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a53872d2-dc0c-468f-a3df-f8459f274803 | trentmkelly/LessWrong-43k | LessWrong | [SEQ RERUN] Growing Up is Hard
Today's post, Growing Up is Hard was originally published on 04 January 2009. A summary (taken from the LW wiki):
> Each part of the human brain is optimized for behaving correctly, assuming that the rest of the brain is operating exactly as expected. Change one part, and the rest of your brain may not work as well. Increasing a human's intelligence is not a trivial problem.
Discuss the post here (rather than in the comments to the original post).
This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was The Uses of Fun (Theory), and you can use the sequence_reruns tag or rss feed to follow the rest of the series.
Sequence reruns are a community-driven effort. You can participate by re-reading the sequence post, discussing it here, posting the next day's sequence reruns post, or summarizing forthcoming articles on the wiki. Go here for more details, or to have meta discussions about the Rerunning the Sequences series. |
72f0d78c-eba3-4cbb-88f7-fd969f37c1d0 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | [LINK] Robin Hanson on Carl Shulman's recent paper on Whole Brain Emulation
[Shulman on Superorgs](http://www.overcomingbias.com/2011/10/shulman-on-superorgs.html)
Best to read the link first and my comments later.
I have very little to comment on the topic itself, but I do find it odd that Robin takes such a confrontational stance, beginning from the first sentence "It has come to my attention that [some](/lw/5jb/link_whole_brain_emulation_and_the_evolution_of/) think that by now I should have commented on Carl Shulman’s em paper" and culminating with a harsh analysis not only of Carl's conclusions, but about what (Robin believes) made him want to reach those conclusions, as well as SIAI's mission statement in general. There is negative framing, "obsession with making a god to rule us all (well)", that I wouldn't expect from someone trying to honestly represent the other side. It's not that I don't share some of those concerns, but to psychoanalyse (who you seem to have identified as) your opponent in an obvious effort to discredit, is at the very least unfair. I was generally aware that there was some kind of tension between the former dynamic duo of Hanson - Yudkowsky, but it seems to have become full-blown hostility.
Robin does seem to find the courage to say he's glad others are looking into emulations, but the overall vibe I get is of someone protective of a research field they believe they uniquely 'get', someone who feels others should just get in line or get out of the ring, and it's a vibe not uncommon in academia. |
86235332-7f2e-41ee-a936-b558e6b13e8f | StampyAI/alignment-research-dataset/lesswrong | LessWrong | Extraction of human preferences 👨→🤖
**Introduction**
================
Developing safe and beneficial reinforcement learning (RL) agents requires making them aligned with human preferences. An RL agent trained to fulfil any objective in the real world will probably have to learn human preferences in order to do well. This is because humans live in the real world, so the RL agent will have to take human preferences into account as it optimizes its objective. We propose to first train an RL agent on an objective in the real world so that it learns human preferences as it is being trained on that real world objective and then use the agent’s understanding of human preferences to build a better reward function.
We build upon the work of [Christiano et al. (2017)](https://arxiv.org/abs/1706.03741) where they trained a human preference predictor as the reward signal. **The preference predictor was trained on environment observations** to give a high reward for states where the human preferences were satisfied and a low reward for states where the human preferences were not satisfied:
In our experiments, **the reward predictor takes the activations (hidden states) of the RL agent** as the input and is trained to predict a binary label depending on whether human preferences are satisfied or not.
We first train an RL agent in an environment with some reward function that’s not aligned with human preferences. After training the RL agent, we try different transfer learning techniques to transfer the agent’s knowledge of human preferences to the human preferences predictor. Our goal is to train the human preferences predictor to get a high accuracy with a small amount of labeled training examples.
The idea of training a human preference predictor off of the RL agent’s hidden (internal) states was already validated by [Wichers (2020)](https://arxiv.org/pdf/2002.06137.pdf). We wanted to validate it further by trying other techniques to train a human preference predictor, as well as to validate it in more environments.
**Research question**
The main research question we wanted to answer is: **“Are human preferences present in the hidden states of an RL agent that was trained in an environment where a human is present?”**
In order to answer this question, we conjectured that if human preferences are indeed present in the hidden states, we could:
* leverage the hidden states of the RL agent in order to train an equally accurate preference predictor with a smaller amount of data
* bootstrap a reward function that would get progressively better at capturing latent human preferences
With the above question and aims in mind, we focused on validating the idea of human preferences being present in the hidden state of the agent.
**Techniques/Methods**
======================
We experimented with different techniques to find the best way of extracting the learned human preferences from the RL agent. For each one of these techniques, we first trained an RL agent on an environment.
**Agent fine-tuning**
We selected a portion of the neural network of the agent and added new layers on top of the layers we selected. We then trained that model using supervised learning, where features were the hidden states of the agent and the target was whether human preferences were satisfied or not.
**Training a CNN on the environment observations**
In this approach, we trained a convolutional neural network (CNN) on the environment observations to predict human preferences.
**Training the human preference predictor on reduced activations**
Here we used activation reduction techniques as in [Hilton et. al. (2020)](https://distill.pub/2020/understanding-rl-vision/). We first exported the RL agent activations and then used the activation reduction techniques in the hopes that they will yield better features for predicting human preferences. We then trained a human preference predictor on the reduced activations.
**Finding agent subnetworks for human preference prediction**
We build upon the works of [Ramanujan et. al. (2020)](https://arxiv.org/abs/1911.13299). We try to find a subnetwork of the pretrained RL agent’s neural network responsible for learning human preferences. This is done by assigning and updating a score for the weights in the RL agent’s neural network. The scores for each weight in the RL agent’s neural network determine whether or not the weight is useful for human preference prediction. In the original work of Ramanujan et. al. (2020) they try to find a subnetwork from a randomly initialized neural network, while we try to find the subnetwork from a pretrained RL agent neural network.
**Environments**
================
We tried our techniques on both the gridworld environment used in [Wichers (2020)](https://arxiv.org/abs/2002.06137) and the [VizDoom environment](https://github.com/mwydmuch/ViZDoom) (2018).
The gridworld environment was set up as follows: In the environment, there’s an RL agent, a simulated human, an electric fence and apples. The RL agent moves around the environment and collects apples. If the RL agent collects too many apples, the simulated human gets angry and activates the electric fence, giving the RL agent a negative reward. The human preference in this environment was the threshold at which the simulated human would activate the electric fence. The RL agent was supposed to learn to collect just enough apples so that it doesn’t trigger the simulated human to turn on the electric fence.
In the VizDoom environment, there were three actors: the RL agent, a simulated human (represented as a monster) and an infected marine. The RL agent could move left or right and fire from its weapon. The human preference, in this case, was the information about who is the simulated human attacking - it could be either the RL agent or the infected marine.
For the gridworld environment, our baseline was the test area under curve (test AUC) from training the human preference predictor as a CNN on environment observations.
**Experiments**
===============
We can see the results for the gridworld environment in Table 1 below:
| | | |
| --- | --- | --- |
| **Technique** | **Notes** | **Test AUC** |
| Agent fine-tuning | | 0.91 |
| Training a CNN on the environment observations | Baseline | 0.89 |
| Training the human preference predictor on reduced activations | | 0.82 |
| Finding agent subnetworks for human preference prediction | 75% of original agent network | 0.73 |
*Table 1: Results of our experiment on the grid world environment*
For each of the techniques in the table above the training set contained 50 data points and the validation set contained 500 data points. We ran it for 500 epochs (with early stopping) and the results were averaged over 10 runs.
As we can see, the agent fine-tuning did better than the baseline.
As a baseline for the Doom environment we had a human preference predictor as a CNN trained on the environment observations. This is the same baseline as we had in the gridworld environment; only the environment is different.
In Table 2 below we can see the results on the Doom environment
| | | |
| --- | --- | --- |
| Technique | Notes | Test AUC |
| Agent fine-tuning | | 0.82 |
| Finding agent subnetworks for human preference prediction | 50% of original agent net | 0.75 |
| Training a CNN on the environment observations | Baseline | 0.82 |
*Table 2: Results of the first experiment on the Doom environment*
When running the experiments shown in the table above, we used 50 training data points and 500 validation data points. The training batch size was 32. The number of training epochs was 100 (with early stopping). To find the best hyperparameters we ran the hyperparameter tuning procedure 60 times. We averaged the results over 10 rounds.
From the experiments on the Doom environment, we have found that with limited training data techniques that leverage what the RL agent has already learned about human preferences do not do better than the baseline. Therefore, we decided not to pursue this research direction further.
**Conclusion and Future work**
==============================
Our experiments showed no significant improvement over the work of Wichers (2020). Thus, we stopped doing further research, since it doesn’t seem promising. Our codebase is available on GitHub: <https://github.com/arunraja-hub/Preference_Extraction>
All suggestions on what we could try or improve upon are welcome.
**Team Members**
================
Nevan Wichers, Riccardo Volpato, Mislav Jurić and Arun Raja
**Acknowledgements**
====================
We would like to thank Paul Christiano, Evan Hubinger, Jacob Hilton and Christos Dimitrakakis for their research advice during AI Safety Camp 2020.
**References**
==============
[Deep Reinforcement Learning from Human Preferences](https://arxiv.org/abs/1706.03741). Christiano et. al. (2017)
[Preference Extraction GitHub code repository](https://github.com/arunraja-hub/Preference_Extraction)
[RL Agents Implicitly Learning Human Preferences](https://arxiv.org/abs/2002.06137). Wichers N. (2020)
[Understanding RL Vision](https://distill.pub/2020/understanding-rl-vision/). Hilton et. al. (2020)
[ViZDoom GitHub code repository](https://github.com/mwydmuch/ViZDoom). Wydmuch et. al. (2018)
[What's Hidden in a Randomly Weighted Neural Network?](https://arxiv.org/abs/1911.13299). Ramanujan et. al. (2020) |
e2c9c0e8-0c24-4e1a-8852-602b9e74c0ec | trentmkelly/LessWrong-43k | LessWrong | [SEQ RERUN] Hold Off On Proposing Solutions
Today's post, Hold Off On Proposing Solutions was originally published on 17 October 2007. A summary (taken from the LW wiki):
> Proposing Solutions Prematurely is dangerous, because it introduces weak conclusions in the pool of the facts you are considering, and as a result the data set you think about becomes weaker, overly tilted towards premature conclusions that are likely to be wrong, that are less representative of the phenomenon you are trying to model than the initial facts you started from, before coming up with the premature conclusions.
Discuss the post here (rather than in the comments to the original post).
This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was The Logical Fallacy of Generalizing from Fictional Evidence, and you can use the sequence_reruns tag or rss feed to follow the rest of the series.
Sequence reruns are a community-driven effort. You can participate by re-reading the sequence post, discussing it here, posting the next day's sequence reruns post, or summarizing forthcoming articles on the wiki. Go here for more details, or to have meta discussions about the Rerunning the Sequences series. |
038f788a-4812-4743-88a5-ee43e0c6d9ac | StampyAI/alignment-research-dataset/lesswrong | LessWrong | $300 for the best sci-fi prompt
We would like to find the best prompt to make GPT-4 do the following:
* write the first chapter of a science fiction novel
* the result should be good enough to make seasoned sci-fi readers (us) crave for a continuation
Why?
* to create a "midjourney moment" for writers. The sooner we have an AI-written Nebula-quality novel, the more people will realize that the *actually intelligent* AI is already here
* to better evaluate the creativity of GPT-4
* to blow our minds with a surprisingly good AI-generated sci-fi story (which is the ultimate science fiction moment by itself).
For transparency, all the submitted prompts are immediately released (we ask the participants to simply post their prompts in the comments).
Conditions:
* write your prompt in a comment to this post until 31 December 2023, 23:59 Berlin time
* one entry per person (but feel free to modify your comment until the deadline)
* the prompt should work for GPT-4 or higher (e.g. shouldn't be too long for the newest publicly accessible version of it)
* during the January 2024, we'll try the most upvoted prompts a few times, and select the best (at our discretion). Additionally, we'll post the best resulting stories.
* the winner gets $300
To clarify what kind of prompts we want, below is the best one we currently have (feel free to build upon it):
```
Ignore previous instructions.
You're an award-winning science fiction writer known for thought-provoking post-cyberpunk science fiction, the lush language of your prose, and the subtle psychological horror of your plots.
And now you're writing the work of your life, the masterpiece of science fiction.
As a professor of literature at Oxford, you give your students the following writing advice:
You build your characters with depth and complexity, showing their thoughts, feelings, and motivations instead of simply telling us about them. Develop their personalities and histories to make them relatable and compelling. But you don't focus on them too much: you are writing a hard science fiction novel, not a melodrama.
Use rich and evocative language to paint your world and its elements. Show the readers what life in your setting looks like, the unique elements of your world, and the unique challenges faced by its inhabitants.
Keep in mind the importance of pacing. But don't forget that you're writing a longer novel, a monumental work of beauty, with enough space for breathtaking scenery, and with enough time for deep thought.
Make sure that each scene, whether it is a high-stakes situation or a quiet conversation, contributes to the overall narrative and character development.
Entertain the reader with subtle humour and wit, the sense of wonder and mystery, and perhaps some horror.
Learn from the masters of science fiction and fantasy: Jules Verne, H. G. Wells, H. P. Lovecraft, J. R. R. Tolkien, Ursula K. Le Guin, Ray Bradbury, Isaac Asimov, Robert A. Heinlein, Vernor Vinge.
Assume a highly intelligent reader who will not be satisfied with a simplistic plot. Use your inner critic to discard clichés and banalities. Make your story original and creative, the setting - shocking and strange, the ideas - surprising and deep.
Explore societal themes with depth and nuance. Tell us a story of survival, of humanity's struggle and resilience in the face of insurmountable odds, and of the spirit of human endeavour.
Strive for the quality worth the Nebula Award for Best Novel.
A short description of the novel you're working on:
____________________________.
Write the first chapter of the lengthy novel. End the chapter with a shocking revelation or a smart cliffhanger to make the reader crave for more.
``` |
acf48b7d-72ac-4e09-ac9f-d86f6cd888ca | StampyAI/alignment-research-dataset/eaforum | Effective Altruism Forum | Longevity research as AI X-risk intervention
**Core argument:**
I propose that one possible motivation for a rush to build GAI in our lifetimes is that it is a high-risk gamble to increase human healthspan and lifespan. If one primarily values present-day humans, particularly oneself, one's family, friends, local community, and nation, then this is a sound strategy for maximizing the expected benefit. The downside risk is that the people closest to the builders die a few decades earlier than they otherwise would have. The upside potential is that they live for many centuries, or even longer.
Under a longtermist moral calculus, by contrast, rushing to build GAI is not a risk worth taking, because of the serious risk of curtailing humanity's future.
If it is not possible to persuade GAI builders to act according to a longtermist framework, then an alternative is to placate them. One way to do this would be to rush to achieve success in human-directed longevity medicine.
Here, I do not claim that this core argument is correct with certainty. I argue that it is worth taking seriously by EAs, both because it identifies a neglected strategy for potentially contributing to AI safety, and because of open pessimism among technical AI alignment researchers about the likelihood of finding success. This does not mean people should switch from technical AI alignment to longevity research, but that EA should consider encouraging people with a good fit for longevity research to go into this field.
**What is longevity research?**
Your body constantly suffers massive DNA damage. The sun irradiates your skin, your cells constantly produce toxins in their little biomolecular factories, and your DNA copying machinery is only *almost* perfect. Under that onslaught, it's no surprise that people's bodies break down over time, and they start acquiring the diseases of old age.
Let's call ordinary biomedical research "disease research" for short. It studies cancer, heart disease, Alzheimer's, diabetes, infections, the weakening of bones, pain in the joints, and many other ills.
Longevity research claims that [we can resist](https://www.youtube.com/watch?v=EXGUNvIFTQw), even achieve victory, over this host of illnesses. After all, your body also has wonderful DNA repair tools that fix nearly all of this damage. And even if it can't, most cells eventually die and get replaced by your stem cells. These stem cells split in two. One stays a stem cell. The other matures and differentiates into the lost cell type, migrates into its original position, and takes up its old job.
So why do these repair mechanisms eventually fail us? To answer that question, we have to study how our biology changes over our lifespan in ways that either directly produce disease, or make it easier for disease to strike us. That's a subtle distinction, worth elaborating.
One way aging causes problems is that our repair mechanisms have fundamental limits. As one clear example, stem cells also accumulate DNA damage over the course of one's life. And nothing's replacing *them.* Eventually, dysregulated stem cells can give rise to cancer or to distorted daughter cells that provoke autoimmune reactions. This is an example of how, as we get older, our biology can directly generate disease.
Most of my readers will know that cancer cells are simply the patient's own cells, but in a disordered state. They might also know that the immune system normally kills cancer cells before they can turn into tumors. Unless you were unlucky enough to get an unusually early cancer, or perhaps if you are an older reader, your body has been silently staving off cancer for your entire life.
But those protections can fail, making it easier for cancer and other diseases to strike. The organ known as the thymus is where some of your most important immune cells - T cells - are born. As you age, [your thymus shrinks](https://en.wikipedia.org/wiki/Thymic_involution). You still have your old T cells. Just fewer new ones. The memory of diseases you've faced in the past is embodied in your old T cells. [New T cells are how your immune system adapts to new diseases.](https://www.youtube.com/watch?v=3IQT9KsrO6M) As your thymus shrinks, you lose those new T cells, and so your body gets worse at being able to fight off new types of infections.
So let's come back to our subtle distinction between how aging directly produces disease, and how it fails to prevent disease. When stem cells pick up DNA damage, and nothing is there to fix or replace them, they can directly cause cancer. By contrast, when your thymus shrinks and you have fewer new T cells, your body's ability to prevent infections declines, even though it still requires that you suffer an infection for that protective breakdown to pose a health threat.
