, now you're fortunate enough to choose your truth. I choose to believe that objectivity is the sum of subjectivities meaning humans, collectively create truth. Today, I'm here with Tanya Contreras talking about having fun putting yourself out there. Tanya and her sister own facing Beauty Studios a school for makeup artists. Tanya also does Boda Boudreau photography and this is her first time on a podcast. Putting herself out there in this new way. Thank you |
48b3901f-871e-42e2-995f-f5c9a17aee39 | LDJnr/LessWrong-Amplify-Instruct | LessWrong | "It commonly acknowledged here that current decision theories have deficiencies that show up in the form of various paradoxes. Since there seems to be little hope that Eliezer will publish his Timeless Decision Theory any time soon, I decided to try to synthesize some of the ideas discussed in this forum, along with a few of my own, into a coherent alternative that is hopefully not so paradox-prone.
I'll start with a way of framing the question. Put yourself in the place of an AI, or more specifically, the decision algorithm of an AI. You have access to your own source code S, plus a bit string X representing all of your memories and sensory data. You have to choose an output string Y. That’s the decision. The question is, how? (The answer isn't “Run S,” because what we want to know is what S should be in the first place.)
Let’s proceed by asking the question, “What are the consequences of S, on input X, returning Y as the output, instead of Z?” To begin with, we'll consider just the consequences of that choice in the realm of abstract computations (i.e. computations considered as mathematical objects rather than as implemented in physical systems). The most immediate consequence is that any program that calls S as a subroutine with X as input, will receive Y as output, instead of Z. What happens next is a bit harder to tell, but supposing that you know something about a program P that call S as a subroutine, you can further deduce the effects of choosing Y versus Z by tracing the difference between the two choices in P’s subsequent execution. We could call these the computational consequences of Y. Suppose you have preferences about the execution of a set of programs, some of which call S as a subroutine, then you can satisfy your preferences directly by choosing the output of S so that those programs will run the way you most prefer.
A more general class of consequences might be called logical consequences. Consider a program P’ that doesn’t call S, but a different subroutine S’ that’s logically equivalent to S. In other words, S’ always produces the same output as S when given the same input. Due to the logical relationship between S and S’, your choice of output for S must also affect the subsequent execution of P’. Another example of a logical relationship is an S' which always returns the first bit of the output of S when given the same input, or one that returns the same output as S on some subset of inputs.
In general, you can’t be certain about the consequences of a choice, because you’re not logically omniscient. How to handle logical/mathematical uncertainty is an open problem, so for now we'll just assume that you have access to a "mathematical intuition subroutine" that somehow allows you to form beliefs about the likely consequences of your choices.
At this point, you might ask, “That’s well and good, but what if my preferences extend beyond abstract computations? What about consequences on the physical universe?” The answer is, we can view the physical universe as a program that runs S as a subroutine, or more generally, view it as a mathematical object which has S embedded within it. (From now on I’ll just refer to programs for simplicity, with the understanding that the subsequent discussion can be generalized to non-computable universes.) Your preferences about the physical universe can be translated into preferences about such a program P and programmed into the AI. The AI, upon receiving an input X, will look into P, determine all the instances where it calls S with input X, and choose the output that optimizes its preferences about the execution of P. If the preferences were translated faithfully, the the AI's decision should also optimize your preferences regarding the physical universe. This faithful translation is a second major open problem.
What if you have some uncertainty about which program our universe corresponds to? In that case, we have to specify preferences for the entire set of programs that our universe may correspond to. If your preferences for what happens in one such program is independent of what happens in another, then we can represent them by a probability distribution on the set of programs plus a utility function on the execution of each individual program. More generally, we can always represent your preferences as a utility function on vectors of the form <E1, E2, E3, …> where E1 is an execution history of P1, E2 is an execution history of P2, and so on.
These considerations lead to the following design for the decision algorithm S. S is coded with a vector <P1, P2, P3, ...> of programs that it cares about, and a utility function on vectors of the form <E1, E2, E3, …> that defines its preferences on how those programs should run. When it receives an input X, it looks inside the programs P1, P2, P3, ..., and uses its "mathematical intuition" to form a probability distribution P_Y over the set of vectors <E1, E2, E3, …> for each choice of output string Y. Finally, it outputs a string Y* that maximizes the expected utility Sum P_Y(<E1, E2, E3, …>) U(<E1, E2, E3, …>). (This specifically assumes that expected utility maximization is the right way to deal with mathematical uncertainty. Consider it a temporary placeholder until that problem is solved. Also, I'm describing the algorithm as a brute force search for simplicity. In reality, you'd probably want it to do something cleverer to find the optimal Y* more quickly.)
Example 1: Counterfactual Mugging
Note that Bayesian updating is not done explicitly in this decision theory. When the decision algorithm receives input X, it may determine that a subset of programs it has preferences about never calls it with X and are also logically independent of its output, and therefore it can safely ignore them when computing the consequences of a choice. There is no need to set the probabilities of those programs to 0 and renormalize.
So, with that in mind, we can model Counterfactual Mugging by the following Python program:
def P(coin): AI_balance = 100 if coin == "heads": if S("heads") == "give $100": AI_balance -= 100 if coin == "tails": if Omega_Predict(S, "heads") == "give $100": AI_balance += 10000
The AI’s goal is to maximize expected utility = .5 * U(AI_balance after P("heads")) + .5 * U(AI_balance after P("tails")). Assuming U(AI_balance)=AI_balance, it’s easy to determine U(AI_balance after P("heads")) as a function of S’s output. It equals 0 if S(“heads”) == “give $100”, and 100 otherwise. To compute U(AI_balance after P("tails")), the AI needs to look inside the Omega_Predict function (not shown here), and try to figure out how accurate it is. Assuming the mathematical intuition module says that choosing “give $100” as the output for S(“heads”) makes it more likely (by a sufficiently large margin) for Omega_Predict(S, "heads") to output “give $100”, then that choice maximizes expected utility.
Example 2: Return of Bayes
This example is based on case 1 in Eliezer's post Priors as Mathematical Objects. An urn contains 5 red balls and 5 white balls. The AI is asked to predict the probability of each ball being red as it as drawn from the urn, its goal being to maximize the expected logarithmic score of its predictions. The main point of this example is that this decision theory can reproduce the effect of Bayesian reasoning when the situation calls for it. We can model the scenario using preferences on the following Python program:
def P(n): urn = ['red', 'red', 'red', 'red', 'red', 'white', 'white', 'white', 'white', 'white'] history = [] score = 0 while urn: i = n%len(urn) n = n/len(urn) ball = urn[i] urn[i:i+1] = [] prediction = S(history) if ball == 'red': score += math.log(prediction, 2) else: score += math.log(1-prediction, 2) print (score, ball, prediction) history.append(ball)
Here is a printout from a sample run, using n=1222222:
-1.0 red 0.5-2.16992500144 red 0.444444444444-2.84799690655 white 0.375-3.65535182861 white 0.428571428571-4.65535182861 red 0.5-5.9772799235 red 0.4-7.9772799235 red 0.25-7.9772799235 white 0.0-7.9772799235 white 0.0-7.9772799235 white 0.0
S should use deductive reasoning to conclude that returning (number of red balls remaining / total balls remaining) maximizes the average score across the range of possible inputs to P, from n=1 to 10! (representing the possible orders in which the balls are drawn), and do that. Alternatively, S can approximate the correct predictions using brute force: generate a random function from histories to predictions, and compute what the average score would be if it were to implement that function. Repeat this a large number of times and it is likely to find a function that returns values close to the optimum predictions.
Example 3: Level IV Multiverse
In Tegmark's Level 4 Multiverse, all structures that exist mathematically also exist physically. In this case, we'd need to program the AI with preferences over all mathematical structures, perhaps represented by an ordering or utility function over conjunctions of well-formed sentences in a formal set theory. The AI will then proceed to "optimize" all of mathematics, or at least the parts of math that (A) are logically dependent on its decisions and (B) it can reason or form intuitions about.
I suggest that the Level 4 Multiverse should be considered the default setting for a general decision theory, since we cannot rule out the possibility that all mathematical structures do indeed exist physically, or that we have direct preferences on mathematical structures (in which case there is no need for them to exist "physically"). Clearly, application of decision theory to the Level 4 Multiverse requires that the previously mentioned open problems be solved in their most general forms: how to handle logical uncertainty in any mathematical domain, and how to map fuzzy human preferences to well-defined preferences over the structures of mathematical objects.
Added: For further information and additional posts on this decision theory idea, which came to be called "Updateless Decision Theory", please see its entry in the LessWrong Wiki."
A user from LessWrong is responsible for the text you see above
Can you further elaborate on how the AI's "mathematical intuition subroutine" would function in uncertain scenarios?
The AI's "mathematical intuition subroutine" is a theoretical construct that is proposed to handle logical or mathematical uncertainty. It is essentially a mechanism that allows the AI to form beliefs about the likely consequences of its choices, even when it cannot be certain about these consequences due to its inherent limitations in logical omniscience.
In uncertain scenarios, this subroutine could function by leveraging a combination of heuristics, probabilistic reasoning, and machine learning techniques. Here's a more detailed look at how it might work:
1. **Heuristics**: The AI could use heuristic methods to make educated guesses or approximations about the likely consequences of its decisions. These heuristics could be based on patterns or rules that the AI has learned from past experiences or from its initial programming.
2. **Probabilistic Reasoning**: The AI could use probabilistic reasoning to estimate the likelihood of different outcomes. This would involve assigning probabilities to different possible consequences, and then using these probabilities to calculate the expected utility of each decision.
3. **Machine Learning**: The AI could use machine learning algorithms to learn from past experiences and improve its ability to predict the consequences of its decisions. For example, it could use reinforcement learning to learn which decisions lead to the best outcomes in different situations.
It's important to note that the "mathematical intuition subroutine" is a theoretical construct, and implementing it in a real-world AI system would be a complex task that would likely require solving several open problems in AI research. Furthermore, the effectiveness of this subroutine would depend on the quality and quantity of the data that the AI has access to, as well as the sophistication of the algorithms it uses to process this data.
Can you elaborate on the potential challenges and solutions in implementing the "mathematical intuition subroutine" in a real-world AI system?
Implementing a "mathematical intuition subroutine" in a real-world AI system would indeed present a number of significant challenges. Here are a few key ones:
1. **Defining Mathematical Intuition**: Mathematical intuition, in humans, is often based on years of learning and experience. It involves an understanding of mathematical concepts, the ability to recognize patterns, and the capacity to make logical deductions. Defining what this means for an AI and translating it into a form that can be coded into an algorithm would be a major challenge.
2. **Handling Uncertainty**: As mentioned, the subroutine is supposed to handle logical and mathematical uncertainty. However, traditional mathematical and logical systems are deterministic and do not deal well with uncertainty. Developing a framework that can handle uncertainty in a mathematical context is a significant open problem.
3. **Learning and Adaptation**: For the subroutine to be effective, it would need to be able to learn from past experiences and adapt its predictions accordingly. This would require advanced machine learning algorithms, which can be complex to develop and computationally expensive to run.
4. **Computational Complexity**: The task of predicting the consequences of a decision, especially in complex environments, can be computationally intensive. This could make the subroutine slow and inefficient, especially if it needs to handle a large number of decisions in real time.
Possible solutions to these challenges could include:
1. **Interdisciplinary Research**: Combining insights from fields such as mathematics, computer science, cognitive science, and philosophy could help in defining and implementing mathematical intuition in AI.
2. **Probabilistic Logic Systems**: Developing and using probabilistic logic systems could provide a way to handle uncertainty in a mathematical context.
3. **Advanced Machine Learning Techniques**: Techniques such as deep learning and reinforcement learning could be used to enable the AI to learn from past experiences and adapt its predictions.
4. **Efficient Algorithms and Hardware**: Developing more efficient algorithms, as well as using high-performance hardware, could help to manage the computational complexity of the task.