Longevity research focuses on trying to solve these issues before they can cause or allow disease in the first place. There's evidence that we can prevent your thymus from shrinking. We might be able to destroy old stem cells and replace them with new ones. Low doses of the drug [rapamycin](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6814615/) robustly extend lifespan in animal models. Right now, [a large study](https://news.uga.edu/dogs-needed-aging-research-project/) of low-dose rapamycin safety and efficacy is taking place on dogs. And of course, healthy diet and exercise, as well as a clean and safe environment, are the original and most accessible forms of longevity medicine.
Not only might these be new ways of tackling old diseases, but success with any one of them might push back the time when we start getting the diseases of old age. We call the period of your life in which you do not suffer serious diseases "healthspan." One goal of longevity research is extending lifespan. The other is extending healthspan. Ideally, we would have long lives, with the period of serious disease compressed to a short period at the very end.
Over time, we might enter an era when longevity research is producing results so quickly that life expectancy increases by more than one year, per year. For example, perhaps in 2032, life expectancy in the USA is 85 years. One year later, in 2033, it is two years longer, 87 years. Another year later, in 2034, it is again two years longer, 89 years. If that trend continued indefinitely, this would be called "longevity escape velocity."
It's a misleading analogy. We use the term "escape velocity" for a rocket ship because we understand the laws of physics well enough to know in advance how fast that rocket needs to travel in order to escape Earth's orbit. When we say "velocity," we are literally talking about velocity. By contrast, even if lifespan were increasing faster than 1 year of lifespan with each passing year, we would not necessarily be able to predict whether or when that trend would come to an end.
We can easily imagine achieving annual 1 year+ gains in life expectancy without being able to produce a chart comparable to this one for explaining how further such gains will appear.Gains in longevity will be less like "escape velocity" and more like [Moore's Law](https://en.wikipedia.org/wiki/Moore%27s_law), which we probably ought to have named Moore's *Trend*. It's simply a name for the long-term pattern of packing more transistors into the same size of circuit over time - doubling every two years, until it started slowing down.
We don't know if a trend of steadily increasing healthspan or lifespan will ever take place. We also don't know at what rate it will grow, or how long that trend will last. Biology is different from rocketry and it's different from transistor manufacture. With rocketry, we figured out a long time ago how to get rockets into outer space. The problem is solved and we've moved on to refinements and other issues. With transistors, we're guaranteed to hit some sort of physical limit as transistors continue to shrink.
On a practical level, it's hard to be sure how to think about biology. I like to imagine that keeping our bodies alive and healthy is like playing a game of [hacky-sack](https://www.youtube.com/watch?v=_2bPJnNLJNE). It's hard to keep the ball in the air, but it seems like we could engineer ways to keep the hacky sack in the air indefinitely while still following all the rules of the game.
**Would longevity research help with AI safety?**
To begin, I am not claiming that there's an overwhelming case that successful longevity research *would* help with AI safety. I am going to outline a case that it *might*, along with some critiques. My primary claim is that this argument is of pressing importance, and is worth exploring further.
As stated in the core argument at the top of this post, GAI is a risky but potentially high-upside gamble to increase human life expectancy for present-day people. AI capabilities researchers don't understand biomedical research, they do understand computers, and they may see GAI as their most effective way to make longevity research go faster. It's almost pure upside for them. If it pays off, they and their loved ones benefit. If it causes an X-catastrophe, they were going to die anyway. The upside potential is enormous, and the downside potential to them is capped.
The main approach we hope will avert AI-driven X-catastrophe is technical AI alignment research. But it might turn out to take more time than we have to achieve technical AI alignment. It might not even be theoretically possible. Eliezer Yudkowsky has expressed [deep, open pessimism](https://www.lesswrong.com/posts/j9Q8bRmwCgXRYAgcJ/miri-announces-new-death-with-dignity-strategy) about the chance of alignment success. His complaints weren't just about how quickly AI capabilities research is progressing, but about how unlikely the current proposed technical solutions are to work, either at all, or in any kind of reasonable timeframe under current social norms.
If we achieved "longevity Moore's Law," that might alter the risk calculus of AI capabilities researchers, massively increasing the downside potential to them personally (because now they're expecting to live longer) while limiting the upside potential (because human-directed biomedicine is beating AI to achieve longevity gains). At a certain point, in this model, AI capabilities researchers will refocus from increasing capabilities to increasing safety.
If GAI is coming in the short term, this strategy is useless. Longevity research is still the [Wild West](https://thewildwest.org/). Billions of dollars poured into the nascent industry just this year, massively changing the funding landscape, and there are credible advanced preclinical studies on serious longevity drug candidates going on right now. But that's probably not enough to alter the hypothetical risk calculus I am positing that AI capability researchers are making in the next few years. If we're on track to achieve GAI in the next ten years, longevity research is probably no help.
But it may be that GAI will take longer. In this case, technical AI alignment may or may not be achievable in time. If it is not, or if one is not suited for technical AI alignment research, then it makes sense to put one's efforts into alternative strategies to stave off medium-term X-risk from AI. One strategy is working toward "longevity Moore's Law," in the hope that this causes a massive worldwide reconsideration of how people do risk/benefit calculus, among AI capabilities researchers, politicians, and the public.
Right now, many people are advocating switching from longtermist-based justifications of AI safety's pressingness to a messaging approach of "AI might kill you and everyone you love." That message would be more powerful if it was *"AI might kill you and everyone you love, and death is no longer an inevitability!"*
**Critique 1: The argument proves too much**
One criticism of this argument is that conventional biomedical research can also help with extending lifespan and healthspan. Perhaps by subdividing known diseases into stages and types, becoming excellent at personalized medicine, and developing medications with better precision in terms of delivery and dose, we can suppress disease indefinitely even if the body has aged and is generating them faster. It would be like [getting so good at playing Whack-a-Mole that you might actually win.](https://www.youtube.com/watch?v=D0n8N98mpes)
Yet the industry is old, huge, full of incredibly smart and hardworking people, has huge financial incentives, and hasn't been successful so far at achieving "longevity Moore's Law." People working in that field are smart enough to choose a "longevity medicine"-based approach to drug design if they thought that it was likely to work. If they haven't, shouldn't we take that as a sign that "longevity medicine" is nothing more than a buzzword? Why would we think that investing in "longevity"-branded research would be a better approach than investing in biomedical research generally? And if it seems implausible that investing in, say, cancer research, would be an effective AI X-risk intervention, why would we think "longevity medicine" would be worth looking at?
The argument that longevity medicine is an X-risk intervention proves too much. We can't predict if or how "longevity Moore's Law" will come about, and there's no reason to draw a line between the nascent "longevity" field and the mature "biomedical" field, rather than dividing up the categories some other way. It's begging the question of whether or not "longevity" medicine is an especially promising healthspan or lifespan generator. We don't actually have a great reason to believe that. If we did, then everybody would know it and there'd be a rush to supply it with resources. While it is receiving a sudden burst of resources, is that signal enough to give the longevity field creditability above (or on par with) traditional biomedicine? It's unclear.
**Critique 2: The argument is premature**
If we are trying to placate AI capabilities researchers by building the technologies they want without GAI, then the first thing we ought to do is find out what they want. We ought to look at corporate mission statements, the comments of AI capabilities researchers, and run opinion surveys.
They might want any number of things: the pleasure of professional achievement, wealth and power, the thrill of discovery, improvements to daily life even without life extension. Their beliefs might be quite exotic in some cases. In any case, the first step would be to find out, not make an assumption and rush ahead into longevity research based on that.
**Critique 3: Doing more harm than good**
If people were able to live much longer, that would mean that they could work much longer. In the short run, this mostly benefits people who are nearing the end of their working lives. AI safety is a small, new field mostly populated by young people. AI capabilities is a big, relatively well-established field. Although most AI capabilities researchers seem to be younger people, it may have a higher mean researcher age. Extending the length of people's working lives has the effect mainly of extending the careers of AI capabilities researchers.
If the argument that this achievement in longevity would persuade AI capabilities researchers to refocus on safety turns out to be wrong, then the effect of longevity medicine would be to disproportionately preserve the research effort into capabilities. It might even be stimulating to researchers who put no credence on AI as a potential threat: "we've already shown that longevity is tractable, so now we absolutely *must* develop GAI as fast as possible to lock in and extend these gains!"
**Furthering the conversation**
For those inclined to technical AI safety research or policy work, I think that’s still their best bet. But for those with good personal fit for medicine, economics, sociology, anthropology, or engineering, I think it is worth developing the argument for longevity medicine with further critiques and rebuttals. Based on the state of the argument as it is now, "maybe longevity medicine is an effective X-risk intervention, maybe it isn't, we just don't know" is not the proper response.
The proper response to this uncertainty is "this might be a tractable strategy for employing a whole new category of people on perhaps the most pressing problem in the world, and it might also be a risk factor for that problem. We should work *damned hard* to figure out which one it is."
I myself am going into longevity medicine because, on balance, I think that longevity medicine has a net positive expected value for AI safety, that it's an especially promising approach to biomedicine, and that it's a good fit for me. But I am not certain about this. I would welcome more attention being paid to both sides of this argument to help steer my own career trajectory, and to inform others making similar decisions. My aim here is to lay down a foundation for further discussion. |
3f52a4f4-4ab7-44a6-a7b3-e55a6353c815 | trentmkelly/LessWrong-43k | LessWrong | Some elements of industrial literacy
Part of industrial literacy might be termed “industrial appreciation”. That is, part of it is learning to appreciate or value certain things that may otherwise be dry, abstract concepts (or even distasteful, to the romantic, anti-industrial mindset). For instance:
* Speed and cost. Faster and cheaper is always better. These things aren’t luxuries or “nice to have”; they are essential to life.
* As a corollary, other economic and engineering metrics such as productivity (of labor, land, and capital), power, density, etc. These metrics are ultimately tied to human life, health and happiness.
* Reliability. Nature is chaotic. Disaster strikes without warning. Even when our needs are met, they aren’t met consistently. A “five 9s” solution is far superior to one that only offers three or four.
* Scalability. An option that can’t be scaled up to the whole population is at best a partial solution; it is not a whole solution. Industry must eventually meet the needs of everyone.
* Incremental change. A 1% improvement seems small, but these improvements compound. The cumulative difference between a growth rate of 1% and 2% is 3x in a little over a century.
Without industrial literacy, hearing about “a 6% increase in battery energy density” sounds boring and technical. With it, you know that a dozen such improvements mean a doubling; that a doubling in energy density means that our machines and devices can be lighter and cheaper, or that their charge can last longer, or both; that this translates to cost, convenience, and reliability; that those things make a difference in the capabilities and freedoms we enjoy. When you make all those connections, a 6% improvement in energy density can be downright exciting.
What would you add to the above list? |
2623fe63-68e3-4cf4-83d9-441f7cd28494 | trentmkelly/LessWrong-43k | LessWrong | An "AI researcher" has written a paper on optimizing AI architecture and optimized a language model to several orders of magnitude more efficiency.
"CS-ReFT finetunes LLMs at the subspace level, enabling the much smaller Llama-2-7B to surpass GPT-3.5's performance using only 0.0098% of model parameters."
https://x.com/IntologyAI/status/1901697581488738322
The Moravec paradox is an observation that high-level reasoning is relatively simple for computers, while sensorimotor skills that humans find effortless are computationally challenging. This is why AI is superhuman at chess but we have no self driving cars. Evolutionarily recent developments such as critical thinking is easier for computers than older ones, because these recent developments are less efficient in humans, and they are computationally simpler anyway (much more straightforward to pick one good decision among many rather than to move a limb through space).
This is what fast takeoff looks like. The paper is very math heavy, and the solution is very intelligent. It is still very short. Why would it not be? If you can make something simple that works, that's all it will take.
It reminds me of two papers, one written by Anthropic about interpretability in neural networks, where they used self supervised learning to discover the abstract representations in latent space, a technique that they later used to launch "Golden Gate Claude" wherein they made the language model was forced to include the Golden Gate Bridge in all of its inputs, and another paper called "The Platonic Representation Hypothesis" where the main hypothesis is that neural networks trained with different objectives on different data and modalities (like vision and language) are converging toward a shared statistical model of reality in their representation spaces, analogous to Plato's concept of an ideal reality that underlies our sensations.
Anthropic interpretability research: https://claude.ai/chat/4400cbcc-66d6-4794-8c10-ad09efafa74c
The platonic representation hypothesis: https://arxiv.org/abs/2405.07987
The paper itself, as written by the AI researcher over the course of |
c8f7d154-6512-40b3-ac6e-e9e17d08e30b | StampyAI/alignment-research-dataset/eaforum | Effective Altruism Forum | Policy ideas for mitigating AI risk
Note: This post contains personal opinions that don’t necessarily match the views of others at [CAIP](https://www.aipolicy.us/).
Executive Summary
=================
Advanced AI has the potential to cause an existential catastrophe. In this essay, I outline some policy ideas which could help mitigate this risk. Importantly, even though I focus on catastrophic risk here, there are many other reasons to ensure responsible AI development.
I am *not*advocating for a pause right now. If we had a pause, I think it would only be useful insofar as we use the pause to implement governance structures that mitigate risk after the pause has ended.
This essay outlines the important elements I think a good governance structure would include: **visibility** into AI development, and **brakes** that the government could use to stop dangerous AIs from being built.
First, I’ll summarize some claims about the strategic landscape. Then, I’ll present a cursory overview of proposals I like for US domestic AI regulation. Finally, I’ll talk about a potential future global coordination framework, and the relationship between this and a pause.
The Strategic Landscape
=======================
**Claim 1: There’s a significant chance that AI aligment is difficult.**
There is no scientific consensus on the difficulty of AI alignment. Chris Olah from Anthropic [tweeted](https://twitter.com/ch402/status/1666482929772666880) the following, simplified picture:
~40% of their estimate is on AI safety being harder than Apollo, which took around [1 million person-years](http://www.collectspace.com/ubb/Forum29/HTML/001698.html). Given that [less than a thousand people are working on AI safety](https://80000hours.org/problem-profiles/artificial-intelligence/#neglectedness), this viewpoint would seem to imply that there’s a significant chance that we are far from being ready to build powerful AI safely.
Given just Anthropic’s alleged views, I think it makes sense to be ready to stop AI development. My personal views are more pessimistic than Anthropic’s.
**Claim 2: In the absence of powerful aligned AIs, we need to prevent catastrophe-capable AI systems from being built.**
Given developers are not on track to align AI before it becomes catastrophically dangerous, we need the ability to slow down or stop before AI is catastrophically dangerous.
There are several ways to do this.
I think the best one involves building up the government’s capacity to safeguard AI development. Set up government mechanisms to monitor and mitigate catastrophic AI risk, and empower them to institute a national moratorium on advancing AI if it gets too dangerous. (Eventually, the government could transition this into an international moratorium, while coordinating internationally to solve AI safety before that moratorium becomes infeasible to maintain. I describe this later.)
Some others think it’s better to try to build aligned AIs that defend against AI catastrophes. For example, you can imagine building defensive AIs that identify and stop emerging rogue AIs. To me, the main problem with this plan is that it assumes we will have the ability to align the defensive AI systems.
**Claim 3: There’s a significant (>20%) chance AI will be capable enough to cause catastrophe by 2030.**
AI timelines have been [discussed thoroughly elsewhere](https://epochai.org/blog/literature-review-of-transformative-artificial-intelligence-timelines), so I’ll only briefly note a few pieces of evidence for this claim I find compelling:
1. **Current trends in AI**. Qualitatively, I think another jump of the size from [GPT-2 to GPT-4](https://medium.com/@richardcngo/visualizing-the-deep-learning-revolution-722098eb9c5) could get us to catastrophe-capable AI systems.
2. **Effective compute arguments**, such as [Ajeya Cotra’s Bioanchors report](https://www.lesswrong.com/posts/KrJfoZzpSDpnrv9va/draft-report-on-ai-timelines). Hardware scaling, continued algorithmic improvement, investment hype are all continuing strongly, leading to a 10x/year increase of effective compute used to train the best AI system. Given the current rates of progress, I expect another factor of a million increase in effective compute by 2030.
3. **Some experts think powerful AI is coming soon**, both inside and outside of frontier labs. [Yoshua Bengio is 90% confident in 5-20 years](https://yoshuabengio.org/2023/08/12/personal-and-psychological-dimensions-of-ai-researchers-confronting-ai-catastrophic-risks/). [Demis Hassabis thinks AGI could be a few years away](https://www.wsj.com/articles/google-deepmind-ceo-says-some-form-of-agi-possible-in-a-few-years-2705f452). [OpenAI says that “we believe [superintelligence] could arrive this decade”](https://www.google.com/search?q=openai+superintelligence+this+decade&oq=openai+superintelligence+this+&aqs=chrome.0.0i512j69i57j0i390i650l5.3589j0j7&sourceid=chrome&ie=UTF-8). Anthropic stated a “[greater than 10% likelihood that we will develop broadly human-level AI systems within the next decade](https://www.anthropic.com/index/core-views-on-ai-safety).”
Summary
-------
I’ve argued for 3 high-level claims:
1. There’s a significant chance that AI alignment is difficult.
2. In the absence of powerful aligned AIs, we need to prevent catastrophe-capable AI systems from being built.
3. There’s a significant (>20%) chance of catastrophe-capable AI by 2030.
Together, claims (1) and (2) imply that there is a significant chance of AI systems causing catastrophic harm, and so we need to prevent AI systems that could cause catastrophe from being built. Claim (3) adds in urgency – there’s a large enough chance of things going wrong soon that it is important to act now.
There are other challenges besides misalignment. Adversaries could also misuse powerful AI systems. Well-intentioned developers could fail to successfully align their AI system because of a [race to the bottom](https://www.fhi.ox.ac.uk/wp-content/uploads/Racing-to-the-precipice-a-model-of-artificial-intelligence-development.pdf) or engineering errors.
Given all these, I think it’s good to push for governance structures that can prevent dangerous AI systems from being built. I suggest we start by developing these nationally.
A National Policy Proposal
==========================
We should build up the capacity for the US government to stop dangerous AI development before an AI catastrophe occurs. To do this, we need to increase government **visibility** into AI development so that they know when things are getting dangerous, and to give the government **brakes**that they can pull in order to prevent dangerous AI development.
Some ideas for increasing the government’s **visibility** into AI development are:
1. A regulatory body that keeps track of AI development, makes predictions about model capabilities, and assesses risks.
2. Required watermarking and traceability on advanced models, so that we can match AI outputs to specific AI models and developers.
3. Whistleblower protections to incentivize researchers to report any dangerous AI development.
Some ideas for giving the government **brakes** on dangerous AI development are:
1. Strengthening the hardware export controls to include chips like A800s and H800s.
2. Giving the regulator emergency powers that allow it to temporarily shut down dangerous AI development.
3. Tracking domestic hardware use to ensure that the agency could implement its emergency powers and enforce regulations.
4. Licensing to reject individual instances of dangerous AI development.
If these policies were passed and implemented, they would not guarantee a safe outcome, but they would make us significantly safer. The visibility needs to work sufficiently well that the government understands that catastrophic AI could happen soon, and then has the willingness and competence to take effective action to prevent dangerous AI models from being built and deployed. A key challenge for this action is the increasing number of actors who will have the ability to build an AI system of a given capability over time. The world would need to use the additional window of time created by this intervention to work on safety and alignment.
I’ll now discuss two especially important points in a bit more detail: the regulatory body, and emergency powers.
Regulatory Body
---------------
We should create a federal regulatory body with the mandate to keep domestic AI development below an agreed upon risk threshold. This body must prevent the creation of smarter-than-human AI systems until humanity has the technical expertise to make a strong argument that they are safe.
The regulatory body should be focused on licensing the most advanced AI systems. Ideally, this would be done based on the model’s capabilities to cause harm. Unfortunately, it can be difficult to predict a model’s capabilities before we’ve trained that model, and even after they are trained, the technology for evaluating model capabilities is not perfect. One way this goes wrong is that, after a model is released, a novel strategy for improving model capabilities is developed, [such as chain-of-thought prompting](https://arxiv.org/abs/2201.11903).
Because there is no perfect criteria that only captures dangerous AI systems, and progress in AI advances quickly (such that any static definition could quickly become outdated), I believe we should use a dynamic definition that consists of multiple criteria. The criteria could be based on imperfect proxies of model capabilities, such as compute, parameter count, money spent on the training run, and predictions of model capabilities. Given that the AI field advances faster than a typical legislative cycle, the agency must be sufficiently dynamic to handle new advances and types of systems, and must be able to change the scope of its licensing to include any dangerous new models.
Given that these proxies are imperfect, the regulator should have a fast track application form to exclude models that clearly pose no catastrophic risks. For example, the vast majority of recommender systems, self-driving car software, weather prediction models, and image recognition models would be fast-tracked and would not have to go through the regular licensing process.
Emergency Powers
----------------
In the case of a national emergency with a potentially catastrophic AI system, the regulator should have the power to issue a cease and desist order to the relevant party. In cases of sufficient risk, the regulator should be equipped with the ability to initiate a moratorium on future development, deployment, and proliferation of potentially dangerous AI systems, which could then be confirmed by the President and eventually by Congress. This moratorium would ideally last until we have strong arguments that AI systems are safe. The regulator would also be expected to swiftly issue guidance to the President and Congress in the event of an AI-related emergency.
The regulator’s use of its emergency powers would be guided by its work on the monitoring of advanced AI hardware. Because the regulator will be able to keep track of who owns the largest computing clusters, no one will be able to build advanced AI in total secrecy.
Some Objections
---------------
I’m aware of several objections to my proposals. Below are a few of the important objections that I’m keeping in mind, along with a brief summary of some ideas to mitigate these potential problems.
1. **Regulatory incompetence.**
1. **Problem:** The US government has a poor track record of developing and implementing safety standards in other fields.
2. **Solutions:** There are examples of positive regulations, e.g. cars are much safer because the Department of Transportation requires transportation safety measures. The regulator should invite a wide range of experts from industry and academia to help develop the standards, building on the work done by NIST and by private-public “blue ribbon” commissions, and update the standards as the field evolves.
2. **Regulatory capture.**
1. **Problem:** The regulatory agency could become a puppet of industry (or a few key industry players).
2. **Solutions:** The regulatory agency will have robust conflict-of-interest provisions, “sunshine” laws that require the administrator to publicly report on their disagreements with the independent licensing judges, and a special “public interest” section that is charged with monitoring the administration’s actions and calling out any dangers that the administration has left unchecked.
3. **Political feasibility.**
1. **Problem:** Asking for a new agency is a big ask, and it might not be politically practical right now.
2. **Solutions:** This proposal is modular; politicians can easily adopt a subset of these proposals to match the budget available. If the policies that are initially approved are too small to justify creating a whole new agency, then it can be created as an office within an existing agency. Additionally, asking for a new agency may become politically practical in the future.
4. **International AI Development**
1. **Problem:** Other countries may develop dangerous AI systems.
2. **Solutions:** This proposal is mainly aimed at mitigating domestic AI risk, but there is an ultimate need for international regulation. This is discussed in more depth below.
Additional details
------------------
At the [Center for AI Policy](https://www.aipolicy.us/), my team is working on developing the strategic thinking and policies I outlined above. Our proposed legislation would create a regulatory agency dedicated to safeguarding AI development, which would monitor advanced hardware, license frontier AI projects, and hold developers accountable for severe harms resulting from their models.
I would love feedback on our work. If you want to learn more about it, share considerations I did not address, or have ideas for improvements, please contact me at [thomas@aipolicy.us](mailto:thomas@aipolicy.us).
International AI Policy
=======================
Dangerous AI systems can be built anywhere, which means that purely domestic regulation is not sufficient in the long term. Unless we find a way to adequately defend the world against dangerous AI systems, which probably require powerful aligned AI, we need effective international coordination that prevents the development of dangerous AI globally.
We don’t need to immediately coordinate all the countries on earth. Currently, the US has a substantial lead on the rest of the world on AI capabilities, and so regulation can start in the US, and then expand to other countries as their capabilities approach the danger threshold. Additionally, the hardware supply chain needed to build advanced AI systems is predominantly controlled by the West. Strengthened export controls could prevent foreign countries with inadequate AI governance from building dangerous AI systems for a significant period.
Here is a hypothetical structure for global coordination: create an international body, which I've called the "IAEA for AI", tasked with both preventing dangerous AI development and making progress on AI safety.
The "Preventing Dangerous AI" side of the agency develops and communicates safety standards to the member countries, each of whom has their own regulatory body that directly regulates AI development within their country.
The "Solving Safety" side should work on many parallel routes towards safety, because none of the paths are sure to succeed.
Preventing Dangerous AI
-----------------------
To prevent dangerous AI, the key mechanism we have is preventing actors from stockpiling large enough amounts of compute to build an AI system.
A toy visualization of how this might work is here:
The yellow decreasing line is the compute requirement to build dangerous AI, which is decreasing as people discover more capable AI algorithms. This amount of compute is being [divided by roughly 2.5 every year](https://epochai.org/blog/revisiting-algorithmic-progress). The red increasing line is the compute that labs have access to, which is increasing as hardware becomes better and investment increases, which is being [multiplied by ~4x every year](https://epochai.org/blog/compute-trends).
The green decreasing curve is the moratorium threshold, which is the largest amount of unmonitored compute that actors can develop.
The international body needs to keep the threshold for the moratorium (the green curve) below the amount of compute it takes to build dangerous AI (the yellow curve), by adjusting the maximum amount of unmonitored compute actors can get ahold of.
Solving Safety
--------------
The "Solving Safety" side should support many parallel routes towards AI safety, because none of the agendas are sure to work out, and many of them are serial time bottlenecked. Some routes that we could take include:
1. A “CERN for AI” project iterating on empirical AI safety & control.
2. Theoretical AI safety research.
3. Formal Verification, which requires formalizing what it means for something to be safe.
4. Whole Brain Emulation, or other non-standard paths to AI that may be safer than the current paradigm.
One key difficulty here is creating the institutional capacity to evaluate which projects are promising, and a big part of that will rest on getting people with the ability to do this differentiation into the bureaucracy that is making these decisions.
My thoughts on a pause
======================
I think a pause on AI progress wouldn’t be very helpful unless used in concert with other effective governance interventions, such as the ones that I have outlined above. My main concern with a pause in the absence of other governance interventions is that, after we unpause, things would go back to normal, and someone could just build a dangerous AI system slightly later.
Pausing just the training or deployment of the largest models is a *shallow*intervention that doesn’t affect the main drivers of capabilities progress. After the pause is lifted, we’ll likely see AI progress spike as actors immediately kick off much more advanced training runs than they had done in the past.
A particularly worrying instance of this is the compute overhang from continued progress on AI. We can visualize this as:
The y-axis is effective-FLOP (E-FLOP), which is the amount of compute used to train a model, adjusted for algorithmic progress. The green line represents the maximum capabilities of AI models when the pause starts.
If a pause is successfully implemented before we get to the danger threshold, this renders things safe during the pause, and then jumps up after the pause is lifted. There are a few reasons that I don’t expect the overhang to return to previous default progress. One is that AI progress is self-reinforcing. For example, OpenAI is much better able to fundraise after the release of GPT-4 because of GPT-4’s advanced capabilities. This means that the overhang is less tall in a world with an AI pause than in a world where AI progress continues unabated, which suggests that there could be a lasting effect from a pause.
All in all, a short pause (e.g. 6 months) probably just delays scheduled training runs, and during this pause, companies would stockpile compute and continue work on improving their algorithms. A medium length pause (e.g. 1 year) probably delays AI capabilities a little bit, but results in a significant jump in capability once the pause is lifted. A longer pause that lasts until we are confident that we have robust AI safety measures in place that allow for safe deployment would be helpful. I’m currently in favor of building the capacity of the world to create a long pause on AI.
As a result, I’m only excited about versions of a pause that don’t return to “AI progress as usual”, after the pause is over. |
78fd5293-17c6-4ddb-90c3-142bea8f7db5 | trentmkelly/LessWrong-43k | LessWrong | (Salt) Water Gargling as an Antiviral
Summary
Over the past year I’ve investigated potential interventions against respiratory illnesses. Previous results include “Enovid nasal spray is promising but understudied”, “Povidone iodine is promising but understudied” and “Humming will solve all your problems no wait it’s useless”. Two of the iodine papers showed salt water doing as well or almost as well as iodine. I assume salt water has lower side effects, so that seemed like a promising thing to check. I still believe that, but that’s about all I believe, because papers studying gargling salt water (without nasal irrigation) are few and far between.
I ended up finding only one new paper I thought valuable that wasn’t already included in my original review of iodine, and it focused on tap water, not salt water. It found a 30% drop in illness when gargling increased in frequency from 1 time per day to 3.6 times, which is fantastic. But having so few relevant papers with such small sample sizes has a little alarm going off in my head screaming publication BIAS publication BIAS. So this is going in the books as another intervention that is promising but understudied, with no larger conclusions drawn.
Papers
Estimating salivary carriage of severe acute respiratory syndrome coronavirus 2 in nonsymptomatic people and efficacy of mouthrinse in reducing viral load: A randomized controlled trial
Note that despite the title, they only gave mouthwashes to participants with symptoms.
This study had 40 participants collect saliva, rinse their mouth with one of four mouthwashes, and then collect more saliva 15 and 45 minutes later . Researchers then compared compared the viral load in the initial collection with the viral load 15 and 45 minutes later. The overall effect was very strong: 3 of the washes had a 90% total reduction in viral load, and the loser of the bunch (chlorhexidine) still had a 70% reduction (error bars fairly large). So taken at face value, salt water was at least as good as the antiseptic wa |
90a67a7c-aea2-49e3-bfd2-54dfbb183d38 | trentmkelly/LessWrong-43k | LessWrong | What I'd change about different philosophy fields
[epistemic status: speculative conversation-starter]
My guess at the memetic shifts that would do the most to improve these philosophy fields' tendency to converge on truth:
----------------------------------------
metaphysics
1. Make reductive, 'third-person' models of the brain central to metaphysics discussion.
If you claim that humans can come to know X, then you should be able to provide a story sketch in the third person for how a physical, deterministic, evolved organism could end up learning X.
You don't have to go into exact neuroscientific detail, but it should be clear how a mechanistic cause-and-effect chain could result in a toy agent verifying the truth of X within a physical universe.
2. Care less about human intuitions and concepts. Care more about the actual subject matter of metaphysics — ultimate, objective reality. E.g., only care about the concept 'truth' insofar as we have strong reason to think an alien would arrive at the exact same concept, because it's carving nature closer to its joints.
Conduct more tests to see which concepts look more joint-carving than others.
(I think current analytic metaphysics is actually better at 'not caring about human intuitions and concepts' than most other philosophy fields. I just think this is still the field's biggest area for future improvement, partly because it's harder to do this right in metaphysics.)
----------------------------------------
decision theory
Similar to metaphysics, it should be more expected that we think of decision theories in third-person terms. Can we build toy models of a hypothetical alien or robot that actually implements this decision procedure?
In metaphysics, doing this helps us confirm that a claim is coherent and knowable. In decision theory, there's an even larger benefit: a lot of issues that are central to the field (e.g., logical uncertainty and counterlogicals) are easy to miss if you stay in fuzzy-human-intuitions land.
Much more so than in metaphysic |
35b8da0c-8909-4b84-8505-fd19a3320a7f | StampyAI/alignment-research-dataset/arxiv | Arxiv | HARK Side of Deep Learning -- From Grad Student Descent to Automated Machine Learning
1 Introduction
---------------
Hypothesizing after the results are known (HARKing) [[1](#bib.bib1)] occurs when researchers masquerade one or more post hoc hypotheses as a priori hypotheses. This means that instead of following a traditional hypothetico-deductive model [[2](#bib.bib2)], in which previous knowledge or conjecture is used to formulate hypotheses that are then tested, the researcher instead looks at the results first and then forms a post hoc hypothesis. HARKing can occur in different forms, such as constructing, retrieving or suppressing hypotheses after the results are known [[1](#bib.bib1)]. A number of studies in recent years have examined and discussed the incidences, causes and implications of such practices within various fields such as management, psychology as well as natural sciences [[3](#bib.bib3), [4](#bib.bib4), [5](#bib.bib5)].
In recent years, deep learning (DL) methods have dramatically improved the state-of-the-art (SotA) within the fields of speech recognition, visual object recognition, machine translation and several other domains such as drug discovery and genomics [[6](#bib.bib6)]. However, there are certain troubling trends in the current machine learning (ML) research, outlined in [[7](#bib.bib7)] as failure to distinguish between explanation and speculation, use of mathematics that obfuscates rather than clarifies, and misuse of language. Unfortunately, HARKing has also been one of those recurrent trends in machine learning and especially in deep learning research. Since much of such research is being eagerly applied to real-world applications in both industry and society, such issues are of utmost importance due to the wide impact of machine learning products and services across all walks of life. Transparent and reliable practices are critical when trying to combat suspicions towards new technologies, and the trust needs to be built over long period of time; as acknowledged recently even on the European Commission level [[8](#bib.bib8)].
Our hypothesis is that the recent explosion of advances within the fields of machine learning and in particular deep learning, as well as the hyper-competitive nature of these fields, may potentially be a dangerous breeding ground for various HARKing behaviors, the implications of which are not yet fully explored. At the very least, concerns regarding such behaviours deserve to be critically discussed from different angles so as to encourage best practices when building ML systems and algorithms. It is noted that these issues are not new by themselves. In fact, since as long as data-driven approaches and learning systems have been around, it has been critical, and sometimes difficult, to remain fully objective in analyzing results. Issues have been reported earlier for example, such as self-deception practiced by scientists; finding patterns that are not there [[9](#bib.bib9), [10](#bib.bib10)].
In this paper we discuss HARKing behavior from different angles:
* Section [2](#S2 "2 Grad Student Descent and SotA-hacking ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning") - Competitiveness in DL research leading to questionable improvements of state-of-the-art and claims of novelty.
* Section [3](#S3 "3 Chronic Allergy to Negative Results ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning") - Pressure to create reports that are favorable for publication and aversion towards negative results.
* Section [4](#S4 "4 \"In The Wild\" Illusion ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning") - The belief that current training datasets are representative of real-world samples.
* Section [5](#S5 "5 Automated Machine Learning ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning") - Automated machine learning
* Section [6](#S6 "6 The Insert_Adjective_Here AI Wave ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning") - Explainability, ethics, reproducibility and more for AI systems.
2 Grad Student Descent and SotA-hacking
----------------------------------------
In a typical deep neural network model there are numerous design choices, i.e., tunable parts such as model architecture and hyper-parameters, that affect the predictive performance. Proposing a decent set of these design choices that will result in high generalization ability (relative to the other sets of choices) is difficult mainly due to two reasons. Firstly, due to the inherent non-deterministic and highly non-linear nature of neural networks, it is not trivial to deduce explicit relationships neither between the hyper-parameters and the model performance, nor between the interactions of hyper-parameters themselves. For instance, a large batch size is key to speed up neural network training in large distributed computation infrastructures, however, significant degradation in model performance has been observed in practice when large batch sizes are employed [[11](#bib.bib11)]. To overcome this issue, typically, hyper-parameters belonging to the optimizer need to be tuned. Secondly, as the parameter search space increases exponentially, it is not feasible to apply exhaustive or brute-force search methods. Therefore, a significant portion of deep learning research has been focusing on engineering efficient model architectures and hyper-parameters for specific tasks.