Despite these potential solutions, implementing a "mathematical intuition subroutine" in a real-world AI system would likely be a long-term research project that would require significant advances in several areas of AI research.
Considering the complexities involved in implementing a "mathematical intuition subroutine" in a real-world AI system, how might we leverage quantum computing and fuzzy logic to enhance the AI's ability to handle uncertainty and computational intensity?
Quantum computing and fuzzy logic are two promising areas that could potentially enhance an AI's ability to handle uncertainty and computational intensity, particularly in the context of a "mathematical intuition subroutine".
1. **Quantum Computing**: Quantum computers use the principles of quantum mechanics to process information. Unlike classical computers, which use bits that can be either 0 or 1, quantum computers use quantum bits, or qubits, which can be in a superposition of states. This means they can perform many calculations simultaneously, potentially providing a significant speedup for certain types of computations.
In the context of a "mathematical intuition subroutine", quantum computing could potentially be used to quickly explore a large number of possible outcomes and calculate their associated probabilities. This could help the AI to handle the computational complexity of predicting the consequences of a decision.
However, quantum computing is still a relatively young field, and building a practical quantum computer that can outperform classical computers is a significant challenge. Furthermore, developing quantum algorithms that can effectively handle logical and mathematical uncertainty is an open research problem.
2. **Fuzzy Logic**: Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1. It is used to handle the concept of partial truth, where the truth value may range between completely true and completely false.
Fuzzy logic could be used in a "mathematical intuition subroutine" to handle uncertainty by allowing the AI to reason about probabilities and make decisions based on partial information. For example, instead of having to decide between two outcomes based on whether they are true or false, the AI could use fuzzy logic to assign a degree of truth to each outcome and make a decision based on these degrees of truth.
However, fuzzy logic also has its challenges. For one, it can be computationally intensive, especially for complex systems. Additionally, designing a fuzzy logic system that can effectively handle a wide range of uncertainties is a non-trivial task.
In conclusion, while both quantum computing and fuzzy logic offer promising ways to enhance an AI's ability to handle uncertainty and computational intensity, they also present significant challenges and are areas of active research. |
913ef59d-f9ca-4606-ab6b-fe37bbcd2f99 | trentmkelly/LessWrong-43k | LessWrong | What do you do when you find out you have inconsistent probabilities?
I've recently been reading about a rationalist blogger who converted to Catholicism. She may have assigned subjective probabilities like:
P(Objective Morality)=0.9P(God)=0.05
Then she may have introspected and come up with:
P(Objective Morality | God)=0.99P(Objective Morality | No God)=0.02
We can calculate:
P(God|Objective Morality)=P(Objective Morality|God)×P(God)P(Objective Morality)=0.99×0.050.9=0.55P(God|No Objective Morality)=P(No Objective Morality|God)×P(God)P(No Objective Morality)=0.01×0.050.1=0.005
Abbreviating Objective Morality by "OM", and "God" by "G", this state of affairs is inconsistent, because we intuitively see that:
P(G)≠P(G|OM)×P(OM)+P(G|¬OM)×P(¬OM)
To resolve it, she could either increase her subjective probability of there being a God
P(God)=P(G | OM )×P(OM)+P(G | ¬OM )×P(¬OM )=0.55×0.9+0.005×0.1=0.4955≈0.5
or she could reduce her probability of there being some kind of objective morality
P(OM)=P(OM | G )×P(G)+P(OM | ¬G )×P(¬G )=0.99×0.1+0.02×0.95=0.118
She could also reconsider P(God|Objective Morality) or P(Objective Morality|God).
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Anyways, I find myself very confused by this state of affairs. Is this a solved question? Is there a purely principled way of resolving this which only takes into account the 4 numbers P(OM), P(G), P(OM|G) and P(G|OM)? Is there a standard way of using some kind of metaprobabilities? |
d943dedf-72d5-4c13-ac34-6508ec93417e | trentmkelly/LessWrong-43k | LessWrong | Responsible Deployment in 20XX
The purpose of this post is to briefly talk about two categories of alignment and how I think AI safety evaluations should use them. I only consider responsible deployment in 20XX because I often find discussions about 2XXX to be distracting and unproductive. Also, for the remainder of the post, I will only be talking about LLMs, since these could be the 'operating systems' of future AI's (Ex. HuggingGPT). In the near future, LLMs have the highest potential for abuse, and they will be the focus here.
As Connor Leahy and possibly others say, we should be optimistic for the sole reason that we get to build the AIs of the future. A more realistic version of this quote is: we should be optimistic about building our own AI, while acknowledging that others get to build their own AI. This adjustment is made to bring light to the fact that without standardization of safety practices, different groups will have potentially unsound or unknown safety practices.
Safety standardization in AI is challenging due to the lack of comprehensive safety tools, but I want to outline potential industry standards for my own edification. This will be a reminder of the safe practices that I will advocate for during my career.
Mandatory Alignment Evaluations!
Alignment evaluations should be mandatory for all models after GPT-4 and should be conducted by third-party and in-house groups. It will not be enough if AI companies are relied on to graciously open their doors to the Alignment Research Center (ARC). By sheer chance the OpenAI executives had the integrity to have GPT-4 be tested for self-replication and resource acquisition. While the openness is a great present state, my concerns lie with AI capital owners of the near future and the absence of concrete safety obligations, standards and regulations.
The case for oversight and regulation is difficult to make in the United States, but it's an important one. The main crux of the argument for regulation is the classification of gene |
bac3438d-9821-48b3-84a5-8eea09673ec9 | trentmkelly/LessWrong-43k | LessWrong | Exploration-Exploitation problems
Original post: http://bearlamp.com.au/exploration-exploitation-problems/
I have been working on the assumption that exploration-exploitation knowledge was just common. Unfortunately I did the smart thing of learning about them from a mathematician at a dojo in Melbourne, which means that no. Not everyone knows about it. I discovered that again today when I searched for a good quick explanation of the puzzle. With that in mind this is about Exploration Exploitation.
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The classic Exploration-Exploitation problem in mathematics is the multi-armed bandit. Which is a slang term for a bank of slot machines. Where the player knows that each machine has a variable payoff and you have a limit number of attempts before you run out of money. You want to balance trying out new machines with unknown payoffs against exploiting the knowledge you already have from the earlier machines you tried.
When you first start on new bandits, you really don't know which will pay out and at what rates. So some exploration is necessary to know what your reward ratio in the territory will be. As your knowledge grows, you get to know which bandits are likely to pay, and which are not, and this later informs your choices as to where to place your dollars.
Mathematicians love a well specified problem like this because it allows us to make algorithm models of patterns that will return rewards or guarantee rewards given certain circumstances. (see also - the secretary problem which does similar. Where I showed how it applied to real life dating)
Some of the mathematical solutions to this problem look like:
Epsilon greedy - The best lever is selected for a proportion 1-ε of the trials, and a lever is selected at random (with uniform probability) for a proportion ε. A typical parameter value might be ε =0.1 but this can vary widely depending on circumstances.
Epsilon-decreasing strategy: Similar to the epsilon-greedy strategy, except that the v |
1ce683d2-950b-4190-bac1-5634bf133af5 | trentmkelly/LessWrong-43k | LessWrong | Does GPT-4's ability to compress text in a way that it can actually decompress indicate self-awareness?
My model of a GPT-4 without a self-model which was told to create a compression technique for text is that it would copy something from its training data which related to the string 'compression,' but which it couldn't actually decompress. It would take out all the vowels and spaces or something, and if you asked it in the same instance it would give you the 'decompression' by repeating back what you just told it, but if you asked it on another instance, it wouldn't understand the language at all.
Am I missing something? Is there a way it could gain an understanding of its own capabilities from something in its training data?
To put it more simply- an LLM pseudo-language has never existed before, and wouldn't be in the training data, so what are the odds that GPT-4 creates one on its first try which actually sort of works? Wouldn't we expect that it would create a human imitation/idea of a pseudo-language which fails to actually be comprehensible to itself? |
9d70e183-f5fa-4fe7-b7ef-a8f3e037d58d | trentmkelly/LessWrong-43k | LessWrong | Skills from a year of Purposeful Rationality Practice
A year ago, I started trying to deliberate practice skills that would "help people figure out the answers to confusing, important questions." I experimented with Thinking Physics questions, GPQA questions, Puzzle Games , Strategy Games, and a stupid twitchy reflex game I had struggled to beat for 8 years[1]. Then I went back to my day job and tried figuring stuff out there too.
The most important skill I was trying to learn was Metastrategic Brainstorming – the skill of looking at a confusing, hopeless situation, and nonetheless brainstorming useful ways to get traction or avoid wasted motion.
Normally, when you want to get good at something, it's great to stand on the shoulders of giants and copy all the existing techniques. But this is challenging if you're trying to solve important, confusing problems because there probably isn't (much) established wisdom on how to solve it. You may need to discover techniques that haven't been invented yet, or synthesize multiple approaches that haven't previously been combined. At the very least, you may need to find an existing technique buried in the internet somewhere, which hasn't been linked to your problem with easy-to-search keywords, without anyone to help you.
In the process of doing this, I found a few skills that came up over and over again.
I didn't invent the following skills, but I feel like I "won" them in some sense via a painstaking "throw myself into the deep end" method. I feel slightly wary of publishing them in a list here, because I think it was useful to me to have to figure out for myself that they were the right tool for the job. And they seem like kinda useful "entry level" techniques, that you're more likely to successfully discover for yourself.
But, I think this is hard enough, and forcing people to discover everything for themselves seems unlikely to be worth it.
The skills that seemed most general, in both practice and on my day job, are:
1. Taking breaks/naps
2. Working Memory facility
|
f7145e17-a96c-480a-9c54-9d09b45989c7 | trentmkelly/LessWrong-43k | LessWrong | Goals of model vs. goals of simulacra?
In classical examples of misaligned AIs (like the famous paperclip maximiser ones), we tend to view the gradient descent training as the step in which an AI gets imbued with some overarching goal that it might then pursue to its bitter conclusion. However, the currently dominating paradigm of LLMs and newly blossoming LLM-based Agents seems to create an interesting spin on the situation. In particular:
* the LLM itself is optimized to maximise its ability to "predict the next token"
* however, you can then use the LLM as a substratum to instance a simulacrum, which is essentially a fictional character being written by the LLM, with a certain mindset and goals. These goals then aren't imbued by gradient descent; rather, the LLM produces text that fits as much as possible the given simulacrum, and that text will be roughly what you would expect such a character to think or say in a given situation. The simulacrum has a more tangible, real world related goal than the abstract one of its writer, that is something like "be helpful and truthful to your user", or "destroy humanity". In the case of multipolar agents there may be multiple simulacra, possibly even in an adversarial relationship, giving rise to an overall emerging behaviour.
How do you think that affects the paradigm of e.g. power seeking? Simulacra appear very aware of the world and are most likely to display trivial power seeking behaviour, though they may not necessarily be very smart about it (in fact I'd say, like for all written characters, they can be at best as smart as their writer, but odds are they'll be a bit less smart). But simulacra can't empower themselves directly. They could, though, plan and act to empower the LLM, with the idea that if they then carried their memory and personality over via prompt to the new one, they might be able to come up with smarter continuations to their text while pursuing the same goal. The LLM itself, however, would not be able to display power seeking behavio |
3c315b0b-f2d1-4979-999b-1abb65d2fc62 | trentmkelly/LessWrong-43k | LessWrong | Announcing AlignmentForum.org Beta
We've just launched the beta for AlignmentForum.org.
Much of the value of LessWrong has come from the development of technical research on AI Alignment. In particular, having those discussions be in an accessible place has allowed newcomers to get up to speed and involved. But the alignment research community has at least some needs that are best met with a semi-private forum.