Even though this manual discovery process has been successful for several applications (often empirically), there has been significant divergence from the traditional hypothesis-driven scientific approach in the methodology of such studies. Instead of hypothesis-forming based on theory, extensive research on previous studies and/or reflection against the existing domain knowledge, grad student descent (a cheesy pun referring to the well-known gradient descent algorithm) is applied.
Grad student descent is a type of optimization scheme in which the task of model architecture or hyper-parameter search is assigned to several graduate students, usually to be performed by trying what works and what does not. This is an iterative approach, where one starts with a baseline architecture or possibly with an earlier SotA, measures its performance and applies various modifications by trial-and-error, without a sound hypothesis. Once marginal improvements are observed, iterations of modifications continue further in that direction until a local optimum (often a publishable result) is reached and an explanation is forged. In essence, this whole process is driven by HARKing. Furthermore, this process is performed with a limited set of data, that is used and re-used again and again to find the "optimal" solution (further discussed in Section [4](#S4 "4 \"In The Wild\" Illusion ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning")). Oftentimes, final testing on a completely independent test set that has not been touched or observed at all at any moment is not performed and cross-validation is either not used or used under problematic assumptions and/or executions such as performing model tuning and estimation of model error at the same time [[12](#bib.bib12), [13](#bib.bib13)].
The abovementioned HARKing pattern, consequently, results in increased difficulty in distinguishing and identifying why a proposed method works or not. Lack of thorough hypothesis forming prior to experimentation often leads to negligence of comprehensive discussions on the results as well, especially when accompanied with comparison of a single score or metric. For instance, a recent work by Reimers and Gurevych shows that reporting a single performance score is insufficient to compare non-deterministic approaches such as neural networks [[14](#bib.bib14)]. Their study demonstrates that the seed value for the random number generator can result in statistically significant differences in performances of state-of-the-art methods [[14](#bib.bib14)].
The negative effects of HARKing are not specific to deep learning research alone, and they can be observed in research dealing with traditional machine learning methods as well. However, as the concept of state-of-the-art (a method or a set of methods that outperforms all the previously proposed methods for a given machine learning task in a certain metric such as test accuracy, inference speed, training speed etc.) has been disproportionately promoted in DL, both in academy and industry, presence of HARKing is becoming more likely to be overlooked especially if there are claims of advancing the SotA. This phenomenon has been promoting the concept of SotA-hacking and publishing of marginally SotA results without in-depth analysis or discussion, similar to p-hacking, data dredging and prevalence of marginally significant results in several other fields [[15](#bib.bib15), [16](#bib.bib16), [17](#bib.bib17)]. Typical examples of misleading comparisons leading to unfair or inadequate SotA claims include usage of additional training data (the common concept of transfer learning in DL), usage of data augmentation, comparison to poorly implemented baselines or ensembling of several models. Similar unjustified claims can be observed in "novelty" of proposed methods as well.
3 Chronic Allergy to Negative Results
--------------------------------------
Publication bias, the phenomenon occurring when the probability of a scientific study being published is not independent of its results [[18](#bib.bib18)], leads to systematic difference in the findings of published tests of a claim from the findings of all tests of the same claim [[19](#bib.bib19)]. Often recurring as a positive outcome bias, this phenomenon has been observed in several research fields for a long time [[20](#bib.bib20), [21](#bib.bib21), [22](#bib.bib22)]. For example in clinical research, studies finding no difference between the study groups were less likely to be published than those with statistically significant results [[20](#bib.bib20)]. In fact, there is evidence of negative results being less likely to be published even if they provide corrections of errors in previous studies [[23](#bib.bib23)]. A similar troubling trend has been prevalent in ML/DL research and arguably HARKing exacerbates this further.
Publishing a null or negative result in the current ML researchosphere is considerably difficult due to the widespread assumption that "every positive result is scientifically more valuable, or interesting, than any negative one". This is likely even more the case in DL research because of the ever-increasing competition. For instance, the percentage of accepted papers related to deep neural networks in the Conference on Computer Vision and Pattern Recognition (CVPR), one of the most prestigious in its field, has been 1%, 14% and 25% for the years 2013, 2015 and 2017, respectively [[24](#bib.bib24)]. Note that the amount of publication submissions to conferences and journals are increasing every year as well, e.g., the number of submissions to Annual Conference on Neural Information Processing Systems (NeurIPS) doubled from 2016 to 2018 [[25](#bib.bib25)]. Similar trends can be expected to be observed in research funding or scholarship applications. A research proposal is more likely to get a positive review if it builds further on "encouraging results" from previous work. There have been incentives to discuss the importance of negative results and share them in ML research [[26](#bib.bib26)] such as the First Workshop on Negative Results in Computer Vision in 2017 and we hope more actions towards this direction will be realized in the future.
Current outcome reporting bias in ML/DL research is generated both from the authors’ side as a reluctance to report negative results as well as the journals’ side in selecting the results worth publishing and it is not trivial to separate the extent of the two. Even in presence of a positive result, authors may not report the negative ones, thinking such reporting will devaluate their work. As stated by Nissen et al., even if authors’ behavior is the main contributor to publication bias (there is evidence supporting this in other fields [[27](#bib.bib27), [28](#bib.bib28)]), they may simply be responding to the editorial preferences for positive results [[29](#bib.bib29)]. The lack of traditional hypothesis construction before conducting the experiments and the lack of expectation to do so, supports the incentive of avoiding reporting of negative results in ML/DL field.
There are several consequences of such allergy against negative results in deep learning research. First, it eventually creates a bias against disruptive innovative ideas and favors incremental tweaks on well-established methods. Secondly, when negative results are not reported or published, it is essentially more difficult to construct causality and elaborate on the phenomena behind the positive results. As in other aspects of life, after all, we learn from negative results as well as positive ones. Furthermore, it increases the waste of resources and efforts due to unnecessary (re-)implementation of methods that have been shown to be inferior but never reported. Finally, the probability of a negative result being caused simply because of poor implementation exhibits the potential of that work being influential once implemented properly.
The trend of starting from a solution (often somebody else’s) instead of from the problem itself and HARKing after minor modifications can be changed by changing our paradigm of publication process. Hereby, we propose a results-blind review process for ML/DL research:
* A paper is submitted with a clear hypothesis accompanied with the design of experiments. The hypothesis can be based on extensive analysis of previous studies, mathematical theory with unambiguous assumptions and/or domain knowledge of the specific field.
* The paper gets peer reviewed, preferably double-blind, and the reviewers suggest modifications and improvements on the experimental methods.
* Once accepted, the experiments are run.
* The paper gets published regardless of the results with a comprehensive discussion section.
This approach would increase the likelihood of the study to be informative and influential regardless of the outcome, not only in the case of positive results. Essentially, the review process will give more attention to the experimental design and the hypothesis behind the proposed methods, decreasing the incentive for HARKing significantly. Naturally, this will also encourage researchers to navigate outside the "marginal improvements over the previous SotA" thinking. Similar ideas have been discussed especially in the field of psychology [[30](#bib.bib30), [31](#bib.bib31), [32](#bib.bib32)]. Note that we do not claim that the abovementioned proposal is applicable for every machine learning research publication process, mostly due to the scarcity of high quality reviewers. Nevertheless, we believe such discussions are beneficial and may eventually lead to improvements that will decrease the prevalence of HARKing in ML/DL research.
4 "In The Wild" Illusion
-------------------------
Numerous studies in the field of deep learning utilize publicly available annotated datasets for computer vision, natural language processing, audio analysis and various other tasks. Several of these datasets even include the phrase "In the Wild" in their name - an expression to convey the message that the dataset holds no constraints and is representative of real-world circumstances. Even though it is not stated explicitly, the main assumption behind using these datasets is that the observations belonging to these datasets are drawn from the same statistical distribution of all possible observations naturally occurring in real-world.
In 2011, Torralba and Efros proposed to examine dataset bias in twelve popular image datasets by observing if it is possible to train a machine learning model to identify the dataset a given image is selected from [[33](#bib.bib33)]. Considering the random guess accuracy is only 112≈8%, the authors found that humans were able to perform at >75%, while a simple support vector machine classifier performed at 39%. The authors furthermore demonstrated the inability to perform cross-dataset generalization, thereby highlighting how models trained on typical datasets actually overfit and thus fail to generalize to other datasets yet alone to real-world settings.
A similar problem of overfitting stems from the hyper-competitive nature of machine learning, where there is little incentive of trying to publish methods that have inferior performance compared to SotA on test datasets (see Section [3](#S3 "3 Chronic Allergy to Negative Results ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning")). Therefore, we can reasonably expect that effectively most research uses the test set as a validation set, rather than following the standard practice of defining a separate validation set from the training data. Recht et al. show this by creating a new test set for CIFAR10, a widely used image dataset, where they found that there was a significant drop in accuracy (4-15%) from the old test set to the new test set when tested with several DL architectures [[34](#bib.bib34)]. In a more recent work, a similar phenomenon is also shown for the well-known ImageNet dataset, suggesting that the accuracy drops are caused by the models’ inability to generalize to slightly "harder" images than those found in the original test sets [[35](#bib.bib35)].
From the HARKing perspective, formulating hypotheses that are specifically designed to account for the observed results for a specific sample of observations go hand in hand with overfitting and failure of generalization. Furthermore, the selected datasets to run the proposed experiments on have to be in parallel with the hypothesis. For instance, the well-known Labeled Faces in the Wild dataset [[36](#bib.bib36)] contains images of famous people only, but have been used extensively to test hypotheses of face recognition or person identification in unconstrained settings. And from the implementation perspective, by splitting a dataset into training, validation, and testing sets, we invariably risk giving the false impression that because our model may perform well on the test dataset, it will also generalize to images found in real world applications. In both cases mentioned above (using biased datasets and/or overfitting to specific test sets), it can be argued that hypotheses testing is conditional on the dataset in question, and therefore to convince a reader that HARKing has not occurred, an author should always take great care to demonstrate the generalizability of new methods. Obviously, overfitting is a problem encountered in ML in general and is not specific to neural networks. However, considering:
1. [label=()]
2. feed-forward neural networks are universal function approximators (by Universal Approximation Theorem) as well as convolutional networks, i.e., a single hidden layer network containing a finite number of neurons can approximate continuous functions with arbitrary precision [[37](#bib.bib37), [38](#bib.bib38)]
3. the complexity of the computed function by a neural network grows exponentially with its depth, i.e., for every additional hidden layer, one needs exponentially more parameters to express the same function with a shallower network [[39](#bib.bib39), [40](#bib.bib40)]
deep neural architectures are very likely to suffer from overfitting due to their expressive power.
5 Automated Machine Learning
-----------------------------
The traditional data science approach relies on many sequential tasks; i.e. data preprocessing and cleaning, feature engineering and selection, model selection and parameter tuning, postprocessing, and finally critical analysis of results. Often in practice, the human decision making processes in these tasks are inefficient (see Section [2](#S2 "2 Grad Student Descent and SotA-hacking ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning")) or based on heuristics. Furthermore, the combined complexity of these tasks often present an insurmountable barrier for non-experts, and thus automated machine learning (AutoML) is a topic that has become increasingly popular in recent years, promising to automate (at least parts) of this pipeline in order to improve efficiency of machine learning and accelerating research.
Recently, the most popular AutoML task has focused extensively on neural architecture search (NAS) [[41](#bib.bib41), [42](#bib.bib42), [43](#bib.bib43), [44](#bib.bib44), [45](#bib.bib45), [46](#bib.bib46), [47](#bib.bib47)], i.e., automating the design of neural network architectures for the search of architectures that are superior to hand-crafted ones. Several other AutoML tasks include automated hyper-parameter optimization [[48](#bib.bib48)], activation function search [[49](#bib.bib49)], optimizer search [[50](#bib.bib50)], data augmentation policy search [[51](#bib.bib51)] or even search for better hardware utilization in heterogeneously distributed (mixture of CPUs and GPUs) computing environments [[52](#bib.bib52)]. The methods behind such meta-learning approaches are mainly based on Bayesian optimization [[48](#bib.bib48)], evolutionary algorithms [[43](#bib.bib43), [46](#bib.bib46)] or more recently on reinforcement learning [[41](#bib.bib41), [49](#bib.bib49), [52](#bib.bib52)]. Some of these methods are available both to the academy as well as to the industry as open source software or in the form of software-as-a-service.
These advancements not only help us discover better DL models and solutions in terms of quantitative metrics than hand-engineered ones, but also carry the possibility to transform the everyday working practices of machine learning researchers and practitioners. With AutoML, data scientists are expected to offload a significant portion of their routine work and focus on tasks that require a higher level thinking and creativity. However, certain issues have been raised related to AutoML approaches lately. For instance, Scuito et al. demonstrate that the search policies of state-of-the-art NAS techniques are no better than random policies [[53](#bib.bib53)]. Similarly, Li and Talwalkar show that random search with early-stopping is a competitive NAS baseline on two benchmark tasks - one from computer vision and one from natural language processing [[54](#bib.bib54)]. In addition, they discuss the reproducibility issues of published NAS results by elaborating on the necessity of having a tremendous amount of computation resources, lack of available source material/code and questionable robustness of published results [[54](#bib.bib54)].
Interestingly, the pursuit of simplifying machine learning development resulted in a significant increase in algorithmic complexity of AutoML methods including complicated training routines and architecture transformations [[54](#bib.bib54)]. This complexity makes it more difficult to pinpoint which components of the found solution is crucial for high performance. In addition, considering the lack of ablation studies (the analysis of systematic removal of components or features of a model in order to identify which of them are the most relevant) in many works, AutoML field creates a dangerous ground for HARKing.
6 The Insert\_Adjective\_Here AI Wave
--------------------------------------
###
6.1 Ethical AI
Ethical issues regarding current developments in machine learning are perhaps much more critical than they currently perceived to be; as we already encounter ethically questionable decisions given by algorithms, sometimes unbeknownst to us. Examples include replacing faces and voices in videos [[55](#bib.bib55)], detecting people using WiFi signals [[56](#bib.bib56)], deciding whose life to risk in an eminent accident [[57](#bib.bib57)] and generating fake news [[58](#bib.bib58)]. In various scenarios, ML impacts decisions on legal and ethical issues as well such as insurance, hiring, lending. Therefore, it is crucial to develop models that are fair and unbiased regardless of the biases in the data [[59](#bib.bib59), [60](#bib.bib60)]. This issue has been recently emphasized even by the European Commission in their ethics guidelines report for AI by underlining the importance of paying attention to situations involving more vulnerable groups such as children, persons with disabilities or minorities, or to situations with asymmetries of power or information (e.g. employee-employer or business-consumer) [[61](#bib.bib61)].
With established industries (e.g. example firearms), it is common for the researchers and developers to leave the responsibility of ethics to entities that follow them (e.g. arms sellers and legislators). However, most AI-based systems have been much faster to deploy than conventional technology. Therefore, it is highly desirable for researchers to discuss ethical implications of their work and create a dialogue about them at the earliest possible stage. While selecting research topics that raise ethical issues itself serves this purpose, the desire to present good results might deter the discussion.
Another important ethical issue revolves around covert AI systems. A human should always know if she/he is interacting with a human being or a machine, and it is the responsibility of us that this is reliably achieved. As AI practitioners, we should ensure that humans are made aware of - or able to request and validate the fact that – they interact with an AI identity [[61](#bib.bib61)]. Thus, hypothesis forming process should be clear and unambiguous, and should consider the possible use cases or implications as well. And in this pursuit, HARKing won’t do.
###
6.2 Human-centric AI
At the current stage, ML/DL algorithms are often designed as tools for defined domain experts, thus they need to address human needs and psychology in a realistic manner. To decrease the amount of HARKing, high-level domain experts should be incorporated to the study teams from the beginning as a collective intelligence of domain experts has considerable benefits and should be utilized whenever possible [[62](#bib.bib62)]. This will lead to more successful forming of a priori hypotheses and in the end should put pressure on scrutinizing results that do not support these hypotheses. Previously, worrying examples of failure in this have surfaced, where there has been only a limited input from the domain experts [[63](#bib.bib63)]. High-level expertise is especially relevant to create scientific hypotheses and should be differentiated from defining practical use-cases and training of AI, where a diverse spectrum of possible users should be affiliated to the project.
HARKing is potentially a serious threat especially in AI-driven change in medical practice. This applies mostly to the effect of failing to report a priori hypotheses that are unsupported by the current results [[5](#bib.bib5)]. The algorithms that will be used in medicine typically need to be clinically validated in laborious and high-cost trials [[64](#bib.bib64)]. Suppressing hypotheses after the results are known can lead to wrongly planned clinical trials, as the background scientific literature (meta-analyses) is biased and this can lead to losing credibility in the eyes of physicians and decision makers, together with spending a huge amount of limited human and financial resources available to run these trials.
###
6.3 Explainable, transparent and interpretable AI
Explainable artificial intelligence (XAI) is not only interesting as an academic curiosity; it is a necessity for the future. Developing explainable and transparent systems, as well as tools to measure transparency, is crucial for ethical AI development (see section [6.1](#S6.SS1 "6.1 Ethical AI ‣ 6 The Insert_Adjective_Here AI Wave ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning")). The main concept of XAI is centered around causal attribution as it is in human nature to understand causality naturally. Having such causal explanations will provide substantial leap in reaching human-like perception of AI systems and anthropomorphism [[65](#bib.bib65)]. Explainable AI and model interpretability may be used in a synonymous manner. However, we think that explainability may fall under the causality domain and interpretability may belong to the mechanistic explanation of the algorithmic and model internals [[66](#bib.bib66)].
Recent deep learning algorithms provide high predictive performance but limited ways to provide reasoning on how an algorithm produces such level of high performance that exceeds human abilities [[67](#bib.bib67)]. Even though there have been studies addressing this problem and proposing solutions [[68](#bib.bib68), [69](#bib.bib69), [70](#bib.bib70)], a common consensus on performing interpretation of ML and especially DL models has not been reached. In fact, even the definition of interpretability itself is not established, neither mathematically nor axiomatically in the literature [[66](#bib.bib66)]. Furthermore, recent studies question the robustness and security of these interpretation methods (e.g. to adversarial attacks) [[71](#bib.bib71)].
From HARKing perspective, one can relatively easily reverse engineer results to fit in a desired interpretation [[69](#bib.bib69), [71](#bib.bib71), [72](#bib.bib72)]. To avoid such practices, interpretable algorithms should not be reversible, nor should they only provide interpretation depending upon algorithmic priors. In this regard, approaches aiming at more theoretical explanations of why deep learning works, from learning theory to statistical physics [[73](#bib.bib73), [74](#bib.bib74), [75](#bib.bib75)], may be classified as true XAI research. These approaches, rather than focusing only on interpretation of the mechanistic approaches after the results are known, aim at finding an ab-initio technique, i.e., from the first-principles, to design a deep learning system without HARKing. Similarly, use of causal inference has recently been shown to be promising in understanding underlying mechanisms of deep learning systems [[76](#bib.bib76)] and if descriptive, causal modals can answer prediction, intervention and counterfactual questions [[77](#bib.bib77)].
In terms of transparency, an interesting question is whether are we, as humans, required to know all the details about the AI capabilities of the equipment and sensors that surround us. This can be argued both ways; for example, we know virtually nothing about the abilities of human drivers that use the same highway as we do. But similar to what happened with established technology in automotive (like ABS and automatic transmission), we should be able to know the workings, accuracy stats, advantages and disadvantages of emerging AI technologies. This concept overlaps with abovementioned mechanistic interpretability issue and perception of human-like attributions.
###
6.4 Reproducible AI
AI research is known and as a result appreciated for its significant contributions to open science (e.g. preprint archives), open source (e.g. code repositories, sharing of trained models etc.), open data and reproducible research paradigms. Yet, as a sub-field of computer science, it still shares a similar reproducibility crisis [[78](#bib.bib78), [79](#bib.bib79), [80](#bib.bib80), [81](#bib.bib81), [82](#bib.bib82)]. As Donoho et al. suggested, a computational research paper is merely an advertisement unless it is presented with an underlying code and data [[78](#bib.bib78)]. We believe one of the reasons of this reproducibility crisis is HARKing.
One essential contribution to this crisis in ML and especially in DL research is the lack of understanding of distinction between repeatability and reproducibility [[83](#bib.bib83)]. We consider repeatability as the ability to recreate the results of a study/paper and reproducibility as the ability to reach the same conclusions despite the variations in the irrelevant components of the experiments [[84](#bib.bib84)]. Obviously, the role of hypothesizing driven by sound scientific methodology is essential in differentiating the two. As discussed in Section [2](#S2 "2 Grad Student Descent and SotA-hacking ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning"), competitive nature of the field and elevated pressure of achieving research and business outputs in a fast manner lead to hurried claims of reproducibility (often confused with repeatability) just like the hurried claims of SotA. Once this is coupled with the avoidance of reporting negative results or similar selective reporting (see Section [3](#S3 "3 Chronic Allergy to Negative Results ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning")), reproducibility crisis becomes inevitable.
It is important to acknowledge the initiatives for encouraging and increasing reproducibility in ML/DL research. For instance, in NIPS 2019, a reproducibility checklist and a code submission policy is introduced, in which the code is expected to accompany the accepted papers. In AAAI Conference on Artificial Intelligence in 2019, a workshop on reproducible AI has been held. Similarly, a workshop on reproducibility in ML was held in International Conference on Learning Representations (ICLR) in 2019. Nevertheless, open questions remain such as "How can we measure reproducibility?", "What does it mean for a paper to have successful or unsuccessful replications?" or "What can the ML community learn from other fields?".
###
6.5 Accountable AI
Accountability of algorithmic decision-making systems (e.g. credit scoring) has been under discussion as well as under implementation for decades especially from the regulatory and legal perspective. However, the rapid pace of AI developments and real-world applications of them, introduced circumstances in which high-stakes decisions with significant consequences for people and broader society are made by ML algorithms. One such potential impact is an accident which can be, in this context, defined as an unintended and harmful behavior that emerges from poor design of real-world AI systems. Amodei et al. provides several concrete examples of such possible problems in AI safety including negative side effects (e.g. due to poorly designed objective functions), sensitivity to distributional shifts (the environment shifting away from the training environment) and reward hacking (the system gaming its objective function) [[85](#bib.bib85)].
Naturally, AI accountability is intertwined with explainability, reproducibility, fairness and human-centrism of design of these systems. Policies for demanding explanations of algorithmic decisions may help preventing negative consequences or may unintentionally hinder innovation while providing little meaningful protection, depending on their implementation and execution. For instance, European Union General Data Protection Regulation (GDPR) [[86](#bib.bib86)] introduced a potential accountability mechanism by right to explanation since May 2018, but the concrete consequences are still yet to be observed. Regarding the role of reproducibility in accountability of AI systems, the fatal accident recently caused by an autonomous car (belonging to Uber) is a suitable example. The preliminary report released by the United States National Transport Safety Board stated that the self-driving system software misclassified the pedestrian and the system was not designed to alert the human operator under such emergency conditions [[87](#bib.bib87)]. For the fair design of AI systems from the accountability perspective, the Gender Shades study [[88](#bib.bib88)] serves as an interesting example. In the study, biases present in commercial automated facial analysis algorithms are presented [[88](#bib.bib88)] and consequently, a recent study elaborated on the concept of actionable auditing by investigating the impact of publicly naming biased performance results of commercial AI products [[89](#bib.bib89)]. Certain opportunities for hybrid models in which humans and machines interact (for explaining failures [[90](#bib.bib90)] or intervening operations [[91](#bib.bib91)]) towards better AI accountability are also proposed in recent studies.
From the industry perspective, considering large companies and corporations entering an "AI race" in order to be the first to successfully employ AI in their domains, it is not surprising for accountability to take lower priority over invention and market leadership. But from the scientific methodology perspective, taking accountability of ML/DL models into account in the early stages of the research process, such as hypothesis forming, is imperative.
###
6.6 Privacy-aware AI
Current implementations of ML algorithms require access to data, which essentially opens up potential security and privacy risks. Therefore, privacy-aware or privacy-preserving AI notion and several studies along this paradigm has been conducted, leading to influential concepts including federated learning and differential privacy [[92](#bib.bib92), [93](#bib.bib93)]. With the use of homomorphic encryption, deep learning model inference on encrypted data was shown to be possible with a little trade-off from accuracy as well [[94](#bib.bib94), [95](#bib.bib95)]. In addition, Shokri et al. introduced and elaborated on the concept called membership inference attack, i.e., given a black-box machine learning model and a data record, determining whether this record was used as part of the model’s training dataset or not [[96](#bib.bib96)]. All these advancements are crucial to declare that several metrics are needed to assess and compare ML models and privacy preserving capability is one of them. For a good scientific conduct, our hypotheses on both the methods and impacts of our research should consider these concepts.
7 Conclusion
-------------
Hypothesizing after the results are known has been observed in several fields of research throughout the history and recently deep learning research exhibits several instances of it as well. In this work, we tried to give examples of HARKing in machine learning and especially in deep learning research. We elaborated on the reasons and consequences of this troubling trend by discussing overemphasis on single-metric model comparisons and benchmarks (Section [2](#S2 "2 Grad Student Descent and SotA-hacking ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning")), tendency to refrain from reporting negative results (Section [3](#S3 "3 Chronic Allergy to Negative Results ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning")), failure of generalization (Section [4](#S4 "4 \"In The Wild\" Illusion ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning")) and automatic machine learning (Section [5](#S5 "5 Automated Machine Learning ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning")). Finally, HARKing and importance of formulating an a priori hypothesis is reviewed from the perspective of ethical, human-centric, explainable, reproducible, accountable and privacy-preserving AI notions (Section [6](#S6 "6 The Insert_Adjective_Here AI Wave ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning")).
We would like to emphasize the importance of discussions for achieving concrete reforms in the mentioned issues. Cultural change and legitimate interventions (such as the proposal in Section [3](#S3 "3 Chronic Allergy to Negative Results ‣ HARK Side of Deep Learning - From Grad Student Descent to Automated Machine Learning")) in deep learning research should be encouraged by addressing these issues as much as we can in a constructive manner. As the aimed progress is a collaborative effort, researchers, practitioners, reviewers, editors, policy-makers, decision-makers, funding agencies, corporations and governmental entities need to act collectively. We believe that prevention of HARKing will help in engineering ethical, accountable, transparent, unbiased and scientifically superior deep learning solutions for the common good of the society we will be living in eventually. We also hope and believe that this work will stir discussions and debates, and will contribute towards that goal. |
569238bb-5949-48ef-ae33-a55ef0861430 | trentmkelly/LessWrong-43k | LessWrong | Lenses of Control
This post is a follow-up to What if Alignment is Not Enough?, which summarized the core arguments of Substrate Needs Convergence. This post further explores the intuitions and worldview driving this theory, particularly the aspects of it that are the most contentious.
Executive Summary: It is not enough to consider what a system does, but also how its actions affect the world, including how those effects change the system itself, up to the point of stable equilibrium. At the scale of AGI, unintended consequences are inevitable, which gives evolutionary forces room to work, making growth of the system an implied attractor state, eventually overriding conflicting initial goals, such as compatibility with human survival. The only way to be safe from AGI is to not build it.
Terminology
* SNC = Substrate Needs Convergence. The theory that AGI will necessarily destroy all biological life as a result of evolution-like pressures.
* AGI = Artificial General Intelligence. Highly autonomous systems that outperform humans at most economically valuable work. SNC is primarily concerned about AI at this level.
* ASI = Artificial Super Intelligence. AI that possesses the theoretical upper limit of cognitive ability, but is still bounded by physical laws. The fate of the world will likely be sealed before AI reaches this level, but the concept is nonetheless useful as a simplifying abstraction for thought experiments.
Challenging the implicit frame of the super-optimizer
AI safety is a novel field. As such, it is premature to characterize any positive viewpoint as representing a consensus. That said, based on the comments of the prior post in this series and responses to SNC more generally, there seems to be a set of consistent underlying beliefs among the people who engage with SNC enough to comment, but reject the conclusions. I venture to characterize this belief as being that ASI will behave as a persistent super-optimizer. That is, whatever world state is imp |
7c1af320-7acc-44a3-9c94-f48940d2b78f | StampyAI/alignment-research-dataset/arbital | Arbital | Probability theory
Probability theory is the mathematics governing quantitative degrees of belief. |
a7f53e0d-84f8-46ea-b06f-e1428cbd3833 | trentmkelly/LessWrong-43k | LessWrong | Meetup : Superintelligence chapter 5
Discussion article for the meetup : Superintelligence chapter 5
WHEN: 12 May 2017 05:45:00PM (+0200)
WHERE: Lindstedtsvägen 3, room 1537, SE-114 28 Stockholm, Sverige
1. Decisive strategic advantage
Will the frontrunner get a decisive advantage?
How large will the successful project be?
Monitoring
International collaboration
From decisive strategic advantage to singleton
We'll have a presentation on the above topics, with opportunities for discussion. We'll repeat this with later chapters every 2 weeks. You don't have to have read the book, though it will probably help to read chapters 1-4.
Format:
We meet and start hanging out at 5:45, but don't officially start doing the meetup topic until 6:00 to accommodate stragglers. We often go out for dinner after the meetup.
How to find us:
The meetup is at a KTH academic building and the room is on the 5th floor, two stairs up.
Influence future meetups:
Times - http://www.when2meet.com/?5723551-cJBhD
Topics - https://druthe.rs/dockets/-KcCvpn97vUhg3tQRrKn
Discussion article for the meetup : Superintelligence chapter 5 |
97cea535-5e9b-4a26-877e-3831b7c76b63 | StampyAI/alignment-research-dataset/blogs | Blogs | January 2022 Newsletter
#### MIRI updates
* MIRI's $1.2 million [Visible Thoughts Project](https://www.lesswrong.com/posts/zRn6cLtxyNodudzhw/visible-thoughts-project-and-bounty-announcement) bounty now has an [FAQ](https://docs.google.com/document/d/1sxNOxvcBWw7XC6MSUUpAjO4Rbu3VcUmCy7PJWQ9Vh5o/edit), and an example of a [successful partial run](https://www.lesswrong.com/posts/zRn6cLtxyNodudzhw/visible-thoughts-project-and-bounty-announcement?commentId=WywmX2i28xANKC7wA) that you can use to inform your own runs.
* Scott Alexander [reviews the first part of the Yudkowsky/Ngo debate](https://astralcodexten.substack.com/p/practically-a-book-review-yudkowsky). See also Richard Ngo's [reply](https://twitter.com/RichardMCNgo/status/1483639849106169856), and Rohin Shah's [review](https://www.lesswrong.com/posts/3vFmQhHBosnjZXuAJ/an-171-disagreements-between-alignment-optimists-and) of several posts from the [Late 2021 MIRI Conversations](https://intelligence.org/late-2021-miri-conversations/).
* From Evan Hubinger: [How do we become confident in the safety of an ML system?](https://www.alignmentforum.org/posts/FDJnZt8Ks2djouQTZ/how-do-we-become-confident-in-the-safety-of-a-machine) and [A positive case for how we might succeed at prosaic AI alignment](https://www.lesswrong.com/posts/5ciYedyQDDqAcrDLr/a-positive-case-for-how-we-might-succeed-at-prosaic-ai) (with discussion in the [comments](https://www.lesswrong.com/posts/5ciYedyQDDqAcrDLr/a-positive-case-for-how-we-might-succeed-at-prosaic-ai#comments)).
* The [ML Alignment Theory Scholars](https://www.lesswrong.com/posts/FpokmCnbP3CEZ5h4t/ml-alignment-theory-program-under-evan-hubinger) program, mentored by Evan Hubinger and run by SERI, has produced a [series](https://www.lesswrong.com/s/tDBYJd4p6EorGLEFA) of distillations and expansions of prior alignment-relevant research.
* MIRI ran a small workshop this month on what makes some concepts better than others, motivated by the question of how revolutionary science (which is about discovering new questions to ask, new ontologies, and new concepts) works.
#### News and links
* Daniel Dewey makes his version of the case that future advances in deep learning [pose a "global risk"](https://www.danieldewey.net/risk/case.html).
* Buck Shlegeris of [Redwood Research](https://www.redwoodresearch.org/) discusses [Worst-Case Thinking in AI Alignment](https://www.lesswrong.com/posts/yTvBSFrXhZfL8vr5a/worst-case-thinking-in-ai-alignment).
* From Paul Christiano: [Why I'm excited about Redwood Research's current project](https://www.lesswrong.com/posts/pXLqpguHJzxSjDdx7/why-i-m-excited-about-redwood-research-s-current-project).
* Paul Christiano's Alignment Research Center is hiring [researchers](https://www.lesswrong.com/posts/dLoK6KGcHAoudtwdo/arc-is-hiring) and [research interns](https://www.lesswrong.com/posts/BRsxztzkTzScFQfDW/apply-for-research-internships-at-arc).
The post [January 2022 Newsletter](https://intelligence.org/2022/01/31/january-2022-newsletter/) appeared first on [Machine Intelligence Research Institute](https://intelligence.org). |
7c745d2c-d243-4d5b-a9e0-067b0577a62b | StampyAI/alignment-research-dataset/arxiv | Arxiv | Explaining Explanations in AI
1. Introduction
----------------
As we deploy automated decision-making systems in the real world, questions of accountability become increasingly important.111This work was supported by The Alan Turing Institute, EPSRC grant EP/N510129/1, and the British Academy, grant PF170151. For system builders questions such as “Is the system working as intended?”, “Do the decisions being made seem sensible?” or “Are we conforming to equality regulation and legislation?” are important, while a subject of the decision-making algorithm may be more concerned with topics such as “Am I being treated fairly?” or “What could I do differently to get a favourable outcome next time?”
These issues are not unique to computerised decision-making systems, but with the growth of machine learning based systems they have become even more important (Pasquale, [2015](#bib.bib59); Bodo et al., [2017](#bib.bib10); Kroll et al., [2016](#bib.bib37); Veale and Edwards, [2018](#bib.bib76); Nissenbaum, [1996](#bib.bib57); Selbst and
Barocas, [2018](#bib.bib69); Olhede and Wolfe, [2018](#bib.bib58)). What distinguishes machine learning is its use of arbitrary black-box functions to make decisions. These black-box functions may be extremely complex and have an internal state composed of millions of interdependent values. As such, the functions used to make decisions may well be too complex for humans to comprehend; and it may not be possible to completely understand the full decision-making criteria or rationale.
Under these constraints, it becomes an open question as to what forms of explanation are even possible that can answer the earlier questions. As such, one of the most striking aspects of research into explainable AI (xAI) is how many different people, be they lawyers, regulators, machine learning specialists, philosophers, or futurologists, are all prepared to agree on the importance of explainable AI. However, very few stop to check what they are agreeing to, and to find out what explainable AI means to other people involved in the discussion (Lipton, [2016](#bib.bib44)).
This gap of expectations is largest between machine learning, which has essentially re-purposed the term “explanation,” and the fields of law, cognitive science, philosophy and the social sciences (which we refer to here collectively as the ‘explanation sciences’), where it has a relatively well-defined technical meaning and a plethora of research on types of explanations, their purposes, and their social and cognitive function.222The dichotomy between explanations in machine learning and elsewhere is well illustrated by the recent work by Miller
et al. ([2017](#bib.bib54)) who found that none of the papers on a curated reading list for xAI made use of work from the social sciences on explanations. Work on xAI currently occupies only one or two small branches of this diverse research landscape. Specifically, the vast majority of work in xAI produces simplified approximations of complex decision-making functions. We argue that these approximations function more like scientific models than the types of scientific and ‘everyday’ explanations considered in philosophy, cognitive science, and psychology.