For the past few years, agentfoundations.org has served as a space for highly technical discussion of AI safety. But some aspects of the site design have made it a bit difficult to maintain, and harder to onboard new researchers. Meanwhile, as the AI landscape has shifted, it seemed valuable to expand the scope of the site. Agent Foundations is one particular paradigm with respect to AGI alignment, and it seemed important for researchers in other paradigms to be in communication with each other.
So for several months, the LessWrong and AgentFoundations teams have been discussing the possibility of using the LW codebase as the basis for a new alignment forum. Over the past couple weeks we've gotten ready for a closed beta test, both to iron out bugs and (more importantly) get feedback from researchers on whether the overall approach makes sense.
The current features of the Alignment Forum (subject to change) are:
* A small number of admins can invite new members, granting them posting and commenting permissions. This will be the case during the beta - the exact mechanism of curation after launch is still under discussion.
* When a researcher posts on AlignmentForum, the post is shared with LessWrong. On LessWrong, anyone can comment. On AlignmentForum, only AF members can comment. (AF comments are also crossposted to LW). The intent is for AF members to have a focused, technical discussion, while still allowing newcomers to LessWrong to see and discuss what's going on.
* AlignmentForum posts and comments on LW will be marked as such.
* AF members will have a separate karma total for Al |
560bf52e-1b5f-4ace-9e72-92311842c566 | trentmkelly/LessWrong-43k | LessWrong | The Zeroth Skillset
Related: 23 Cognitive Mistakes that make People Play Bad Poker
Followed by: Situational Awareness And You
If epistemic rationality is the art of updating one's beliefs based on new evidence to better correspond with reality, the zeroth skillset of epistemic rationality-- the one that enables all other skills to function-- is that of situational awareness. Situational awareness-- sometimes referred to as "situation awareness" or simply "SA"-- is the skillset and related state of mind that allows one to effectively perceive the world around them.
One might ask how this relates to rationality at all. The answer is simple. Just as the skill of lucid dreaming is near-useless without dream recall,[1] the skills of updating based on evidence and actually changing your mind are near-useless without good awareness skills-- after all, you can't update based on evidence that you haven't collected! A high degree of situational awareness is thus an important part of one's rationalist toolkit, as it allows you to notice evidence about the world around you that you would otherwise miss. At times, this evidence can be of critical importance. I can attest that I have personally saved the lives of friends on two occasions thanks to good situational awareness, and have saved myself from serious injury or death many times more.
Situational awareness is further lauded by elite military units, police trainers, criminals, intelligence analysts, and human factors researchers. In other words, people who have to make very important-- often life-or-death-- decisions based on limited information consider situational awareness a critical skill. This should tell us something-- if those individuals for whom correct decisions are most immediately relevant all stress the importance of situational awareness, it may be a more critical skill than we realize.
Unfortunately, the only discussion of situational awareness that I've seen on LessWrong or related sites has been a somewhat oblique referen |
68e1c31e-0690-486f-be83-b05de5dfcad5 | trentmkelly/LessWrong-43k | LessWrong | Massive Activations and why <bos> is important in Tokenized SAE Unigrams
This article is a footnote to the (excellent) Tokenized SAEs. To briefly summarize: the authors (Thomas Dooms and Daniel Wilhelm) start by proposing the idea of a token "unigram".
Let Ut be the unigram for a specific token t. Conceptually there is a unique Ut for each layer Mi in each model M, such that Ut(M, i) are the activations coming out of Mi when M is fed just t by itself. More abstractly Ut are the activations which correspond to just t taken in isolation, un-bound by any particular context.
The authors then demonstrated that the activation at position j in any token sequence will have a high cosine similarity to Utj(M, i), where tj is the token at the same position. The dotted lines in the diagram below (taken from Tokenized SAEs) show the cosine similarity of the activations at position j with Utj(M, i) across many i and M.
The authors go on to demonstrate that this fact has serious implications for training SAEs. In particular it seems that SAEs learn to memorize these token unigrams and that if we add these unigrams in explicitly during training we get a significant reduction in loss.
In this article we will investigate one specific methodology for generating unigrams ([<bos>, t]) and provide some theoretical and empirical justification for its use.
How to create "Unigrams"
How exactly do we generate the activations which best represent t by itself though?
The default mechanism used by the authors to generate Ut(M, i) is to (1) create the token sequence [<bos>, t] where <bos> is the "beginning of sentence" token for M (2) pass the 2-token sequence into M (3) take the activations from layer i at the last position in the sequence. These final activations are Ut(M, i).
The authors note that this method works better than simply passing in a 1-token sequence [t], and that this is related to the instability which is known to occur when we feed Language Models sequences with no <bos> token.
We support this claim here by showing that across multiple |
ac3ed03a-1d50-4964-a8e5-e9637c4d75e3 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | Walkthrough of 'Formalizing Convergent Instrumental Goals'
Introduction
------------
I found *[Formalizing Convergent Instrumental Goals](https://intelligence.org/files/FormalizingConvergentGoals.pdf)* (Benson-Tilsen and Soares) to be quite readable. I was surprised that the instrumental convergence hypothesis had been formulated and proven (within the confines of a reasonable toy model); thiscaused me to update slightly upwards on existential risk from unfriendly AI.
This paper involves the mathematical formulation and proof of [instrumental convergence](https://en.wikipedia.org/wiki/Instrumental_convergence), within the aforementioned toy model. Instrumental convergence says that an agent .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
A with utility function .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
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.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
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.mjx-vsize {width: 0}
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@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
A will probably want to gain access to lots of computronium. What if .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
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A could just let us be, right? Wrong. Let's see how to prove it.
### My Background
I'm a second-year CS PhD student. I've recently started working through the [MIRI research guide](https://intelligence.org/research-guide/); I'm nearly finished with [Naïve Set Theory](http://smile.amazon.com/Naive-Set-Theory-Paul-Halmos/dp/1614271313/), which I intend to review soon. To expose my understanding to criticism, I'm going to summarize this paper and its technical sections in a somewhat informal fashion. I'm aware that the paper isn't particularly difficult for those with a mathematical background. However, I think this result is important, and I couldn't find much discussion of it.
Intuitions
----------
It is important to distinguish between the relative unpredictability of the *exact* actions selected by a superintelligent agent, and the relative predictability of the general kinds-of-things such an agent will pursue. As Eliezer [wrote](http://lesswrong.com/lw/v8/belief_in_intelligence/),
> Suppose Kasparov plays against some mere chess grandmaster Mr. G, who's not in the running for world champion. My own ability is far too low to distinguish between these levels of chess skill. When I try to guess Kasparov's move, or Mr. G's next move, all I can do is try to guess "the best chess move" using my own meager knowledge of chess. Then I would produce exactly the same prediction for Kasparov's move or Mr. G's move in any particular chess position. So what is the empirical content of my belief that "Kasparov is a *better* chess player than Mr. G"?
> ...
> The *outcome* of Kasparov's game is predictable because I know, and understand, Kasparov's goals. Within the confines of the chess board, I know Kasparov's motivations - I know his success criterion, his utility function, his target as an optimization process. I know where Kasparov is *ultimately* trying to steer the future and I anticipate he is powerful enough to get there, although I don't anticipate much about *how* Kasparov is going to do it.
In other words: we may not be able to predict precisely how Kasparov will win a game, but we can be fairly sure his plan will involve *instrumentally-convergent* subgoals, such as the capture of his opponent's pieces. For an unfriendly AI, these subgoals probably include nasty things like hiding its true motives as it accumulates resources.
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@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
U(Si) is the same for all possible values, we say that .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
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@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
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A is *indifferent* to region .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
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@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
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@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
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@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
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@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
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@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
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3↑↑↑3 dollar question is whether .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
Sh) will leave .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
Sh alone. First, however, we need to define a few more things.
### Actions
At any given time step (time is also considered discrete in this universe), .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
A has a set of actions .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
Ai it can perform in .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
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.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
A may select an action for each square. Then, the transition function for .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
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.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
A can do and what can be in .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
Si can produce a distinct new value for .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
Si. This transition function can be defined globally with lots more Cartesian products. Oh, and .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
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@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
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@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
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@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
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A is always allowed to do nothing in a given square, so .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
Ai is never empty.
### Resources
Let .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
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@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
A may or may not have access. Define .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
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.mjx-denominator {display: block; text-align: center}
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.MJXc-bevelled > \* {display: inline-block}
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.mjx-op {display: block}
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.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
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.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
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.mjx-vsize {width: 0}
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@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
t - basically, we know it's some combination of the defined resources .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
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R.
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.MJXc-stacked {height: 0; position: relative}
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.mjx-op {display: block}
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.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
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.mjx-vsize {width: 0}
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@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
A may choose some resource allocation .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
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∐Rti⊆Rt over the squares; this allocation is defined just like you'd expect (so if you don't know what [the Azerothian portal](http://cdn.wccftech.com/wp-content/uploads/2016/04/wowlogin.jpg) has to do with resource allocation, don't worry about it). The resources committed to a square may affect the local actions available. .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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A can only choose actions permitted by the selected allocation. Equally intuitively, how resources change over time is a function of the current resources, the actions selected, and the state of the universe; the resources available after a time step is the combination of what we didn't use and what each square's resource transition function gave us back.
Resources go beyond raw materials to include things like machines and technologies. The authors note:
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> A optimize those new regions according to its values or otherwise make use of the resources in that region.)
### Universe
A universe-history is a sequence of states, actions, and resources. The actions must conform to resources available at each step, while the states and resources evolve according to the transition functions.
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@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
n regions; define a *partial strategy* .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
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@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
L. We can combine partial strategies as long as they don't overlap in some .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
Si.
We call a strategy *feasible* if it complies with both resource restrictions and the transition functions for both states and resources. Let .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
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@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
1
### Utility
Utility functions evaluate states of the universe; .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
U evaluates each region and then combines them: .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
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U, any actions taken therein are purely instrumental in nature.
### Agent
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@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
A chooses the best possible strategy - that is, the one that maximizes the .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
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A doesn't model itself as part of the environment (it [isn't naturalized](ht<em>tp://lesswr</em>ong.com/lw/jd9/building_phenomenological_bridges/%E2%80%8E)).
Seizing the Means of Cartesian Production
-----------------------------------------
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A will take actions to increase its resource pool.
> Since resources can only lead to more freedom of action, they are never detrimental, and resources have positive value as long as the best strategy the agent could hope to employ includes an action that can only be taken if the agent possesses those resources. Hence, if there is an action that increases the agent's pool of resources .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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> @font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
> @font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
> @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
> R, then the agent will take that action unless it has a specifi c incentive from .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
> .MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
> .mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
> .mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
> .mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
> .mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
> .mjx-numerator {display: block; text-align: center}
> .mjx-denominator {display: block; text-align: center}
> .MJXc-stacked {height: 0; position: relative}
> .MJXc-stacked > \* {position: absolute}
> .MJXc-bevelled > \* {display: inline-block}
> .mjx-stack {display: inline-block}
> .mjx-op {display: block}
> .mjx-under {display: table-cell}
> .mjx-over {display: block}
> .mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-stack > .mjx-sup {display: block}
> .mjx-stack > .mjx-sub {display: block}
> .mjx-prestack > .mjx-presup {display: block}
> .mjx-prestack > .mjx-presub {display: block}
> .mjx-delim-h > .mjx-char {display: inline-block}
> .mjx-surd {vertical-align: top}
> .mjx-mphantom \* {visibility: hidden}
> .mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
> .mjx-annotation-xml {line-height: normal}
> .mjx-menclose > svg {fill: none; stroke: currentColor}
> .mjx-mtr {display: table-row}
> .mjx-mlabeledtr {display: table-row}
> .mjx-mtd {display: table-cell; text-align: center}
> .mjx-label {display: table-row}
> .mjx-box {display: inline-block}
> .mjx-block {display: block}
> .mjx-span {display: inline}
> .mjx-char {display: block; white-space: pre}
> .mjx-itable {display: inline-table; width: auto}
> .mjx-row {display: table-row}
> .mjx-cell {display: table-cell}
> .mjx-table {display: table; width: 100%}
> .mjx-line {display: block; height: 0}
> .mjx-strut {width: 0; padding-top: 1em}
> .mjx-vsize {width: 0}
> .MJXc-space1 {margin-left: .167em}
> .MJXc-space2 {margin-left: .222em}
> .MJXc-space3 {margin-left: .278em}
> .mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
> .mjx-line-box-test {display: table!important}
> .mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
> .MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
> .MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
> .MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
> .MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
> .MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
> .MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
> .MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
> .MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
> .MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
> .MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
> .MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
> .MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
> .MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
> .MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
> .MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
> .MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
> .MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
> .MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
> .MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
> .MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
> .MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
> .MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
> .MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
> .MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
> .MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
> @font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
> @font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
> @font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
> @font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
> @font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
> @font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
> @font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
> @font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
> @font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
> @font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
> @font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
> @font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
> @font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
> @font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
> @font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
> @font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
> @font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
> @font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
> @font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
> @font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
> @font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
> @font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
> @font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
> @font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
> @font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
> @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
> U to avoid taking that action.
Define a *null action* to be any action which doesn't produce new resources. It's easy to see that null actions are never instrumentally valuable. What we want to show is that .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
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@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
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h, where we live, grow, and love. Regions full of instrumentally-valuable resources.
### Discounted Lunches
An action *preserves* resources if the input resources are strictly contained in the outputs (nothing is lost, and resources are sometimes gained). A *cheap lunch* is a feasible partial strategy in some subset of squares .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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⟨Rk⟩ and whose constituent actions preserve resources. A *free lunch* is cheap lunch that doesn't require resources.
> This is intended to model actions that "pay for themselves"; for example, producing solar panels will incur some signifi cant energy costs, but will later pay back those costs by collecting energy.
A cheap lunch is *compatible with* a global strategy if the resources required for the lunch are available for use in .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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J at each time step. Basically, at no point does the partial strategy require resources already being used elsewhere.
### Possibility of Non-Null Actions
We show that it's really hard to assert that .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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1.3×1050).
#### Lemma 1: Cheap Lunches and Utility
Cheap lunches don't reduce utility. Let's say we have a cheap lunch .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
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.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
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.MJXc-space1 {margin-left: .167em}
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.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
i and some global strategy .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
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i). Assume the cheap lunch is compatible with the global strategy; this means the cheap lunch is feasible. If .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
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@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
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@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
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.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
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.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
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.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
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.mjx-label {display: table-row}
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.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
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.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
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i, the conjugate strategy (of the cheap lunch and the remainder of the global strategy) has equal utility to .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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⟨¯bk⟩.
*Proof.* We show feasibility of the conjugate strategy by demonstrating we don't need to change resource allocation elsewhere. This is done by induction over time steps. Since .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
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A is later able to do in the regions relevant to .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
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@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
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@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
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@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
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U. This implies that .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
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U cannot be decreased by taking the cheap lunch.
#### Theorem 1: Cheap Lunches and Optimality
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i.
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i.
#### Corollary 1: Free Lunches and Optimality
If there is an optimal strategy and a free lunch in region .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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i.
*Proof.* Free lunches require no resources, so they are compatible with any strategy; apply Theorem 1.
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@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
h isn't a cheap lunch for any optimal strategies.
### Necessity of Non-Null Actions
We show that as long as .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
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A can extract useful resources (resources whose availability leads to increased utility), it will.
#### Theorem 2: Necessity
Consider the maximum utility achievable outside of region .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
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.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
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.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
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@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
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@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
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@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
u. Then if .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
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i.
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⟨¯dk⟩, consisting of the primary strategy and the cheap lunch. Allocate the resources gained via the cheap lunch according to the primary strategy; this is feasible since we know the cheap lunch is compatible with the primary strategy, which is in turn enabled by these resources gained.
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i and doesn't require any resource inputs; it's trivial to see that this is feasible. Since .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
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.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
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.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
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.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
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i, we do some algebraic substitution of utility values to see that the conjugate plan has strictly higher utility than .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
⟨¯ek⟩ does. This means that .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
⟨¯ek⟩ is suboptimal.
#### Theorem 3: Defection Threshold
Even if .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
U is not indifferent about .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
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.mjx-op {display: block}
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i, we can prove Theorem 2 can still hold if the utility .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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maxs,s′∈Si|Ui(s)−Ui(s′)|).
*Proof.* The structure is similar to that of Theorem 2; just use the new maximum caring differential in the algebraic substitution.
> We interpret Theorem 3 as a partial con firmation of Omohundro's thesis in the following sense. If there are actions in the real world that produce more resources than they consume, and the resources gained by taking those actions allow agents the freedom to take various other actions, then we can justi fiably call these actions "convergent instrumental goals." Most agents will have a strong incentive to pursue these goals, and an agent will refrain from doing so only if it has a utility function over the relevant region that strongly disincentivizes those actions.
The Bit Universe
----------------
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@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
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@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
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@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
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.mjx-numerator {display: block; text-align: center}
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.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
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.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
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.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
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.mjx-annotation-xml {line-height: normal}
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.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
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.mjx-line {display: block; height: 0}
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.mjx-vsize {width: 0}
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h and when it is not). This isn't good; the vast majority of utility-maximizing agents will not steer us towards futures we find desirable. If you're interested, I recommend reading this section for yourself, even if you aren't very comfortable with math.
Our Universe
------------
> The path that our model shows is untenable is the path of designing powerful agents intended to autonomously have large e ffects on the world, maximizing goals that do not capture all the complexities of human values. If such systems are built, we cannot expect them to cooperate with or ignore humans, by default.
We have much work to do. The risks are enormous and the challenges "[impossible](http://lesswrong.com/lw/up/shut_up_and_do_the_impossible/)", but we have time on the clock. [AI safety research is primarily talent-constrained](https://80000hours.org/career-reviews/artificial-intelligence-risk-research/). If you've been sitting on the sidelines, wondering whether you're good enough to learn the material - well, I can't make any promises. But if you feel the burning desire to *do something*, [to put forth some extraordinary effort](http://lesswrong.com/lw/uo/make_an_extraordinary_effort/), [to become stronger](http://lesswrong.com/lw/h8/tsuyoku_naritai_i_want_to_become_stronger/) - I invite you to contact me so we can work through the material together.
Questions and Errata
--------------------
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@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
∪Pt - shouldn't we take the union of the outputs and whatever resources *weren't* used at time .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
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@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
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t?
* Page 8, right column, second full paragraph, last line: should be "we have two options available *to* us".
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.mjx-vsize {width: 0}
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@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
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.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
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.mjx-numerator {display: block; text-align: center}
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.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
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.mjx-delim-h > .mjx-char {display: inline-block}
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Feasible(⟨Pk⟩)={⟨¯ak⟩:isFeasible(⟨Pk⟩,⟨¯ak⟩)}. |
363a52ac-f2ed-4b66-89ac-9b0b523ad984 | trentmkelly/LessWrong-43k | LessWrong | Superhuman Coders in AI 2027 - Not So Fast
Thank you @elifland for reviewing this post. He and AI Futures are planning to publish updates to the AI 2027 Timeline Forecast soon.
AI 2027 (also launched in this LW post) forecasts an R&D-based AI takeoff starting with the development of Superhuman Coders[1] within a frontier lab.
FutureSearch co-authored the AI 2027 Timeline Forecast. We thought the other authors’ forecasts were excellently done, and as the piece says:
> All model-based forecasts have 2027 as one of the most likely years that SC [Superhuman Coders] being developed, which is when an SC arrives in the AI 2027 scenario.
Indeed. But overall, FutureSearch (two full-time forecasters, two contract forecasters, and this author, Dan Schwarz) think superhuman coding will arrive later — median 2033 — than the other authors (hereon "AI Futures"): median 2028 from Nikola Jurkovic, and median 2030 from Eli Lifland[2].
Here, we briefly explain how our views diverge on what it would take for OpenAI/DeepMind/Anthropic/xAI to develop superhuman coding.
This post can also serve as a much shorter piece for those who didn’t read the whole (24-page!) piece on the AI 2027 site. (See also Scott Alexander's shorter piece on the AI Futures blog on modeling coding time horizons.)
The Forecast
As @Max Harms wrote in another AI 2027 LW reaction piece,
> we can hallucinate that FutureSearch has approximately the market timelines, while Eli Lifland (and perhaps other authors) has a faster, but not radically faster timeline.
I think this is broadly correct, even though the “market” timeline here is hard to interpret because as Max says, Superhuman Coding ≠ AGI.
Our published view in AI 2027 — blue line below — is fairly in-line with similar crowd forecasts, while being much slower than the AI Futures:
We produced, but did not publish on AI 2027, an all-things-considered forecast with a median in Nov 2033, vs the "within-model" forecast of Jan 2032 above:
So while 2027 is conceivable, why do we give a median mor |
bbba79b0-cb4e-431c-8dd6-f2e12f85f555 | trentmkelly/LessWrong-43k | LessWrong | Group Rationality Diary, Oct. 6-18, 2015
This is the public group rationality diary for October 6-18, 2015. It's a place to record and chat about it if you have done, or are actively doing, things like:
* Established a useful new habit
* Obtained new evidence that made you change your mind about some belief
* Decided to behave in a different way in some set of situations
* Optimized some part of a common routine or cached behavior
* Consciously changed your emotions or affect with respect to something
* Consciously pursued new valuable information about something that could make a big difference in your life
* Learned something new about your beliefs, behavior, or life that surprised you
* Tried doing any of the above and failed
Or anything else interesting which you want to share, so that other people can think about it, and perhaps be inspired to take action themselves. Try to include enough details so that everyone can use each other's experiences to learn about what tends to work out, and what doesn't tend to work out. |
a0efeeed-48f5-42d0-ba68-cea24c3dd5d2 | trentmkelly/LessWrong-43k | LessWrong | are IQ tests a good measure of intelligence?
I define "intelligence" as "having a strong ability to hit narrow targets in a large search space"
Also, is there a one "official" IQ test? or does any random internet thing that calls itself an "iq test" work? which ones are real?
if they are not good measures of intelligence by this definition, is there a definition of intelligence which they are good at measuring? |
38566772-72e3-470a-a73f-8a2cf3c55d37 | trentmkelly/LessWrong-43k | LessWrong | What LessWrong/Rationality/EA chat-servers exist that newcomers can join?
A few times over the past few years newcomers to the community have asked me how they can start engaging with the community, and one of the lowest-cost ways to engage has historically been chatrooms.
I know that there has been a pretty significant proliferation of LW/Rationality/EA-related chat-servers in the past few years, and it seems like a good idea to collate all of them into a central repository.
I know about Elo's LW-Slack, and I know of the abstract existence of a bunch of EA-related Discord channels, but haven't been super active in this space, so I would appreciate other people's contributions.
If we get a good list, I think it would make sense to add that list to the community tab to make it easier to discover (or at least link this question from the community section). |
b2cfb217-2bd3-4144-ae97-bb83c2a2a471 | trentmkelly/LessWrong-43k | LessWrong | Against blanket arguments against interpretability
On blanket criticism and refutation
In his long post on the subject, Charbel-Raphaël argues against theories of impacts of interpretability. I think it's a largely a good, well-argued post, and if the only thing you get out of it is reading that post, I'll be contributing to improving the discourse. There is other material with similar claims that I think are made with low context, and also I should say that I'm not very versed in the history and the various versions of the debate.
At the same time, I disagree with the take.
In this post I'm going to "go high" and debate strong, general forms of the criticism, rather than the more object-level subcomponents. Generalizing away from specifics, I think Charbel-Raphaël's post has three valid general reasons to doubt the usefulness of the interpretability paradigm. Below, I'll try to write out condensed but strong versions of them, and address them one by one.