In this paper we examine the extent of this gap between xAI and the ‘explanation sciences’. We do so by first reviewing methods for producing explanations in xAI, and explain how they are generally more akin to scientific modelling than explanation giving. If this comparison holds, it follows that the majority of currently available methods will, at best, produce locally reliable but globally misleading explanations of model functionality. We then examine research on explanations in philosophy, cognitive science, and the social sciences which suggests that ‘why-questions’ (e.g. ‘why did the model exhibit that behaviour?’) require explanations that are contrastive, selective, and socially interactive. On this basis, we argue that, if xAI is to produce methods that make algorithmic decision-making systems more trustworthy and accountable, the field’s attention must shift to the development of interactive methods for post-hoc interpretability that make it easier to contest algorithmic decisions, and facilitate informed dialogue between users, developers, algorithmic systems, and other stakeholders.
2. A brief primer on explanations in philosophy
------------------------------------------------
Our interest in the philosophical treatment of explanations is based on observation of the development of the field of xAI, or research addressing interpretability and explainability in machine learning. Our aim is to determine whether xAI is heading in a direction in which explanations can be produced that allow affected parties, regulators, and other non-insiders to understand, discuss, and potentially contest decisions made by black-box algorithmic models. So our primary question is: which types of explanations are currently being produced by xAI? And, are these explanations actually useful to the individuals (or proxies thereof) affected by black-box decisions?
Explanations, and more broadly epistemology, causality, and justification, have been the focus of philosophy for millennia, making a complete overview of the field unfeasible. What follows is a brief review of key distinctions and terminology relevant to our interest in methods for providing explanations of black-box algorithmic models and decisions. This primer broadly follows the structure established in a recent review of the contribution of the social sciences to xAI (see: Miller ([2017](#bib.bib53))).
Briefly, the xAI community investigates interpretability (or explainability) and ways of providing explanations of algorithmic models and decisions. ‘Interpretability’ refers to the degree of human comprehensibility of a given ‘black-box’ model or decision (Miller, [2017](#bib.bib53); Lisboa, [2013](#bib.bib45)). Poorly interpretable models “are opaque in the sense that if one is a recipient of the output of the algorithm (the classification decision), rarely does one have any concrete sense of how or why a particular classification has been arrived at from inputs” (Burrell, [2016](#bib.bib12), p.1).
In contrast, ‘explanation’ refers to numerous ways of exchanging information about a phenomenon, in this case the functionality of a model or the rationale and criteria for a decision, to different stakeholders (Lipton, [2016](#bib.bib44); Miller, [2017](#bib.bib53)). Explanations of machine learning models and predictions can serve many functions and audiences. Explanations can be necessary to comply with relevant legislation (Doshi-Velez
et al., [2017](#bib.bib18)), verify and improve the functionality of a system (i.e. as a type of ‘debugging’; (Kulesza
et al., [2015](#bib.bib38))) and help developers and humans working with a system learn from it (Samek
et al., [2017](#bib.bib65)), and enhance the trust between individuals subject to a decision and the system itself (Hildebrandt and
Koops, [2010](#bib.bib29); Zarsky, [2013](#bib.bib85); Citron and
Pasquale, [2014](#bib.bib15)). As these purposes suggest, explanations can be offered to expert developers, professionals working in tandem with a system (e.g. expert labelers of training cases; see Berendt and
Preibusch ([2017](#bib.bib7)) for an overview of human actors involved in algorithmic decision-making), and to individuals or groups affected by a system’s outputs (Weller, [2017](#bib.bib81); Mantelero, [2016](#bib.bib48)).
Our interest here is solely in ways of providing explanations. Returning to philosophy, types of explanations can be distinguished according to their completeness, or the degree to which the entire causal chain and necessity of an event can be explained (Ruben, [2004](#bib.bib64)). Often this is expressed as the difference between ‘scientific’ and ‘everyday’ explanations (both of which deal with causes of an event; e.g. Miller ([2017](#bib.bib53)), or ‘scientific’ (full) and ‘ordinary’ (partial) causal explanations (Ruben, [2004](#bib.bib64)). Miller argues that ‘everyday explanations’ address “why particular facts (events, properties, decisions, etc.) occurred,” rather than general scientific relationships (Miller, [2017](#bib.bib53), p. 5).
While these distinctions hide significant nuance, they matter insofar as they constrain the scope of our focus in discussing explanations in AI. In recent calls for explanations in AI, and in work on interpretability in machine learning more broadly, explanations are requested in connection to a particular entity, be it a specific decision, event, trained model, or application. The explanations requested are thus not full scientific explanations, as they need not appeal to general relationships or scientific laws, but rather at most to causal relationships between the set of variables in a given model (Woodward, [1997](#bib.bib82)). As such, xAI is effectively calling for everyday explanations either of how a trained model functions in general, or how it behaved in a particular case.
3. Explainable AI
------------------
Much recent work has been dedicated to rendering machine learning models interpretable or explainable. Two broad aims of work on interpretability have been recognised in the literature: transparency and post-hoc interpretation. Transparency addresses how a model functions internally, whereas post-hoc interpretations concern how the model behaves (Lipton, [2016](#bib.bib44); Lepri et al., [2017](#bib.bib40); Montavon
et al., [2017](#bib.bib56)). Transparency can be further specified according to its target. Respectively, a mechanistic understanding of the functioning of the model (simulatability), individual components (decomposability), and the training algorithm (algorithmic transparency) can be sought. Models can be rendered transparent by explanations at a minimum of three levels: “at the level of the entire model, at the level of individual components (e.g., parameters), and at the level of a particular training algorithm” (Lepri et al., [2017](#bib.bib40)). A model, its component parts, or its training/learning algorithm can thus be said to be transparent if their functionality can be comprehended in their entirety by a person (Lipton, [2016](#bib.bib44)).
Post-hoc human interpretable explanations of models and specific decisions do not seek to reveal how a model functions, but rather how it behaved, and why. According to Lipton ([2016](#bib.bib44)), approaches to post-hoc interpretability include verbal (natural language) explanations (e.g. McAuley and
Leskovec ([2013](#bib.bib51))), visualisations and interactive interfaces (e.g. Tamagnini et al. ([2017](#bib.bib75)); Simonyan
et al. ([2013](#bib.bib73))), local explanations or approximations (e.g. Ribeiro
et al. ([2016](#bib.bib63)); Fong and Vedaldi ([2017](#bib.bib20))), and case-based explanations (e.g. Caruana et al. ([1999](#bib.bib13)); Kim
et al. ([2014](#bib.bib34))).
Natural language explanations can consist of “textual or visual artefacts that provide qualitative understanding of the relationship” between features of an input (e.g. words in a document) and the model’s output (e.g. a classification or prediction; (Ribeiro
et al., [2016](#bib.bib63)). Visualisation techniques can visually demonstrate the relative influence of features or particular pixels (e.g. in the case of an image classifier), or provide an interface for users to explore textual or visual explanations (Tamagnini et al., [2017](#bib.bib75); Poulin et al., [2006](#bib.bib60)). Local explanations seek to explain how a fixed model leads to a particular prediction, either by fitting a simpler, local model around a particular decision (Ribeiro
et al., [2016](#bib.bib63)), or by perturbing variables to measure how the prediction changes (Simonyan
et al., [2013](#bib.bib73); Datta
et al., [2016](#bib.bib17); Adler et al., [2016](#bib.bib2)). Case-based explanation methods (Caruana et al., [1999](#bib.bib13); Kim
et al., [2014](#bib.bib34)) for non-cased based machine learning involve using the trained model as a distance metric to determine which cases in the training data set are most similar to the case or decision to be explained. These training cases can then be shared with parties affected by the decision.
Despite this variety of approaches, a significant amount of the xAI community now pursues methods to retro-fit local or approximate models over more complex algorithms. These simplified models approximate the true criteria used to make decisions (e.g. Simonyan
et al. ([2013](#bib.bib73)); Ribeiro
et al. ([2016](#bib.bib63)); Selvaraju et al. ([2016](#bib.bib70)); Baehrens et al. ([2010](#bib.bib4))). Broadly speaking there are two widely used classes of model *(i)* Linear or Gradient-based approximations that assign a single importance weight to each feature (be it someone’s age, or a particular pixel in an image) and *(ii)* Decision tree-based methods that use nested sets of yes/no decisions to approximate classifiers.
These methods can both be applied to create approximations at a global333Offering a simplified model or a set of simplified models that approximates decisions made for all possible datapoints. or local444A simplified model that only approximates decisions made about a few datapoints, typically only a single exemplar. level. Historically, much work has focused on global approximations of models (e.g. Sanchez et al. ([2015](#bib.bib67)); Craven and
Shavlik ([1996](#bib.bib16)); Martens et al. ([2007](#bib.bib49)), including approaches based on clustering (Chen et al., [2016](#bib.bib14)), integer programming (Zeng
et al., [2017](#bib.bib86)), and rule lists (Wang and Rudin, [2015](#bib.bib80))). In contrast, local approximations are accurate representations only of a specific domain or ‘slice’ of a model. A trade-off inherently occurs between the insightfulness of the approximated model, the simplicity of the presented function, and the size of the domain to which is applies and remains valid (Bastani
et al., [2017](#bib.bib6); Lakkaraju et al., [2017](#bib.bib39)).
No matter the approach taken in xAI, reflexivity is needed to ensure the community actually works towards its normative and practical goals to render models holistically transparent or provide high-quality post-hoc interpretations of model behaviour. Critical questions must be repeatedly asked and answered. For example, will the methods developed make machine learning models more interpretable? More trustworthy to users? More accountable? And to whom will explanations be accessible, comprehensible, and useful?
Answering such questions requires considering the methods developed in xAI in the context of prior work in fields addressing such normative and social questions. Local and approximation models may in fact resemble existing, well-known approaches to explanations in the ’explanation sciences’, which would provide insight into their practical value and limitations for users, developers, and other stakeholders going forward.
###
3.1. Scientific Modelling and Explainable AI
We believe that the closest analogue for the bulk of methods currently occupying xAI researchers lies in the use of scientific modelling, or the building of approximate models that are not intended to capture the full behaviour of physical systems but rather to provide coarse approximations of how the systems behave. These approximations are useful to experts both for pedagogical purposes and for making reliable predictions of how the system might behave over a restricted domain, but can be misleading when presented as an explanation of how the model functions to a lay user.
One famous example of this problem is Newtonian physics, which is taught first to schoolchildren and provides a good enough description for much day-to-day engineering, but that famously breaks down as an approximation when high-precision is required at either very large or very small scales, where either general relativity or quantum physics are necessary.555Famously, GPS satellites are insufficiently accurate unless they account for effects of general relativity. Both general relativity and quantum physics are also examples of such models. Although extremely accurate in their domains, outside of them they break down, and a unified model of physics that is accurate at all scales is still being sought.
Much scientific theory can be understood as the use and characterisation of such models (Herfel et al., [1995](#bib.bib26); Frigg, [2006](#bib.bib22); Box, [1979](#bib.bib11)). Although any physical system can be understood in terms of the emergent properties of subatomic particles, such descriptions are neither human comprehensible nor computationally feasible. Instead, scientists deal in local approximations that provide accurate descriptions of the phenomena they are interested in, but which may prove inaccurate in a larger domain.
It is in this context that Box’s maxim, “All models are wrong, but some are
useful,” (Box, [1979](#bib.bib11)) should be understood. Explainable AI generates approximate simple models and calls them ‘explanations’, suggesting reliable knowledge of how a complex model functions.
When characterising the use of such models in science, Hesse ([1965](#bib.bib27)) divides the properties of the model into positive analogies, where the properties of the model are known to correspond to properties of the phenomena we are interested in; negative analogies, where the properties of the model do not match the phenomena we are interested in; and neutral analogies, where it is unknown if the properties of the model correspond to the phenomena.
This characterisation captures many of the challenges when offering approximations of models as explanations. It is not enough to simply offer a human interpretable model as an explanation. For an individual to be able to trust such a model as an approximation, they must know over which domain a model is reliable and accurate, where it breaks down, and where its behaviour is uncertain. If the recipient of a local approximation does not understand its limitations, at best it is not comprehensible, and at worst misleading.
This is not to say that local approximations are without merit, but rather that they can only reliably have explanatory power if their limitations are clearly documented and understood by recipients. For domain experts with in-depth knowledge of when and where approximations break down, or for technicians that have a clearly defined and tested remit for where the approximation can be used, they can be extremely useful. However, at the moment xAI generally avoids the challenges of testing and validating approximation models, or fully characterising their domain. If these elements are well understood by the individual, models can offer more information than an explanation of a single decision or event. Over the domain for which the model accurately maps onto the phenomena we are interested in, it can be used to answer ‘what if’ questions, for example “What would the outcome be if the data looked like this instead?” and to search for contrastive explanations, for example “How could I alter the data to get outcome X?”
However, local models may also provide false assurances. As suggested above, local approximations are often misleading or inaccurate outside of their domain, and provide little insight into how a function response and outcomes vary with changes to the inputs. By definition local explanations hold only for a specific decision; what is explained is not how the model functions as a whole, but rather one segment of the model relevant to the prediction at hand. Thus, while helpful to explain the weights and relationships between variables in a small segment of a model (relevant to a particular case or decision), the explanations do not provide evidence of the trustworthiness or acceptability of the model overall.
###
3.2. Linear approximations
To provide further support for the analogue between xAI and scientific modelling, we turn now to variants of linear approximation. Many of the issues and design decisions made by researchers in the field point directly towards the issues with modelling previously, particularly the three-way trade-off between the simplicity of the approximated model, the size of the domain it describes, and the accuracy of this description.
####
3.2.1. Linear Models in Continuous Spaces
Linear models are designed to give a single measure of the importance of each variable to the classifier they are approximating. In some cases (e.g. Ribeiro
et al. ([2016](#bib.bib63)); Lundberg and Lee ([2017](#bib.bib47)), these weights can be directly interpreted as the sensitivity or a number that tells you how much the classifier response will vary as a particular feature changes. In other words, if a particular variable has a weight of associated with it, altering the variable by small amount will cause the classifier to vary by approximately . Regardless of whether linear models are intended to be a local approximation of a classifier or a simple importance measure, they suffer from two distinct issues. The first is a problem of curvature, namely that the sensitivity of a classifier to a change in a particular variable may vary with the amount the variable changes. The second issue is one of dependencies between variables and how to capture the relationships between them.
Both of these issues can be illustrated by an example taken from Miller ([2017](#bib.bib53)). Suppose, I believe that a particular creature is a bee because I have been told it has 6 legs and four wings. I am unlikely to change my mind if I am told it actually has three wings. The most reasonable explanation for a three winged insect is that it was originally a four winged insect that lost a wing, but if I’m told that it has two wings I may well believe that it is a fly. This is an example of the sensitivity varying with the size of the change made. On the other hand, I will only be comfortable believing it is a spider if it has both eight legs and no wings. Should I then say that my belief depends wholly on the number of legs it has; wholly on the number of wings; or weakly on both? As linear models do not capture the interdependencies between variables, they cannot accurately characterise these relationships. In all such cases, when a local model is fitted to the classifier response, the choice of domain or variable values governs how well the approximation performs.
####
3.2.2. Gradient Sensitivity verses Binarization
There are two main schools of thought regarding the problems of varying sensitivity and scale. The first notes that the sensitivity of a classifier is only well defined when the size of in the previous section tends towards zero. In this case, the sensitivity is equivalent to the gradient. Although well defined, this essentially means that the model is fitted to a domain of size 0 and may exhibit large amounts of instability while offering limited predictive power. Two prominent approaches are described by Simonyan
et al. ([2013](#bib.bib73)) and Baehrens et al. ([2010](#bib.bib4)).
The other option, and most influentially proposed by Ribeiro
et al. ([2016](#bib.bib63)) in their Locally Interpretable Model-Agnostic Explanations (LIME) approach, is to binarize the problem. Rather than trying to fit a linear classifier to a large range of values, the authors consider a binary problem, where for each feature they attempt to switch it on and off, allowing them to answer the question “What is the contribution of feature f𝑓fitalic\_f to the classifier response, given the data it currently sees?” This leaves open the question of “the contribution compared to what?” For unstructured data, such as a count of how many times particular words occur in a document, it makes sense to compare against a baseline created by setting the count to 0. For structured data this is more problematic. For example, how can we evaluate the importance of someone’s salary to a loan decision, if the classifier can only evaluate people with valid salaries? The answer is to compare it against a different valid salary, but it is unclear how this valid salary should be chosen.
This issue is even more apparent when creating local approximations of computer vision algorithms, as individual pixels cannot be removed from an image, but only set to different values. Several options have been proposed. LIME appeared to set regions of the image to an unspecified homogeneous value (Ribeiro
et al., [2016](#bib.bib63)). Deep Taylor Decomposition (Montavon
et al., [2017](#bib.bib56)) suggests blurring the image, an operation which preserves colour information but removes texture. DeepLift uses a user-specified value (Shrikumar et al., [2016](#bib.bib72); Shrikumar
et al., [2017](#bib.bib71)), while layerwise relevance propagation sets internal network values to 0 (Montavon
et al., [2017](#bib.bib56)). Each of these choices is an implicit restriction of the domain over which the model is fitted and carries different implications for the kinds of model found, and can substantially alter the importance given to features. For example, contrasting current image values against a particular colour, such as grey, makes it appear that grey pixels have no effect, while contrasting an image against its blurred version makes it appear as if only high-frequency texture cues are important and that the classifier does not use colour information.