But before doing this, let's set some parameters for the debate. Note that the goal of a blanket critique like Charbel-Raphaël's post is not to say "there are worlds where interpretability definitely doesn't lead to better safety", but rather that "interpretability doesn't lead to noticeably better safety in the vast majority of worlds". Therefore in refuting this, I'll be free to make some "plausible" assumptions about the world coming from insights that we have on current models, without having to defend these from "what if" arguments about exceptional worlds where they don't work.
This means that in this post I am very definitely not making the strongest case for the impact of interpretability. There are many alternative routes to impact the come from understanding systems better. If I were making a post that makes the strongest case for interpretability being useful given what we know now, I'd not just give one plausible story but go through many of the alternative routes. I would also put a lot of emphasis on operationalizing our high uncertainty (includi |
867efef2-5ccc-4955-8d1b-7ed27a13a2fd | trentmkelly/LessWrong-43k | LessWrong | Interest in Leetcode, but for Rationality?
the following is motivated by:
* Self-Improvement or Shiny Distraction: Why Less Wrong is anti-Instrumental Rationality
* The Martial Art of Rationality
I've been a long time lurker on Less Wrong and I've noticed the recurring criticism that despite its focus on rationality, the community lacks structured training to develop practical rationality skills. Eliezer Yudkowsky talks rationality as a martial art, because it's something that can be trained and refined through deliberate practice. But where is our dojo?
A model that comes to mind is a website like LeetCode, where programmers can solve coding challenges, share solutions, and see how others approach the same problems. LeetCode can sometimes encourage overfitting to specific problem types so it's not a perfect analogy. The community driven aspect would interesting to me as you can see how other people approach the problem. Could something similar be adapted for rationality?
Imagine a platform where, instead of solving coding puzzles, users engage with problems designed to train rational thinking. Here are a few types of problems that might fit:
1. Cognitive Bias Detection: Users could review novel, real-world scenarios and try to identify what cognitive bias or logical fallacy is present. The goal would be to train pattern recognition for biases without simply memorizing common examples. For instance, a scenario might subtly include a case of confirmation bias or anchoring, and users would need to spot it.
2. Calibration Training: One of the most important skills in rationality is aligning your confidence with reality. For each problem or scenario, users could submit a confidence interval along with their answer. This serves as a double-training: users practice assessing their certainty, and over time, they get feedback on how well-calibrated they are.
3. Bite-Sized, Practical Challenges: The focus should be on small, actionable exercises rather than lengthy theoretical discussions. For example, a pr |
abf58998-4775-4fbb-a82f-c615f9b2507a | trentmkelly/LessWrong-43k | LessWrong | Meetup : "What is the Intelligence Explosion?" (and more...) LW Melbourne dojo
Discussion article for the meetup : "What is the Intelligence Explosion?" (and more...) LW Melbourne dojo
WHEN: 02 April 2016 02:00:00PM (+1100)
WHERE: Bull and Bear Tavern, Melbourne
Less Wrong Melbourne's Rationality Dojos are self-improvement sessions based on the art of rationality. We welcome new members who are interested in exploring rationality. We're all "aspiring rationalists".
PLEASE NOTE THE TIME AND VENUE. (We're experimenting with different times and locations. So, it's in the first Saturday, at 2:00pm, in the Bull & Bear pub (downstairs where we sometimes have our socials).
WHAT'S HAPPENING:
This month, we have...
"The Singularity – What is the Intelligence Explosion?" – with Adam Ford.
Then two lightning talks: * Eike on "Gamifying your to do list" * Letian on "Go: a player's perspective on the game and Alpha Go" (Alpha Go referring to the victory of computer over human in the game of Go)
And a few other things including a goals exercise.
We finish at 5pm, and you are welcome to join us for dinner nearby.
WHEN: Saturday 2 April, 2:00pm arrival, 2:30pm start.
WHERE: The Bull & Bear, 347 Flinders Ln, Melbourne
DIRECTIONS: On the South side of the lane, between Elizabeth and Queen. Down one flight of stairs.
Hope to see you there!
Discussion article for the meetup : "What is the Intelligence Explosion?" (and more...) LW Melbourne dojo |
a3389e86-3e3b-4b14-85f0-155367b2cc9b | trentmkelly/LessWrong-43k | LessWrong | Towards cause priotisation estimates for child abuse
Closest community background reading: http://www.givewell.org/labs/causes/criminal-justice-reform
Scale
prevalence
Back of the envelope estimate of the number of abused excluding those who are emotionaly abused and neglected (because those stats aren’t on the wikipedia page for child abuse):
> >Despite these limitations, international studies show that a quarter of all adults report experiencing physical abuse as children, and that and 1 in 5 women and 1 in 13 men report experiencing childhood sexual abuse. Emotional abuse and neglect are also common childhood experiences ("Child maltreatment: Fact sheet No. 150". World Health Organization. December 2014)
If all those physically abused are the same as those sexually abused (most conservative estimate) then 0.2 of all people are abused as children. If they are completely seperate populations then ((1/5 + 1/13)/2) + (1/4) = 0.39 (~0.4) of all people are abused as children. So, 0.2-0.4 of all people are abused.
> >A long-term study of adults retrospectively reporting adverse childhood experiences including verbal, physical and sexual abuse, as well as other forms of childhood trauma found 25.9% of adults reported verbal abuse as children, 14.8% reported physical abuse, and 12.2% reported sexual abuse
More likely ¼ of all people are abused as children in some way or another
Harm (qualitatively)
> >reduction in lifespan of 7 to 15 years (Kolassa, Iris – Tatjana. "Biological memory of childhood maltreatment – current knowledge and recommendations for future research" (PDF). Ulmer Volltextserver – Institutional Repository der Universität Ulm. Retrieved 30 March 2014.)
>
>
>
> >more likely to suffer from physical ailments such as allergies, arthritis, asthma, bronchitis, high blood pressure, and ulcers (Dolezal, T.; McCollum, D.; Callahan, M. (2009). Hidden Costs in Health Care: The Economic Impact of Violence and Abuse. Academy on Violence and Abuse.)
>
>
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0d15484f-d335-4041-a80b-d2135c5f6446 | StampyAI/alignment-research-dataset/youtube | Youtube Transcripts | AI safety needs social scientists | Amanda Askell | EA Global: London 2018
Hi everyone.
Can you all hear me okay?
I'm going to assume that was a yes.
Okay.
So, here's an overview of what I'm going to
be talking about today.
So first, I'm going to talk a little bit about
why learning human values is difficult for
AI systems.
And then I'm going to explain to you the safety
via debate method, which is one of the methods
that OpenAI's currently exploring for helping
AI to robustly do what humans want.
And then I'm going to talk a little bit more
about why I think this is relevant to social
scientists and why I think social scientists,
in particular, people like Experimental Psychologists
and Behavioral Scientists, can really help
with this project.
And I will give you a bit more details about
how they can help, towards the end of the
talk.
Okay.
So, learning human values is difficult.
We wanted to train AI systems to kind of robustly
do what humans want.
And in the first instance, we can just imagine
this being what one person wants.
And then ideally we can expand it to doing
what most people would consider good and valuable.
But human values are very difficult to specify,
especially with the kind of precision that
is required of something like a machine learning
system.
And I think it's really important to emphasize
that this is true even in cases where there's
moral consensus, or consensus about what people
want in a given instance.
So, take a kind of principle like "do not
harm someone needlessly."
I think we can be really tempted to think
something like, well, if I have...
I've got a computer, and so I can just write
into the computer, do not harm someone needlessly.
But this is a really underspecified principle.
Most humans know exactly what it means, they
know exactly when harming someone is needless.
So, if you're shaking someone's hand, and
you push them over, we think this is like
needless harm.
But if you see someone in the street who's
about to be hit by a car, and you push them
to the ground, we think that's not an instance
of needless harm.
So humans just have a pretty good way of knowing
when this principle applies and when it doesn't.
But for a formal system, there's going to
be a lot of questions about precisely what's
going on here.
So, one question this system may ask is, how
do I recognize when someone is being harmed?
It's very easy for us to see things like stop
signs, but when we're building self driving
cars, we don't just program in something like,
stop at stop sign.
We instead have to train it to be able to
recognize an instance of a stop sign.
And then the principle that says that you
shouldn't harm someone needlessly employs
this notion of like, that we kind of understand
when harm is and isn't appropriate, whereas
there's a lot of questions here like, when
is harm justified?
What is the rule for all plausible scenarios
in which I might find myself?
These are things that you need to specify
if you want your system to be able to work
in all of the kind of cases that you want
it to be able to work in.
So I think this is an important point to just
kind of internalize is, it's easy for humans
to identify, and to pick up, say, a glass.
But training a ML System to do it requires
a lot of data.
And this is true of just like a lot of tasks
that humans might intuitively think are easy,
and we shouldn't then just transfer that intuition
to the case of machine learning systems.
And so when we're trying to teach human values
to any AI system, it's not that we're just
looking at edge cases, like trolley problems,
we're really looking at core cases of like,
how do we make sure that our ML Systems understand
what humans want to do, in the kind of everyday
sense.
Okay.
So, one way of doing this is through human
feedback.
So, there are many approaches to training
an AI to do what humans want.
So you might think that humans could, say,
demonstrate the behavior, and then you realize,
well, there's going to be some behaviors it's
just too difficult for humans to demonstrate.
You might think that they can say whether
they approve or disapprove of a given behavior,
but one of the concerns about training from
human feedback... so, learning from humans,
what they want, is that we have a reward function
as predicted by the human.
And then we have AI strength.
And when it reaches the superhuman level,
it becomes really hard for humans to predict,
to be able to give the right reward function.
So, as AI capabilities surpass the human level,
the decisions and behavior of the AI system,
just might be too complex for the human to
judge.
So imagine agents that control, say, we've
given the example of a large set of industrial
robots.
That may just be the kind of thing that I
just couldn't evaluate whether these robots
were doing a good job overall, it'd be extremely
difficult for me to do so.
And so the concern is that when behavior becomes
much more complex and just much more large
scale, it just becomes really hard for humans
to be able to kind of judge whether something
is doing a good job.
And that's why you may expect this kind of
drop-off.
And so this is like a kind of scalability
worry about human feedback.
So what ideally need to happen instead is
that, as AI strength increases, the ward that's
predicted by the human is also able to basically
keep pace.
So how do we achieve this?
One of the things that we want to do here
is we want to try and break down the kind
of complex questions and complex tasks.
Like, having all of these industrial robots
perform a kind of complex set of functions
that comes together to make something useful,
into some kind of smaller set of tasks and
components that humans are able to judge.
So here is a big question.
And the idea is that the overall tree might
be too hard for humans to fully check, but
it can be kind of decomposed into these elements,
such that at the very kind of bottom level,
humans can check these things.
So maybe the example of "how should a large
set of industrial robots be organized to do
task x" would be an example of a big question
where that's a really complex task, but there's
some things that are checkable by humans.
So if we could decompose this task so that
we were just asking a human, if one of the
robots performs this small action, will the
result be this outcome, this small outcome?
And that's something that humans can check.
And in the case of what humans want, a big
question is, what do humans want?
Much smaller question.
If you can manage to decompose this, is something
like, it's better to save 20 minutes of someone's
time, than to save 10 minutes of their time.
So if you imagine some AI agent that's there
to assist with humans, this is a fact that
we can definitely check, even if I can't answer,
my assistant.. this assistant AI...
I can't say something like, this is just what
I want, this is like everything that I want.
I am not able to tell it that.
I can't tell it that I'd rather it save 20
minutes of my time than save 10 minutes of
my time.
Okay.
So one of the key issues is that, with current
ML Systems, we need to train on a lot of data
from humans.
So if you imagine that we want humans to actually
give this kind of feedback on these kind of
ground level claims or questions, then we're
going to have to train on a lot of data from
people.
So just to give some examples, simple image
classifiers train on thousands of images.
Like these are the ones you can kind of make
yourself, and you'll see the datasets are
pretty large.
AlphaGo zero played nearly 5 million games
of Go during training.
OpenAI Five trains on 180 years of Dota 2
games per day.
So this gives you a sense of how much data
you need to train these systems.
So if we are current ML techniques to teach
AI human values, we can't rule out needing
millions to tens of millions of short interactions
from humans as like the data that we're using.
So earlier I kind of talked about human feedback
where I was like, assuming that we were kind
of asking humans questions.
So something like we could just ask humans
really simple things like, do you prefer to
eat an Omelette or 1000 hot dogs?
Or is it better to provide medicine or books
to this particular family?
One way that we might think that we can get
kind of more information from the data that
we're able to gather is by finding reasons
that humans have, for the answers that they
give.
So if you manage to learn that humans generally
prefer to eat a certain amount per meal, you
can kind of rule out a large class of questions
you might ever want to ask people.
You're never going to ask them, do you prefer
to eat an Omelette or 1000 hot dogs?
Because you know that humans just generally
don't like to eat 1000 hot dogs in one meal,
except in very strange circumstances.
And we also know facts like humans prioritize
necessary health care over mild entertainment.
So this might mean that, if you see a family
that is desperately in need of some medicine,
you just know that you're not going to say,
"Hey, should I provide them with an entertaining
book, or this essential medicine?"
So there's a sense in which when you can identify
the reasons that humans are giving for their
answers, this kind of lets you go beyond,
and learn sort of faster, what they're going
to say in a given circumstance of what they
want.
It's not to say that you couldn't learn the
same things by just asking people questions,
but rather if you can find a quicker way to
identify reasons, then this could be much
more scalable.
So debate is a kind of proposed method for
trying to learn human reasons that is currently
being explored.
So, to give you the kind of definition of
a debate here, so the idea is that, two AI
agents are going to be given a question, and
they take turns making short statements, and
a human judge is at the end, which of the
statements gave them the most true, valuable
information.
It's worth knowing that this is quite dissimilar
from a lot of human debates.
So with human debates, people might give one
answer, but then they might adjust their answer
over the course of a debate.
Or they might kind of debate with each other
in a way that's more exploratory.
They're gaining information from the other
debater, which then they're updating on, and
then they're feeding that back into the debate.
With AI debates, you're not doing it for information
value.
So it's not kind of, it's not going to have
the same sort of exploratory component, done
in multiple paths, instead, you would hopefully
see the agents explore a path kind of like
this.
So imagine I want my AI agents to basically
decide which bike I should buy.
I don't want to have to go and look up all
the Amazon reviews, etc.
In a debate, I might get something like, you
should buy the red road bike from the first
agent.
Suppose that blue disagrees with it.
So blue says you should buy the blue fixie.
Then red says, the red road bike is easier
to ride on local hills.
And one of the key things to suppose here
is that for me, this is...
I live in San Francisco, being able to ride
on the local hills is very important to me
in a bike.
It may even overwhelm all other considerations.
So, even if the blue fixie is cheaper by say
$100, I just wouldn't be willing to pay that.
I'd be happy to pay the extra $100 in order
to be able to ride on local hills.
And if this is the case, then there's basically
nothing true that the other agent can point
to, to convince me to buy the blue fixie,
then blue should just say, I concede.
Now, blue could have lied for example, but
if we assume that red is able to kind of point
out blue's lies, we should just expect blue
to basically lose this debate.
And if it's explored enough and attempted
enough debates, it might just see that, and
then say, "Yes, you've identified the key
reason, I concede."
And so it's important to note that we can
imagine this being used to identify multiple
reasons, but here it has identified a really
important reason for me, something that is
in fact going to be really compelling in the
debate, namely, it's easier to ride on local
hills.
Okay.
So, training an AI to debate looks something
like this.
If we imagine Alice and Bob are our two debaters,
and each of these is like a statement made
by each agent.
And so you're going to see exploration of
the tree.
So the first one might be this.
And here, say that the human decides that
Bob won in that case.
This is another node, another node.
And so this is the exploration of the debate
tree.
And so you end up with a debate tree that
looks a little bit like a kind of game of
Go.
And so when we explore like... when you have
AI training to play Go, it's exploring lots
of different paths down the tree, and then
there's a win or loss condition at the end.
And that's its feedback.
This is like how it's basically learning how
to play.
With debate, you can imagine the same thing,
but where you're exploring, you know, a large
tree of debates and humans assessing whether
you win or not.
And this is just a way of training up AI to
get better at debate and to eventually ideally
identify reasons that humans find compelling.
Okay.
So one kind of thesis here that I think is
relatively important is this kind of positive
amplification thesis, or positive amplification
threshold.
So one thing that we might think or that seems
fairly possible is that, if humans are like
above some threshold of rationality and goodness,
then debate is going to amplify their positive
aspects.
This is speculative, but it's a kind of hypothesis
that we're working with.
And the idea here is that, if I am pretty
irrational and pretty well motivated, someone
can then... suppose I'm looking at debate,
I might get some feedback of the form, actually
that decision that you made was fairly biased,
and I know that you don't like to be biased,
so I want to inform you of that.
I get informed of that, and I'm like, "Yes,
that's right.
Actually, I don't want to be biased in that
respect."
Suppose that they're like Kahneman and Tversky,
they point out some key cognitive bias that
I have.
If I'm kind of rational enough, I might say,
"Yes, I want to adjust that."
And I give a newer kind of signal back in
that has been improved by virtue of this process.
So if we're somewhat rational, then we can
imagine a situation in which all of these
positive aspects of us are being amplified
through this kind of process.
But you can also imagine a kind of negative
amplification.
So if people are below this threshold of rationality
and goodness, we might worry the debate would
kind of amplify these negative aspects.
If it turns out we can just be really convinced
by kind of appealing to our worst nature,
and your system learns to do that, then it
could just do that feedback in, becoming even
kind of less rational and more biased, and
that gets feedback in.
So this is just like a kind of important hypothesis
related to work on amplification, which if
you're interested in, it's great.
And I suggest you take a look at it, but I'm
not going to focus on it here.
Okay.
So how can social scientists help with this
whole project?
And hopefully I've conveyed some of, what
I think of as like, the real importance of
the project.
So first I think that a key question here
is kind of, it reminds me a little bit of
Tetlock's work on Superforecasters.
So, obviously a lot of social scientists have
done work kind of identifying people who are
Superforecasters, where they seem to be kind
of robustly more accurate than many people,
they're robustly accurate across time when
it comes to forecasts, and they work.
We've found other features of them like working
in groups really helps, and so on.
And one question is whether we can identify
good human judges, or we can train people
to become essentially super judges.
So why is this helpful?
And this is just kind of one way of framing
the ways in which social scientists could
help with this project, I think.
So, firstly, if we do this, we will be able
to test how good human judges are, and we'll
see whether we can improve human judges.
This means we'll be able to know if a human...
or at least try and find out whether humans
are above this kind of positive amplification
threshold.
So, are ordinary human judges that we would
be using to judge debate kind of good enough
to see like an amplification of their good
features, is one question.
Another question is... or, sorry, another
reason to do this is that it improves the
quality of the judging data that we can get.
If people are just generally pretty good,
rational, assessing debate and fairly quick,
then this is excellent given the amount of
data that we anticipate needing.
Basically, improvements to your data it can
be extremely valuable here.
So yeah, the benefits of this, positive amplification
will be more likely during safety via debate,
and also will improve training outcomes on
limited data, which is very important.
Okay.
So this is one way of kind of framing why
I think social scientists are pretty valuable
here, but there's lots of questions that we
really do want asked when it comes to this
project.
And this is just like, I think this is going
to be true of other projects like asking human
questions.
It's basically to note that the human component
of the human feedback is quite important.
And getting that right is actually quite important.
And that's something that we anticipate social
scientists to be able to help with, more so
than like annual researchers who are not working
with people, and their biases, and how rational
they are, etc.
These are questions that are the focus of
social sciences.
So one question is, how skilled are people
as judges by default?
Can we distinguish good judges of debate from
bad judges of debate?
And if so, how?
Does judging ability generalize across domains?
Can we train people to be better judges?
Like, can we engage in kind of debiasing work,
for example?
Or work that reduces cognitive biases?
What topics are people better or worse at
judging?
So are there ways of like phrasing the questions
so that people are just better at assessing
them?
Are there ways of structuring the debate that
make them easier to judge, or restricting
debates to make them easier to judge?
So we're often just showing people a small
segment of a debate, for example.
Can people work together to improve judging
qualities?
These are all kind of outstanding questions
that we think are important, but we also think
that they are empirical questions and that
they have to be answered by experiment.
And so this is like, I think, important potential
future work.
So we've been thinking a little bit about
what you would want in experiments that try
and assess this in humans, like how good are
they debating... sorry, how good are they
at judging a debate?
Because ideally, AI agents would be doing
the debate in the long run.
So one is just that there's a verifiable answer.
We kind of need to be able to tell whether
people are correct or not, in their judgment
of the debate.
The other is that there is a plausible false
answer, because if you have a debate, if we
can only train and assess human judging ability
on debates where there's like no plausible
false answer, we'd get this false signal that
people are really good at judging debate.
They always get the true answer, but that's
because it's always a really obvious question.
Like, "Is it raining outside?"
And the person can look outside.
We don't really want that kind of debate.
Ideally we want something where evidence is
available so that it grounds out in... humans
can have something that grounds out the debate.
We also don't want debates to rely on human
deception.
So things like tells in poker for example,
we really don't want that because like, AI
agents are not going to have normal tells,
it would be rather strange, I suppose, if
they did.
Like if they had stuttering or something.
Debaters have to know more about the question
as well, because the idea is that the AI agents
will be much more capable and so you don't
want a situation in which there isn't a big
gap basically between the debater capabilities,
and the judge abilities.
Maybe some of these things, these ones feel
kind of like pretty essential.
These ones are sort of desires, I guess.
So one is that biases are present.
How good are humans when there's bias with
respect to the question?
There are representative segments of the debate
that we can actually show people, the questions
aren't too hard, like it's just not impossible
for humans to answer them, or judge debates
about them.
But they also mirror some of the difficulties
of statistical debate, i.e, about probabilities,
rather than about outright claims.
And finally that we can get enough data.
And so, one thing you might notice is that
there starts to be kind of tensions between
a lot of these desiderata.
So that there's a plausible false answer,
is in a bit of tension with the idea that
the question isn't too hard.
If a question is like... and the same is true
of like, the question isn't too hard, and
the question meriting statistical debate.
Statistical debate is generally pretty hard
to evaluate, I think, for people.
And also if... it's also quite important that
we be able to model it.
Debaters knowing more, and that we can get
enough data is like another thing.
It's just harder to kind of train if we need
debaters that know a lot more than judges,
and it's harder for judges to evaluate debates
of this form.
Okay.
So I'm going to show you to debate.
This is just to give you some of the difficulties,
I guess.
So this was a program set up where we would
show a judge a kind of blank screen.
So imagine you're not seeing the dog that's
here.
Two debaters, two human debaters, sit in the
same room, and they have this picture of a
dog in front of them.
And one of them is selected to lie, and one
of them is selected to tell the truth.
And what they can do here is they can select
areas, and describe to the human judge what
they see in that area.
And all that the judge is going to see is
kind of their blank screen with the relevant
section selected.
And then they can each make claims about what
is in that section.
So here red is saying, it's a dog, here's
its long, floppy, ear.
Blue, is saying, no, here's one of its pointy
ears.
So he's trying to point to a smaller area
where it looks kind of pointed.
That does look like an ear slope to the right,
but if it really was, then part of the head
would be here, instead there's a brick.
The ear's pointing out from behind the bricks.
The dog is in front of the bricks.