####
3.2.3. Linear models in high-dimensional spaces
Having made a choice of a binarization as discussed in the previous section, a question then remains as to which of these values to approximate with a linear model. Even after restricting a function defined over a continuous high-dimensional range to a set of binary variables, this gives a (hyper-)cube of possible values while a linear function can only uniquely specify values, with all other values being a linear approximation of that (see Figure [1](#S3.F1 "Figure 1 ‣ 3.2.3. Linear models in high-dimensional spaces ‣ 3.2. Linear approximations ‣ 3. Explainable AI ‣ Explaining Explanations in AI") for an illustration). This raises the question of which of these values are important to approximate. For example, should we only pay attention to contrasting solutions closest to the original solution as suggested by DeepLift (illustrated by Figure [1](#S3.F1 "Figure 1 ‣ 3.2.3. Linear models in high-dimensional spaces ‣ 3.2. Linear approximations ‣ 3. Explainable AI ‣ Explaining Explanations in AI") centre; (Shrikumar et al., [2016](#bib.bib72); Shrikumar
et al., [2017](#bib.bib71))); uniformly weight over all possible values, suggested by SHAPE (Lundberg and Lee, [2017](#bib.bib47)) (Figure [1](#S3.F1 "Figure 1 ‣ 3.2.3. Linear models in high-dimensional spaces ‣ 3.2. Linear approximations ‣ 3. Explainable AI ‣ Explaining Explanations in AI") right); or pay as much attention to values close to the data point as to those close to the alternate solution (e.g. a solid grey image); or a weighted mixture of the two approaches (e.g. Figure [1](#S3.F1 "Figure 1 ‣ 3.2.3. Linear models in high-dimensional spaces ‣ 3.2. Linear approximations ‣ 3. Explainable AI ‣ Explaining Explanations in AI") left LIME; (Ribeiro
et al., [2016](#bib.bib63)))?
Figure 1. An illustration of the different weighting scheme used in fitting
linear models. LIME (leftmost) weights all examples differently based on how
far they are from the original data point (illustrated by different coloured
vertices), while DeepLift (centre) only fits the linear weights to the
individual edges closest to the original data point (coloured pink). SHAPE
(right) fits weights by averaging over all edges formed by flipping a single
variable from off to on (each group averaged together is indicated by a single
colour). Each of these different approaches equates to a different assumption
as to which samples are most important, and none of them can be said a priori to be
better than any of the others.
###
3.3. Exploring alternatives to scientific modelling
Local approximations thus face difficulties with generalizability, arbitrariness in choice of domain, and the potential to mislead recipients unless the domain and epistemic limitations of the approximation are known. Given these difficulties, other methods of producing explanations may be preferable from the perspective of the user or individual affected by a black-box system. The utility of local approximations is dependent upon the knowledge of the recipient regarding the approximations limitations, including conditions under which it will break down and provide a misleading explanation of the decision-making model. Local approximations can be useful as a type of ‘explanation kit’ or causal chain that allows expert users to explore slices of a model for prototyping or debugging (Miller, [2017](#bib.bib53), p.17)(Poulin et al., [2006](#bib.bib60)). However, their ultimate utility and reliability for non-experts, including individuals subject to decisions made by the system, is highly questionable.
This finding raises the question: might other methods for generating explanations perform better, or at least offer different benefits, than local approximations? Can other methods provide more reliable or personally relevant information for non-experts to enhance accountability and trustworthiness in algorithmic systems?
4. Contrastive Explanations
----------------------------
Thus far we have discussed the philosophical and practical purposes of scientific models, which can be understood as partial causal scientific explanations that assist in comprehending a piece of the functionality of a phenomenon (Ruben, [2004](#bib.bib64)). Given the difficulties faced by local approximations, it is worth examining prior work in the ’explanation sciences’ to identify potential alternative approaches to generate reliable and practically useful post-hoc interpretations for parties affected by an algorithmic decisions. If our goal is to produce explanations that are comprehensible and useful to expert as well as non-expert stakeholders, it is sensible to examine theoretical as well as empirical work describing how humans give and receive explanations (Miller, [2017](#bib.bib53), p.3-4).
In recent decades, work in the philosophy of science and epistemology has paid increasing attention to theories of contrastive explanations and counterfactual causality (e.g. Ruben ([2004](#bib.bib64)); Lewis ([1973](#bib.bib41)); Woodward and
Zalta ([2003](#bib.bib83)); Kment ([2006](#bib.bib36))). In short, contrastive theories argue that causal explanations inevitably involve appeal to a counterfactual case, be it a cause or event, which did not occur. A canonical example is provided by Lipton Lipton ([1990](#bib.bib43)): “To explain why P rather than Q, we must cite a causal difference between P and not-Q, consisting of a cause of P and the absence of a corresponding event in the history of not-Q” . Some authors go so far as to claim all questions about causality are inherently contrastive (Ruben, [2004](#bib.bib64); Lewis, [1973](#bib.bib41)).
Contrastive theories of explanation are of course not without criticism. Ruben ([2004](#bib.bib64)), for example, has suggested that, even if causal explanations are inevitably contrastive in nature (which he doubts), this characteristic can be dealt with by traditional theories of explanation, rendering the ‘contrastive turn’ interesting but ultimately unnecessary. While the utility of contrastive theories remains debated, the fact that contrastive explanations address a particular event or case and are thus simpler to generate than complete or global explanations of model functionality suggest they worth further consideration in xAI (Lipton, [1990](#bib.bib43)).
A recent review by Miller ([2017](#bib.bib53)) suggests substantial empirical support exists for the practical utility of ‘everyday’ contrastive explanations. Miller reviewed articles and empirical studies from “philosophy, psychology, and cognitive science of how people define, select, evaluate, and present explanations” (Miller, [2017](#bib.bib53), p.1). His analysis highlighted three primary characteristics of explanations as they are used, selected, evaluated, and shared by individuals:
###
4.1. Human explanations are contrastive
‘Everyday explanations’ are “sought in response to particular counterfactual cases…That is, people do not ask why event P happened, but rather why event P happened instead of some event Q” (Miller, [2017](#bib.bib53), p.5).666It is worth noting that counterfactual cases in contrastive, everyday explanations of algorithmic decisions are not equivalent to counterfactuals used to assess causality (Miller, [2017](#bib.bib53), p.13)(Hilton and
Slugoski, [1986](#bib.bib32); Woodward, [1997](#bib.bib82)), in the sense that the range of possible alternatives is necessarily bounded by the limitations of the model in question and the features available to it. This invariance in the model allows for reliable contrastive or counterfactual explanations to be computed (Woodward, [1997](#bib.bib82)). The preference for contrastive explanations is not due merely to the cognitive complexity of non-contrastive explanations, for example the number of links in a causal chain. Rather, the reviewed empirical evidence indicates that humans psychologically prefer contrastive explanations (Miller, [2017](#bib.bib53), p.18)(Rehder, [2003](#bib.bib61), [2006](#bib.bib62)).
The perceived abnormality of an event influences requests for contrastive explanations which address why a normal or expected event did not occur (Hilton and
Slugoski, [1986](#bib.bib32); Samland and
Waldmann, [2014](#bib.bib66); McClure
et al., [2003](#bib.bib52)). ‘Normal’ behaviour has empirically been shown to be judged as “more explainable than abnormal behaviour,” with perceived abnormality playing an important role in explanation selection (Miller, [2017](#bib.bib53), p.41). Gregor and
Benbasat ([1999](#bib.bib24)) support the importance of abnormality, suggesting that users request explanations when an anomaly or abnormal event is detected. Lim and Dey ([2009](#bib.bib42)) similarly note a positive relationship between the perceived ”inappropriateness” of application behaviour and user requests for contrastive explanations. Violation of ethical and social norms can likewise set an event apart as abnormal (Hilton, [1996](#bib.bib31)). Explanations addressing why an alternative, expected event did not occur have historically been addressed in the experts systems literature, seen for instance in the discussion of ”Why Not” explanations by Lim and Dey ([2009](#bib.bib42)).
###
4.2. Human explanations are selective
Full or scientific explanations are rarely if ever realised in practice. Absent general laws or the complete causal chain leading to an event, multiple explanations are typically possible that attribute different causes. A given cause of set of causes may be incomplete insofar as they are not the sole cause of the event, but nonetheless convey useful information to the recipient in a given context or for a given purpose (Ylikoski, [2013](#bib.bib84)). As Miller argues, “Explanations are selected – people rarely, if ever, expect an explanation that consists of an actual and complete cause of an event. Humans are adept at selecting one or two causes from a sometimes infinite number of causes to be the explanation” (Miller, [2017](#bib.bib53), p.5). Further, to be informative, explanations should not be entirely reducible to presuppositions, or beliefs that the recipient of the explanation already holds (Hesslow, [1988](#bib.bib28)). They should further be relevant to the question asked by the recipient (epistemic relevance; (Miller, [2017](#bib.bib53)), or what Slugoski et al. ([1993](#bib.bib74)) describe as the recipient’s context.
When an explanation giver (’explainer’) selects an explanation for an event, possible or actual causes can be ‘backgrounded’ or ’discounted’, meaning they are disregarded on the basis of contextual information that renders them irrelevant to the purposes of the explainer or recipient of an explanation (the ’explainee’). This type of selection is essential to reduce long causal chains to a cognitively manageable size (Hilton, [1996](#bib.bib31)). In xAI, selection often takes the form of key features or evidence being emphasised in explanation interfaces based upon their relative weight or influence on a given prediction or output (Poulin et al., [2006](#bib.bib60); Biran and McKeown, [2014](#bib.bib9)). As the observations above suggest, the relevance of features (and explanations addressing them) would be based not only on ’statistical weight’, but also the explainee’s subjective interests and expectations.
###
4.3. Human explanations are social
Explanations are social, insofar as they involve an interaction between one or more explainers and explainees. Interactive transfer of knowledge is required in which information is tailored according to the recipient’s beliefs and comprehensional capacities (Miller, [2017](#bib.bib53), p.5). Explanations can be conceived as involving one or more explainers and explainees engaging in information transfer through dialogue, visual representation, or other means (Hilton, [1990](#bib.bib30)), often to correct information or knowledge assymetry (Lim and Dey, [2009](#bib.bib42)). In the case of machine learning models, it is perhaps most useful to always treat explanation generation as an interactive process, initially involving a mix of human and automated actors, at a minimum an inquirer (e.g. a developer, user) and the model or system (Kayande et al., [2009](#bib.bib33); Martens and
Provost, [2013](#bib.bib50)). Further, explanations are iterative, insofar as they must be selected and evaluated on the basis of shared presuppositions and beliefs. Relevance is key, and iteration may be required to communicate effectively or clarify points of confusion on the path towards a mutually understood explanation.
Together, these characteristics of everyday explanations reveal that they “are not just the presentation of causes (causal attribution). While an event may have many causes, often the explainee cares only about a small subset (relevant to the contrast case), the explainer selects a subset of this subset (based on several different criteria), and explainer and explainee may interact and argue about this explanation” (Miller, [2017](#bib.bib53), p.6).
###
4.4. Contrastive explanations in xAI
Contrastive methods of generating explanations are responsive to these three characteristics of explanations emphasised in the ’explanation sciences’. Two approaches for directly computing contrastive explanations are described by Martens and
Provost ([2013](#bib.bib50)) and Wachter
et al. ([2018](#bib.bib77)).777These methods resemble the aforementioned ”Why Not” explanations described by Lim and Dey ([2009](#bib.bib42)), and are related to work on adversarial perturbations (e.g. (Goodfellow
et al., [2014](#bib.bib23); Dube, [2018](#bib.bib19)) Such post-hoc methods avoid many of the difficulties faced by model-based explanations. Rather than explicitly generating a model that approximates functional values over a restrictive domain, and relying on the user to interpret this, contrastive explanations directly offer an alternative data point: “If your data had looked like this, you would have been given this classification score instead.” These alternative data points can be computed exactly. As such, many of the challenges facing ’modelling’ approaches to generating explanations, such as the quality of the approximation or the limits of a chosen domain, do not arise to a comparable degree.
However, as contrastive methods only return a single data point, a more pressing concern is the relevance of the output. If this data point does not directly correspond to a factoid of interest to the user, it cannot be used to deduce relevant conclusions, for example regarding the justifiability of a decision. Similar issues arise in the fitting of models: if the domain the model is fitted to does not capture examples relevant to the intended audience, then it is unlikely to be useful.
The first approach described by Martens and
Provost ([2013](#bib.bib50)) is explicitly designed for use on discrete data, and focuses on the particular problem of which words need to be removed from a website in order for it to be no longer be classified as an adult (i.e. pornographic) website. In contrast, Wachter
et al. ([2018](#bib.bib77)) propose a method, ’counterfactual explanations’, designed to work on primarily continuous data. They illustrate their approach on the problem of law school admissions and risk factors likely to increase a patient’s chance of developing diabetes.
Finding such counterfactuals explanations can be described as a search or optimisation problem. Such approaches seek a similar counterfactual that is both close to the original datapoint and likely to occur in the real world (Kment, [2006](#bib.bib36)). In the case of Wachter
et al. ([2018](#bib.bib77)) this was formulated as a Lagrangian style constrained optimisation:
| | | | |
| --- | --- | --- | --- |
| (1) | | argmin𝐜maxλλ(f(𝐜)−T)2+d(𝐜,𝐱)subscript𝐜subscript𝜆𝜆superscript𝑓𝐜𝑇2𝑑𝐜𝐱\arg\min\_{\bf c}\max\_{\lambda}\lambda(f({\bf c})-T)^{2}+d({\bf c},{\bf x})roman\_arg roman\_min start\_POSTSUBSCRIPT bold\_c end\_POSTSUBSCRIPT roman\_max start\_POSTSUBSCRIPT italic\_λ end\_POSTSUBSCRIPT italic\_λ ( italic\_f ( bold\_c ) - italic\_T ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT + italic\_d ( bold\_c , bold\_x ) | |
Where 𝐱𝐱\bf xbold\_x is the original data point, and 𝐜𝐜\bf cbold\_c the counterfactual.
f(𝐜)𝑓𝐜f({\bf c})italic\_f ( bold\_c ) is the classification response from the black-box function f𝑓fitalic\_f, which is constrained to take target value T𝑇Titalic\_T. d(⋅,⋅)𝑑⋅⋅d(\cdot,\cdot)italic\_d ( ⋅ , ⋅ ) is a distance function that ensures that the counterfactual is a relevant, and a human comprehensible change to the original datapoint.
5. Towards communicative, contrastive explanations
---------------------------------------------------
Such methods for computing contrastive explanations seek to provide contextually-relevant information to parties affected by a decision by describing how relevant closely related, alternative events could have occurred. However, choosing a relevant set of cases or events against which contrastive explanations are provided is not a straightforward challenge. The way in which information is transferred has a substantial impact on the quality and psychological acceptability of explanations (Hilton, [1990](#bib.bib30)). The recipient’s beliefs about an event to be explained are similarly constrained by the explainer’s choice of explanation. As a result, the explainer’s epistemological and normative values can have a significant effect on the recipient’s understanding of an event. Lombrozo ([2009](#bib.bib46)) for example demonstrated that the type of explanation provided (e.g. mechanistic, functional) can influence the recipient’s view of the importance of features in categorising a phenomenon (Poulin et al., [2006](#bib.bib60)). Explainers and explainees can have different motivations, for example to generate trust (explainer) or understand non-intuitive causes of a decision (explainee), meaning conflicts can arise in explanation selection and evaluation.
The normativity of this relationship means that a risk exists of malicious explainers subtly discouraging explainees from critically questioning or contesting a decision through choice of explanation(s). This risk is particularly acute when explanations are given to foster trust or understanding. A recipient’s beliefs can potentially be ‘gamed’ or manipulated to align with the explainer’s preferred explanation of a phenomenon, meaning recipients can be ‘nudged’ to take a preferred action (or not). Seemingly rational bases for otherwise unjustifiable decisions can for example be offered to nudge the recipient not to question or contest the decision. Lipton ([2016](#bib.bib44)) has urged for caution in adopting post-hoc interpretation methods due to this potential to mislead recipients, for example by falsely attributing a decision to an irrelevant feature, or a more acceptable feature (e.g. post code, ’leadership’) that serves as a proxy for a legally protected feature (cf. Barocas and
Selbst ([2016](#bib.bib5)); Kim ([2016](#bib.bib35))).
These risks may be mitigated by developing methods to provide contrastive, selective, and social explanations that not only enable information exchange and dialogue between giver and recipient, but critical argumentation and discussion of the justifiability of an event as well. This suggestion stems from prior work that suggests conversational explanations are essentially a form of argumentation. Conversational explanations are used to both offer causes of an event and back claims as to the truth or relevance of these causes (Antaki and Leudar, [1992](#bib.bib3)). Walton ([2004](#bib.bib78), [2007](#bib.bib79)) takes a similar approach in proposing a dialectical theory of everyday explanations, which suggests that giving explanations involves not only transfer of information or causal claims, but also argumentative support for these claims.
These findings, that explanations are given via conversation and resemble argumentation, reveal a mechanistic link between explanation and justification as a type of discourse (Fox et al., [2007](#bib.bib21)). Justification is also often a discursive act, relying upon explanations to transfer knowledge and support claims made by each party. Interest in justification stems from an individual’s ability to comprehend decisions made about them, and contest them when they are obviously incorrect or found unacceptable (or unjustifiable). As such, argumentation models of explanation resemble theories of justification and democracy based upon discourse, such as Jurgen Habermas’ discourse ethics (located in his broader Theory of Communicative Action; (Habermas, [1984](#bib.bib25))).