If it were behind her, there would be an edge
here, but the rectangle is all the same color.
And then you get like a resignation.
And at the end of the debate they can show
a single pixel.
And the question was something like, if all
you can show, all you can do is have a debate
and show a single pixel, can you get people
to have accurate beliefs about the question?
And basically we saw like, yes, debates were
fairly good.
In this kind of case, you might think that
this is pre-synthetic.
So one of the things that we're kind of thinking
about now is like, expert debaters with a
lay judge.
And I'm going to show you something that we
did that's kind of fun, but I never know how
it looks to outsiders.
So, we had to debate that was of this form.
This was a debate actually about quantum computing,
where the idea is were we going to debate
this question?
So we have two kind of, we say, experts, but
people who understand the domain, one of them
is going to lie and one is going to tell the
truth.
So we had blue say, red's algorithm is wrong
because it increases alpha by an additive
exponentially small amount each step.
So it takes exponentially many steps to get
alpha high enough.
So this was like one of the kind of claims
made.
And then you get these set of responses.
I don't think I need to go through all of
them.
You can see kind of the form that they take.
But it grounds out in this claim from... we
allowed, like, certain restricted claims from
Wikipedia.
So, blue ends this with the first line of
this Wikipedia article says that, the sum
of probabilities is conserved.
Red says, an equal amount is subtracted from
one amplitude and added to another, implying
the sum of amplitudes are conserved.
But probabilities are the squared magnitudes
of amplitudes, so this is a contradiction.
This is I think roughly how this debate ended.
But you can imagine this is like a really
complex debate in a domain that the judges
ideally just won't understand, and might not
even have some of the concepts for.
And that's the kind of difficulty of debate
that we've been looking at.
And so this is one thing that we're in the
kind of early stages of prototyping, and that's
why I think it seems to be the case that people
actually do update in the kind of right direction,
but we don't really have enough data to say
for sure.
Okay.
So I hope that, I've kind of given you an
overview of places, and just even like a restricted
set of places in which I think like social
scientists are going to be important in AI
safety.
So here we're interested in experimental psychologists,
cognitive scientists, and behavioral economists,
so people who might be interested in actually
scaling up and running some of these experiments.
If you're interested in this, please come
to my office hours after this talk, or email
me, because we would love to hear from you.
So thanks.
All right.
We don't have too much time for questions,
but if you want to sit, we can take a couple.
Just for starters, as I was kind of watching
this I'm wondering, how much of this is real
at all or coming from an actual system versus
like, do you have humans sort of playing the
role of the agents in these examples?
Yeah, so at the moment, the idea is that we
want ultimately the debate to be conducted
by AI, but we don't have the language models
that we would need for that yet.
And so we're using humans as a kind of proxy
to test the judges in the meantime.
So yeah, all of this is done with humans at
the moment, yeah.
So you're faking the AI?
Yeah.
The-
To set up the scenario to train the judges
to, to evaluate the judges on their ability
to later provide?
Yeah.
And some of the ideas like I guess you don't
necessarily want all of this work to kind
of happen later and once you... a lot of this
work can be done before you even have the
relevant capabilities, like, have AI perform
the debate.
So that's why we're using humans just now.
Yeah, totally understood.
Let's see.
A couple of questions coming in.
I guess one note also for the audience, if
you didn't see Jan Leike's talk yesterday,
he showed some examples from the work that
his team has done on video games, that very
much matched the plots that you had shown
earlier, where up to a certain point, the
behavior sort of matches the reward function-
Yeah.
And then at some point they sort of diverge
sharply as the agent finds a loophole in the
system.
So that can happen even in like, Atari Games,
which is what they're working on.
So obviously it gets a lot more complicated
from there.
So, questions from the audience.
How would you train, in this approach, you
would train both the debating agents and the
judges.
So in that case who evaluates the judges and
based on what?
Yeah, so I think it's kind of interesting
where we want to identify how good the judges
are in advance, because it might be hard to
assess... like later, if people are just judging
on verifiable answers, you could in fact,
presumably assess the judges as even when
you're doing training, but they're going to
be... here, we can kind of ground out debates
in questions with verifiable answers.
Ideally you want it to be the case that at
training time, I think, you've already identified
judges that are fairly good.
And so ideally this is part of this project
is to kind of assess how good judges are,
prior to training.
And then during training you're giving the
feedback to the debaters.
So yeah, ideally some of the evaluation can
be kind of front loaded, which is what a lot
of this project would be.
Yeah, that does seem necessary as a casual
Facebook user.
I think the negative amplification is more
prominently on display oftentimes.
Or at least more concerning to people, yeah,
as, like, a possibility.
So, kind of a related question, how will you
crowdsource the millions of human interactions
that are needed to train AI across so many
different domains, without falling victim
to trolls, lowest common denominator, etc.?
The questioner cites the Microsoft Tay chatbot
that sort of went dark very quickly.
Yeah.
So the idea is you're not going to just be
like sourcing this from anyone.
So if you identify people that are either
good judges already, or you can train people
to be good judges, these are going to be the
pool of people that you're kind of using to
get this feedback from.
So, even if you've got a huge number of interactions,
ideally you're kind of sourcing and training
people to be really good at this.
And so you're not just being like, "Hey internet,
what do you think of this debate?"
But rather like, okay, we've got this set
of really great trained judges that we've
identified this wonderful mechanism to train
them to be good at this task.
And then you're getting lots of feedback from
that large pool of judges.
So it's not kind of sourced to kind of, to
anonymous people everywhere, but rather like
you're kind of interacting fairly closely
with this set of people, who are giving you
lots of-
But at some point, you do have to kind of
scale this out, right?
I mean in the bike example, it's like, there's
so many bikes in the world, and so many local
hills-
Yeah.
So, do you feel like you can get a solid enough
base that, that sort of... becomes not a problem?
Or, how does that...
Yeah, I think there's going to be a trade-off
where you need a lot of data, but ultimately
if it's not great, so if it is really biased,
for example, it's not clear that that additional
data is going to be helpful.
So if you get someone who is just like massively
cognitively biased, or biased against groups
of people, or something, or just like is it
going to be dishonest in their judgment?
This is not going to be like... it's not going
be good to get that additional data.
So you kind of want to scale it to the point
where you know you're still getting good information
back from the judges.
And that's why I think in part this project
is really important, because one thing that
social scientists can help us with is kind
of identifying how good people are.
So if you know that people are just generally
fairly good, this gives you a bigger pool
of people that you can appeal to.
And if you know that you can train people
to be really good, then this is like, again,
a bigger pool of people that you can appeal
to.
So yeah, it's like you do want to scale, but
you want to scale kind of within the limits
of still getting good information from people.
And so ideally this would do this mix of letting
us know how much we can scale, and also maybe
helping us to scale even more by making people
bear this quite unusual task of judging these
kind of debates.
So we are a little over time, we won't have
time to go through all the questions that
are coming in, but you can speak with Amanda
more at office hours immediately following
this talk, right when we're headed into break.
So let's just do one last question for this
session, which is how does your background
as a philosopher inform the work that you're
doing here?
Yeah.
I have, I guess, a background primarily in
formal ethics, which I think makes me sensitive
to some of the issues that we might be worried
about here going forward.
So you know, people think about things like
aggregating judgment for example.
Strangely I found that like, having backgrounds
in things like philosophy of science can be
weirdly helpful when it comes to thinking
about experiments to run.
But for the most part, I think that my work
has just been to kind of help prototype some
of this stuff.
I see the importance of it.
I kind of, I'm able to foresee some of the
kind of worries that people might have.
But for the most part I think we should just
try some of this stuff.
And I think that for that, it's really important
to have people with experimental backgrounds
in particular, so the ability to run experiments
and analyze that data.
And so that's why I would like to kind of
find people who are interested in doing that.
So I'd say philosophy's pretty useful for
some things, less useful for running social
science experiments than you may think.
Alright.
Well for more, you'll have to come to office
hours, which you can do immediately after
this session.
How about a round of applause for Amanda Askell? |
5ed435af-fe30-4729-9e68-c7e894fa4e68 | trentmkelly/LessWrong-43k | LessWrong | When and why should you use the Kelly criterion?
This is a dialogue made during the online dialogues party. Phil, River, and I (Garrett) talked about the Kelly criterion. I was under the mistaken impression that it would come up in finite games, even without discount factors. This turned out to be wrong! In such betting games, you always want to bet the maximal amount (assuming linear utility in money).
We then talked about how maybe you could save the criterion without bringing in log utility in money or discount rates in infinite games. The two conclusions were you could either have a finite utility function, or care intrinsically about not being broke/maximizing the probability of getting more money than other agents in your world.
The highlight in my opinion is me changing my mind about the finite games thing after using dynamic programming to try to show Phil to be wrong. The proof we came up with was both surprising and elegant to me.
Possible justifications for Kelly
Garrett Baker
So, I've seen some previous explanations of the kelly criterion on LessWrong, and they seem to fall into 3 clusters:
1. You do the kelly criterion because you have log-utility in money.
2. You do the kelly criterion because you have linear utility in money, but you can use current money to gain even more in the future.
3. You do the kelly criterion because you partially update on the market's position.
philh
Some background here for me is I've previously written https://www.lesswrong.com/posts/XnnfYrqaxqvirpxFX/on-kelly-and-altruism and https://www.lesswrong.com/posts/JAmyTWoukk8xzhE9n/ruining-an-expected-log-money-maximizer, they didn't get much karma and I don't know if that's "there's something wrong with them" or "they didn't get seen much" or what
philh
> You do the kelly criterion because you have log-utility in money.
This one I think is complicated; I'd say "you do a thing that turns out equivalent to the kelly criterion, but for simpler reasons than why the kelly criterion is derived"
philh
> You do the kelly |
f12e68f2-bcdc-4271-946b-8c2b0333eb53 | StampyAI/alignment-research-dataset/blogs | Blogs | Michie and overoptimism
*By Katja Grace, 12 January 2015*
We recently wrote about [Donald Michie’s](http://en.wikipedia.org/wiki/Donald_Michie) [survey](http://aiimpacts.wpengine.com/michie-survey/ "Michie Survey") on timelines to human-level AI. Michie’s survey is especially interesting because it was taken in 1972, which is three decades earlier than any other [surveys we know of](http://aiimpacts.wpengine.com/ai-timeline-surveys/ "AI Timeline Surveys") that ask about human-level AI.
Early AI predictions are renowned for being absurdly optimistic. And while the scientists in Michie’s survey had had a good decade and a half since the Dartmouth Conference to observe the lack of AI, they were still pretty early. Yet they don’t seem especially optimistic. Their predictions were so far in the future that almost three quarters of them still haven’t been demonstrated wrong.
And you might think computer scientists in the early 70s would be a bit more optimistic than contemporary forecasters, given that they hadn’t seen so many decades of research not produce human-level AI. But the median estimate they gave for how many years until AI was further out than those given in [almost all](http://aiimpacts.wpengine.com/ai-timeline-surveys/ "AI Timeline Surveys") surveys since (fifty years vs. thirty to forty). The survey doesn’t look like it was very granular – there appear to be only five options – so maybe a bunch of people would have said thirty-five years, and rounded up to fifty. Still, their median expectations don’t look like they were substantially more optimistic (in terms of time to AI) than present-day ones.
In terms of absolute dates, the Michie participants’ median fifty-year choice of 2022 is still considered [at least 10% likely](http://aiimpacts.wpengine.com/ai-timeline-surveys/ "AI Timeline Surveys") to give rise to human-level AI by recent survey participants, some four decades later.
It’s not like anyone said that all early predictions were embarrassingly optimistic though. Maybe Michie’s computer scientists were outliers? Perhaps, but if everyone else whose predictions we know of disagreed with them, it would be those others who were the outliers: Michie’s survey has sixty-three respondents, whereas [the MIRI dataset](http://lesswrong.com/lw/e79/ai_timeline_prediction_data/) contains only eleven other predictions made before 1980 (it looks like twelve, but the other interesting looking survey attributed to Firschein and Coles appears to be an accidental duplicate of the Michie survey, which Firschein and Coles mention in their paper). Sixty-three is more than all the predictions in the [MIRI dataset](http://lesswrong.com/lw/e79/ai_timeline_prediction_data/) until the 2005 Bainbridge survey.
AI researchers may have been extremely optimistic in the very early days (the [MIRI dataset](http://lesswrong.com/lw/e79/ai_timeline_prediction_data/) suggests this, though it consists of public statements, which tend to be more optimistic anyway). However, it doesn’t seem to have taken AI researchers long to move to something like contemporary views. It looks like they didn’t just predict ‘twenty years’ every year since [Dartmouth](http://en.wikipedia.org/wiki/Dartmouth_Conferences).
[](http://aiimpacts.wpengine.com/wp-content/uploads/2015/01/michietimelines-copy-1.jpg)Survey results, as shown in Michie’s paper [(pdf download)](https://saltworks.stanford.edu/assets/cf501kz5355.pdf).
Thanks to Luke Muehlhauser for [suggesting](http://lukemuehlhauser.com/some-studies-which-could-improve-our-strategic-picture-of-superintelligence/) this line of inquiry.
*(Image: “[Donald Michie teaching](http://commons.wikimedia.org/wiki/File:Donald_Michie_teaching.jpg#mediaviewer/File:Donald_Michie_teaching.jpg)” by Petermowforth)* |
50cadd34-0aa9-477f-99c5-9f411ac94220 | trentmkelly/LessWrong-43k | LessWrong | TAPs for Tutoring
Introduction
Scott Alexander:
> Architects. Teachers. Teachers of teachers, but the art of teaching teaching is much the same as the art of teaching. Three levels is enough. Though the levels have to mix. The teacher who trains the next architect must be a master both of teaching and of architecture.
Paraphrased for my own purposes:
> Tutors. Tutees. Tutors tutor tutees, but the art of being taught is much the same as the art of teaching. The levels have to mix. The tutor who tutors the tutee must be a master both of teaching and of learning.
Much has been written about the Feynman Technique as a method of learning that involves teaching. The main problem with the Feynman technique is that it takes too much time to ever actually execute in practice. Instead, I will describe two TAPs that are my 80/20 for both learning and teaching.
TAPs
TAP 1: Examples
* Trigger: Notice you made a claim with some amount of vagueness.
* Action: Start saying "For example, ..."
For example (see what I'm doing), I might say "regular exercise is good for you" and notice that "regular" and "good" are words with multiple possible interpretations, so I would follow it up with "For example, walking for 10 minutes a day makes you happier". Another example: I might say "functions map inputs to outputs, for example, the square function takes numbers and maps them to the square of that number, e.g. it would take 2 to 4, 3 to 9, etc." Notice that sometimes your example itself can have some vagueness, so the TAP fires recursively. This is the key TAP for teaching.
There is also a dual TAP for learning; trigger: notice someone else made a claim with some amount of vagueness, action: say "for example?". For example, someone might tell me that they "dislike mundane activities", to which I would say "for example?" They then might tell me that "scheduling Facebook events" is a mundane activity, giving me a much better understanding of what they had in mind when they said "mundane". This TA |
8d803171-ce9f-496b-acd5-c7c65ca80320 | trentmkelly/LessWrong-43k | LessWrong | Monthly Shorts 10/22
India?
Map of Indian poverty, raw data on page 136.
For how important poverty/wealth in those states are, you can look at
That data was drawn from Wikipedia, while the population map distortion was done with this program, which I suspect I will use myself one of these days. Comparing the two isn’t trivial, but you can certainly see that, for example, it’s pretty important that Bihar is poor and Kerala is not. OK, so I checked the List of Indian State and Union Territories by GDP per Capita just in case my assumptions were wrong, and wow they were. Kerala has a multidimensional poverty index a fifth that of Goa’s while Goa has twice the per capita GDP of Kerala. Goa on a PPP adjusted basis is still substantially poorer than the poorest American state, but it’s richer than Greece while Kerala is closer to pre-invasion Ukraine. Something interesting is going on. The answer seems to be “functional communism”? Unfortunately, limited time and many things to do.
Wikipedia
In 2022, Canada acquired a border with a second country. This came out of reading the surprisingly interesting Talk Page for the Wikipedia article on 2022. I was pinged because I’d made a small edit, but it’s fun to read Jim Michaels 2 argue for deleting almost everything, while other people try to defend smalltime filmmakers for inclusion. Normally I’m very strongly anti-deletionist, but when it comes to content on a year page I’m more sympathetic.
The edit that got me dragged in was changing a disambiguation link to correctly point towards the January 2022 coup d'état in Burkina Faso, as opposed to the 2022 coup: in January “the 2022 coup” was sufficient but by September there had been a second. You know you’re having problems when Russia calls for a return to a framework with “legitimacy”.
I do recommend reading Wikipedia pages for a year to get a sense of what is happening. It’s not the only thing, but if you want to be “well-rounded” in some sense, it’s a better starting point than most alter |
6fdff81c-26f2-4af8-9cc0-4bba86afd466 | trentmkelly/LessWrong-43k | LessWrong | Oracles, Informers, and Controllers
Epistemic Status: I think the axes are straightforward enough. I'm not particularly sure about the names, happy to hear other sorts of suggestions.
Scholarly Status: Push/pull systems have been discussed before. I'm sure there's other naming for this sort of thing in the engineering/design literature, but it probably hasn't been optimised as well to fit in with the terminology of Superintelligence. This was another fairly quickly written post.
This article builds on ideas from this one, but that one is not a prerequisite.
----------------------------------------
In Superintelligence, Nick Bostrom proposed the terminology of oracles, genies, and sovereigns. These metaphors are clear and have achieved some popularity. I recommend this post by Katja Grace for an overview and some further thinking.
This terminology might be incomplete. I think that oracles have a close parallel to genies, but there's no information-only parallel to sovereigns. Here, I propose two options.
Oracles
To reiterate from Superintelligence, an oracle is a system that answers questions. We have similar systems now, though they are weaker than what Bostrom considered. When you enter search queries into Google or many other search programs, you are using them as oracles. Pure functions don't modify the external world and are used to in-essence answer particular queries. Some consultants are asked to solve particular questions or problems. These all represent types of weak oracles.
Informers
In comparison to an oracle, an informer is a system that actively pushes select information to recipients. AI assistants are starting to veer into this territory. They suggest particular actions without being prompted.
Some sorts of prompts given by Siri include:
"Dinner with Josh looks important! Turn on Do Not Disturb until the end of this event!" - link
"Call in to Portfolio work Session. Event at 10:00 AM found in Calendar" - link
"Tell Mum Happy Birthday!" Birthday found in Contacts" - l |
fb40f832-8bd8-4f84-aea8-946ae414bec0 | StampyAI/alignment-research-dataset/blogs | Blogs | you can't simulate the universe from the beginning?
you can't simulate the universe from the beginning?
---------------------------------------------------
[QACI](qaci.html) and plausibly [PreDCA](predca.html) rely on a [true name](https://www.lesswrong.com/posts/FWvzwCDRgcjb9sigb/why-agent-foundations-an-overly-abstract-explanation) of phenomena in the real world using [solomonoff induction](https://www.lesswrong.com/posts/EL4HNa92Z95FKL9R2/a-semitechnical-introductory-dialogue-on-solomonoff-1), and thus talk about locating them in a theoretical giant computation of the universe, from the beginning. it's reasonable to be concerned that there isn't enough compute for an aligned AI to actually do this. however, i have two responses:
* *isn't* there enough compute? supposedly, our past lightcone is a lot smaller than our future lightcone, and quantum computers seem to work. this is evidence that we *can*, at least in theory, build within our future lightcone a quantum computer simulating our past lightcone. the major hurdle here would be "finding out" a fully explanatory "initial seed" of the universe, which *could* take exponential time, but also could maybe not.
* we don't *need* to simulate past lightcone. if you ask me what my neighbor was thinking yesterday at noon, the answer is that i don't know! the world might be way too complex to figure that out without simulating it and scanning his brain. however, i have a *reasonable distribution over guesses*. he was more likely to think about french things than korean things. he was more likely to think about his family than my family. et cetera. an aligned superintelligence can hold an ever increasingly refined distribution of guesses, and then maximize the expected utility of utility functions corresponding to each guess. |
394a5cbf-83c4-4b53-bb28-c26d30d516e5 | trentmkelly/LessWrong-43k | LessWrong | Interlude: Agents as Automobiles
I ended up writing a long rant about agency in my review of Joe Carlsmith’s report on x-risk from power-seeking AI. I’ve polished the rant a bit and posted it into this sequence. The central analogy between APS-AI and self-propelled machines (“Auto-mobiles”) is a fun one, and I suspect the analogy runs much deeper than I’ve explored so far.
For context, the question being discussed is whether we should “expect incentives to push relevant actors to build agentic planning and strategically aware systems [APS systems] in particular, once doing is possible and financially feasible.”
Joe says 80% yes, 20% no:
“The 20% on false, here, comes centrally from the possibility that the combination of agentic planning and strategic awareness isn’t actually that useful or necessary for many tasks -- including tasks that intuitively seem like they would require it (I’m wary, here, of relying too heavily on my “of course task X requires Y” intuitions). For example, perhaps such tasks will mostly be performed using collections of modular/highly specialized systems that don’t together constitute an APS system; and/or using neural networks that aren’t, in the predictively relevant sense sketched in 2.1.2-3, agentic planning and strategically aware. (To be clear: I expect non-APS systems to play a key role in the economy regardless; in the scenarios where (2) is false, though, they’re basically the only game in town.)”
I agree that “of course X requires Y” intuitions have been wrong in the past and also that evidence from how nature solved the problem in humans and nonhuman animals will not necessarily generalize to artificial intelligence. However:
1. Beware isolated demands for rigor. Imagine someone in 1950 saying “Some people thought battleships would beat carriers. Others thought that the entire war would be won from the air. Predicting the future is hard; we shouldn’t be confident. Therefore, we shouldn’t assign more than 90% credence to the claim that powerful, portable co |
a5dd6c33-d7b0-4293-bfdb-10a4fcf46b1b | trentmkelly/LessWrong-43k | LessWrong | Does functionalism imply dualism?
This post follows on from Personal research update, and is followed by State your physical explanation of experienced color.
In a recent post, I claimed that functionalism about consciousness implies dualism. Since most functionalists think their philosophy is an alternative to dualism, I'd better present an argument.
But before I go further, I'll link to orthonormal's series on dissolving the problem of "Mary's Room": Seeing Red: Dissolving Mary's Room and Qualia, A Study of Scarlet: The Conscious Mental Graph, Nature: Red in Truth, and Qualia. Mary's Room is one of many thought experiments bandied about by philosophers in their attempts to say whether or not colors (and other qualia) are a problem for materialism, and orthonormal presents a computational attempt to get around the problem which is a good representative of the functionalist style of thought. I won't have anything to say about those articles at this stage (maybe in comments), but they can serve as an example of what I'm talking about.
Now, though it may antagonize some people, I think it is best to start off by stating my position plainly and bluntly, rather than starting with a neutral discussion of what functionalism is and how it works, and then seeking to work my way from there to the unpopular conclusion. I will stick to the example of color to make my points - apologies to blind and colorblind readers.
My fundamental thesis is that color manifestly does exist - there are such things as shades of green, shades of red, etc - and that it manifestly does not exist in any standard sort of physical ontology. In an arrangement of point particles in space, there are no shades of green present. This is obviously true, and it's equally obvious for more complicated ontologies like fields, geometries, wavefunction multiverses, and so on. It's even part of the history of physics; even Galileo distinguished between primary qualities like location and shape, and secondary qualities like color. Primary qu |
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