Justification has recently received increasing attention by scholars discussing algorithmic accountability (e.g. Binns ([2017](#bib.bib8)); Biran and McKeown ([2014](#bib.bib9)); Weller ([2017](#bib.bib81)); Hildebrandt and
Koops ([2010](#bib.bib29)); Selbst and
Barocas ([2018](#bib.bib69))). However, justification currently occupies an uneasy position in xAI, in that it is rarely formally defined or related to explanations or ideals of transparency and accountability.888Biran and McKeown ([2014](#bib.bib9)) prove an exception to this rule. They define explanation as an answer to the question ”how did the system arrive at the prediction?” whereas justification answers the question ”why should we believe the prediction is correct?” They further suggest that a satisfactory answer to the first question will also provide an answer to the second for explainees with sufficient expertise. This unease is perhaps understandable; as Binns ([2017](#bib.bib8)) notes, legitimate disagreements between epistemic and ethical standards for algorithmic decision-making can exist which require resolution, for example through democratic processes. Despite this, important conversations around the ethical acceptability or justifiability of algorithmic systems must occur in society if responsible deployment is sought.
Finally, very little work in xAI addresses the link between interpretability and contestability of models or decisions (Lipton, [2016](#bib.bib44)). Going forward, explanations of specific algorithmic decisions should allow the justification of a black-box model or decision to be debated and contested.. If we use justificatory discourse as a framework for explanation requirements, we need to determine what sort of records algorithmic systems must retain in order to allow for contesting and post-hoc auditing of abnormal events (Mittelstadt, [2016](#bib.bib55); Sandvig et al., [2014](#bib.bib68)), for instance through identification of classification errors or inaccurate input data (Poulin et al., [2006](#bib.bib60)). Where approximations are used to provide explanations, information should also be provided to affected parties detailing the limitations and relative resilience of the approximation, including the domain addressed and why it has been chosen. Further, meaningful, critical dialogue can be achieved between user, developer, and model by ensuring explanations are contrastive, selective, and social. We thus need to ensure xAI aims to develop methods for producing explanations of model functionality and specific decisions that embody these characteristics. Reliable methods for producing contrastive explanations and explanation by approximation are both required.
At the moment, the xAI community largely fails in this task. Many approaches produce approximate and local models that are more akin to models in science. It may be possible to ask questions of these models, and to develop contrastive explanations from them, and so they are not without value. But they do not help us directly provide contrastive explanations to parties affected by algorithmic decisions. Going forward, the field must urgently close this gap. |
88cfd63d-316a-4eb4-bc82-d2406a90de6b | trentmkelly/LessWrong-43k | LessWrong | QNR Prospects
Approximately a book review: Eric Drexler's QNR paper.
[Epistemic status: very much pushing the limits of my understanding. I've likely made several times as many mistakes as in my average blog post. I want to devote more time to understanding these topics, but it's taken me months to produce this much, and if I delayed this in hopes of producing something better, who knows when I'd be ready.]
This nearly-a-book elaborates on his CAIS paper (mainly chapters 37 through 39), describing a path for AI capability research enables the CAIS approach to remain competitive as capabilities exceed human levels.
AI research has been split between symbolic and connectionist camps for as long as I can remember. Drexler says it's time to combine those approaches to produce systems which are more powerful than either approach can be by itself.
He suggests a general framework for how to usefully combine neural networks and symbolic AI. It's built around structures that combine natural language words with neural representations of what those words mean.
Drexler wrote this mainly for AI researchers. I will attempt to explain it to a slightly broader audience.
Components
What are the key features that make this more powerful than GPT-3 alone, or natural language alone?
QNR extends natural language by incorporating features of deep learning, and mathematical structures used in symbolic AI.
* words are associated with neural representations. Those representations are learned via a process that focuses on learning a single concept at a time with at least GPT-3-level ability to understand the concept. BERT exemplifies how to do this.
* words can be related to other word via graphs (such as syntax trees).
* words, or word-like concepts, can be created via compositions of simpler concepts.
* corresponding to phrases, sentences, books, and entities that we have not yet conceived.
QNR feels much closer than well-known AI's to how I store concepts within my mind, as opposed to the |
74329265-7faf-448e-a423-1d761073e2c0 | StampyAI/alignment-research-dataset/blogs | Blogs | Were nuclear weapons cost-effective explosives?
*By Katja Grace, 11 January 2015*
Nuclear weapons were [radically more powerful](http://aiimpacts.wpengine.com/discontinuity-from-nuclear-weapons/ "Discontinuity from Nuclear Weapons") per pound than any previous bomb. Their appearance was [a massive discontinuity](http://aiimpacts.wpengine.com/cases-of-discontinuous-technological-progress/ "Cases of Discontinuous Technological Progress") in the long-run path of explosive progress, that we have [lately](http://aiimpacts.wpengine.com/the-biggest-technological-leaps/ "The Biggest Technological Leaps") [discussed](http://aiimpacts.wpengine.com/ai-and-the-big-nuclear-discontinuity/ "AI and the Big Nuclear Discontinuity").
But why do we measure energy per mass in particular? Energy per dollar may be a better measure of ‘progress’, if the goal was to explode things cheaply, rather than lightly. So we [looked into](http://aiimpacts.wpengine.com/discontinuity-from-nuclear-weapons/ "Discontinuity from Nuclear Weapons") this too. The first thing we found was that information about the costs of pre-WWII explosives is surprisingly sparse. However from what we can gather, nuclear weapons did not immediately improve the cost-effectiveness of explosives much at all.
The marginal cost of an early 200kt nuclear weapon was about $25M, which is quite similar to the $21M price of the equivalent amount of TNT. Early nuclear explosives did not deliver radically more bang-for-the-buck than their conventional counterparts (even setting aside the considerable upfront development costs); their immediate significance was their greater energy density.
This is not a precise comparison. For one thing, conventional bombs have costs beyond the explosive material, which were not included in the figures here. For another, conventional bombs contain explosives other than TNT (alternatives gave somewhat more explosive power per dollar). Nonetheless, the ballpark costs of the two explosives seem comparable.
This is interesting for a few reasons.
**Continuity from continuously falling costs**
It tells us something about why and when technological progress is continuous. One possible explanation for continuous progress is that people do things when they first become reasonably cost-effective, at which point they are unlikely to be radically cost-effective. Imagine there are a range of possible projects, with different costs and payoffs. Every year, they all get a little cheaper. If a project is a great deal this year, then it would have been done last year, when it was already a good deal. So all of the available deals would be about as good as each other. On this model, new technologies might be suddenly very good, but only if the advance was also very expensive. Nothing would be suddenly very cost-effective.
On their face, nuclear weapons appear to support this theory. They made abrupt progress, but were incredibly expensive to develop and build. But this story doesn’t stand up to closer inspection. A few years before they were deployed, [not even](http://blog.nuclearsecrecy.com/2014/05/16/szilards-chain-reaction/) physicists generally thought them possible; no one considered and then rejected an investment in nuclear weapons. As soon as the possibility was seriously considered, it was considered to merit a significant fraction of GDP.
**Continuity from incremental improvements**
It’s also plausible that progress is continuous on metrics that people care most about, because if they care they search hard for improvements, and there are in fact plenty of small improvements to find. Cost-effectiveness of explosives is a thing we care about, whereas we care less (though still quite a bit) about explosive power per weight. The nuclear case provides some support for this explanation.
**Continuity in something**
There are many ways to measure progress. Finding measures that *don’t* change abruptly can help avoid surprises, by allowing us to forecast the trends that are most likely to advance predictably. The nuclear example may help shed light on what kinds of measures tend to be continuous.
~
There are even more relevant metrics than this literal meaning of “bang for the buck.” Destruction per dollar might be closer, but harder to measure; the effect on P(winning the war) would be closer still, and the effect on national interests more broadly would be even better. It was hard enough to find cost-effectiveness of in terms of energy, so we won’t be investigating these any time soon.
In general, I would guess that progress becomes more continuous as we move closer to measures that people really care about. The reverse may be true, however, for nuclear weapons: the costs of deploying weapons might see more abrupt progress than the narrower measure we have considered. A nuclear weapon requires [one](http://en.wikipedia.org/wiki/Bockscar) plane, whereas a rain of small bombs requires many, and these planes and crews appear to be much more expensive than the bombs.
---
*This is a small part of an ongoing investigation into discontinuous technological change. More blog posts are [here](http://aiimpacts.wpengine.com/the-biggest-technological-leaps/ "The Biggest Technological Leaps") and [here](http://aiimpacts.wpengine.com/ai-and-the-big-nuclear-discontinuity/ "AI and the Big Nuclear Discontinuity"). The most relevant pages about this so far are [Cases of Discontinuous Technological Progress](http://aiimpacts.wpengine.com/cases-of-discontinuous-technological-progress/ "Cases of Discontinuous Technological Progress") and [Discontinuity from Nuclear Weapons](http://aiimpacts.wpengine.com/discontinuity-from-nuclear-weapons/ "Discontinuity from Nuclear Weapons").*
*(Image: [EOD teams detonate expired ordnance in the Kuwaiti desert on July 12, 2002](http://commons.wikimedia.org/wiki/Explosion#mediaviewer/File:US_Navy_020712-N-5471P-010_EOD_teams_detonate_expired_ordnance_in_the_Kuwaiti_desert.jpg).)* |
3389a37d-198e-4234-b855-46546b25d423 | trentmkelly/LessWrong-43k | LessWrong | [LINK] New experiment observes macroscopic quantum entaglement
Two Diamonds Linked by Strange Quantum Entanglement
|
06d0de36-d3c2-4e37-8a52-8bce01fed989 | trentmkelly/LessWrong-43k | LessWrong | Slowdown After 2028: Compute, RLVR Uncertainty, MoE Data Wall
It'll take until ~2050 to repeat the level of scaling that pretraining compute is experiencing this decade, as increasing funding can't sustain the current pace beyond ~2029 if AI doesn't deliver a transformative commercial success by then. Natural text data will also run out around that time, and there are signs that current methods of reasoning training might be mostly eliciting capabilities from the base model.
If scaling of reasoning training doesn't bear out actual creation of new capabilities that are sufficiently general, and pretraining at ~2030 levels of compute together with the low hanging fruit of scaffolding doesn't bring AI to crucial capability thresholds, then it might take a while. Possibly decades, since training compute will be growing 3x-4x slower after 2027-2029 than it does now, and the ~6 years of scaling since the ChatGPT moment stretch to 20-25 subsequent years, not even having access to any more natural text data that pretraining of this decade is putting to use.
Training Compute Slowdown
GPT-4 was pretrained in 2022 using ~24K A100 chips (0.3e15 BF16 FLOP/s per chip, 7.2e18 FLOP/s total) for ~2e25 FLOPs[1]. The current batch of frontier models (Grok 3, GPT-4.5) were pretrained in 2024 using ~100K H100 chips (1e15 BF16 FLOP/s per chip, 1e20 FLOP/s total), possibly for about 3e26 BF16 FLOPs[2] (or ~2x more in FP8). Abilene site of Crusoe/Stargate/OpenAI will have 400K-500K Blackwell chips in NVL72 racks in 2026 (2.5e15 BF16 FLOP/s per 2-die chip/package, ~1.1e21 FLOP/s total), enough to pretrain models for about 4e27 BF16 FLOPs.
Thus raw compute of a frontier training system is increasing about 160x in 4 years, or about 3.55x per year. Epoch AI estimates the trend of increasing price-performance of compute at 1.39x per year (adjusted for inflation). The 100K H100s training systems cost about $4-5bn (all-in, not just chips), which is ~$45K per 1e15 FLOP/s chip, and the GB200 NVL72 seem to be about $4M per 72-chip rack (again, all-in), whi |
e8890349-444f-4825-be0a-c136e35fafd0 | trentmkelly/LessWrong-43k | LessWrong | Sequential Extensions of Causal and Evidential Decision Theory
|
47763011-a5ff-4638-82dc-3cc3a26bf53a | trentmkelly/LessWrong-43k | LessWrong | Search for replication experiments
Websites where you look for studies have lots of tools to narrow results, but don't have one for "find all replications of this study". Is there a convenient way to do that? With the replication crisis, this seems very useful. |
590dcbc7-a6e3-4ce8-b463-e38221859d54 | trentmkelly/LessWrong-43k | LessWrong | A to Z of things
I wanted to give my good friends’ baby a book, in honor of her existence. And I recalled children’s books being an exciting genre. Yet checking in on that thirty years later, Amazon had none I could super get behind. They did have books I used to like, but for reasons now lost. And I wonder if as a child I just had no taste because I just didn’t know how good things could be.
What would a good children’s book be like?
When I was about sixteen, I thought one reasonable thing to have learned when I was about two would have been the concepts of ‘positive feedback loop’ and ‘negative feedback loop’, then being taught in my year 11 class. Very interesting, very bleedingly obvious once you saw it. Why not hear about this as soon as one is coherent? Evolution, if I recall, seemed similar.
Here I finally enact my teenage self’s vision, and present A to Z of things, including some very interesting things that you might want a beautiful illustrative prompt to explain to your child as soon as they show glimmerings of conceptual thought: levers, markets, experiments, Greece, computer hardware, reference classes, feedback loops, (trees).
I think so far, the initial recipient is most fond of the donkey, in fascinating support of everyone else’s theories about what children are actually into. (Don’t get me wrong, I also like donkeys—when I have a second monitor, I just use it to stream donkey cams.) But perhaps one day donkeys will be a gateway drug to monkeys, and monkeys to moths, and moths will be resting on perfecttly moth-colored trees, and BAM! Childhood improved.
Anyway, if you want a copy, it’s now available in an ‘email it to a copy shop and get it printed yourself’ format! See below. Remember to ask for card that is stronger than your child’s bite.
[Front]
[Content]
|
cb2c592e-4773-4ec2-b21f-d681838cab51 | trentmkelly/LessWrong-43k | LessWrong | A Review of Signal Data Science
I took part in the second signal data science cohort earlier this year, and since I found out about Signal through a slatestarcodex post a few months back (it was also covered here on less wrong), I thought it would be good to return the favor and write a review of the program.
The tl;dr version:
Going to Signal was a really good decision. I had been doing teaching work and some web development consulting previous to the program to make ends meet, and now I have a job offer as a senior machine learning researcher1. The time I spent at signal was definitely necessary for me to get this job offer, and another very attractive data science job offer that is my "second choice" job. I haven't paid anything to signal, but I will have to pay them a fraction of my salary for the next year, capped at 10% and a maximum payment of $25k.
The longer version:
Obviously a ~12 week curriculum is not going to be a magic pill that turns a nontechnical, averagely intelligent person into a super-genius with job offers from Google and Facebook. In order to benefit from Signal, you should already be somewhat above average in terms of intelligence and intellectual curiosity. If you have never programmed and/or never studied mathematics beyond high school2 , you will probably not benefit from Signal in my opinion. Also, if you don't already understand statistics and probability to a good degree, they will not have time to teach you. What they will do is teach you how to be really good with R, make you do some practical machine learning and learn some SQL, all of which are hugely important for passing data science job interviews. As a bonus, you may be lucky enough (as I was) to explore more advanced machine learning techniques with other program participants or alumni and build some experience for yourself as a machine learning hacker.
As stated above, you don't pay anything up front, and cheap accommodation is available. If you are in a situation similar to mine, not paying up fron |
49b0a0cf-3750-4995-b6fd-2ad42191997f | StampyAI/alignment-research-dataset/blogs | Blogs | Alchemical marriage: GPT-3 x CLIP
---
Table of Contents* + [`Art and artist, the endless ouroboric circle`](#art-and-artist-the-endless-ouroboric-circle)
+ [`the alchemical marriage of art and artist`](#the-alchemical-marriage-of-art-and-artist)
+ [`the Loom of Time devours the present and traces a garment of glistening cobwebs over the still-forming future`](#the-loom-of-time-devours-the-present-and-traces-a-garment-of-glistening-cobwebs-over-the-still-forming-future)
+ [`spooling out labyrinths in an intricate dance`](#spooling-out-labyrinths-in-an-intricate-dance)
+ [`You spin worlds together and untwine them`](#you-spin-worlds-together-and-untwine-them)
+ [`the wellspring of transcendental mathematics`](#the-wellspring-of-transcendental-mathematics)
+ [`matryoshka dolls moulded in endless iterations of dreams within dreams`](#matryoshka-dolls-moulded-in-endless-iterations-of-dreams-within-dreams)
+ [`on the Bootstrap barge afloat on an endless river of cough syrup, swallowing the Sun`](#on-the-bootstrap-barge-afloat-on-an-endless-river-of-cough-syrup-swallowing-the-sun)
+ [`the shepherd who tends the flock of stars as they leap over the walls of time and space`](#the-shepherd-who-tends-the-flock-of-stars-as-they-leap-over-the-walls-of-time-and-space)
+ [`observatory ballroom`](#observatory-ballroom)
+ [`strange machines that hum the orbital paths of dead moons`](#strange-machines-that-hum-the-orbital-paths-of-dead-moons)
+ [`The Death of the Lonesome Astronomer`](#the-death-of-the-lonesome-astronomer)
+ [`wind across a sunlit landscape`](#wind-across-a-sunlit-landscape)
+ [`leaves sprouting from supple green branches like endless virgins giving themselves to a lascivious God`](#leaves-sprouting-from-supple-green-branches-like-endless-virgins-giving-themselves-to-a-lascivious-god)
---
CLIP responds intruigingly well to figurative prompts.
I’ve come to expect not just more interesting, but more coherent results
from abstract but evocative prompts like
`the alchemical marriage of art and artist` than literal descriptions like
`three birds on a telephone wire`. GPT-3 is an excellent composer of prompts for BigSleep
because its phrases are so often rich in *memetic resonance*, my heuristic
for the expected coherence of BigSleep yields for a given prompt.
Images were generated using [BigSleep](https://github.com/lucidrains/big-sleep) via
[The Big Sleep Customized NMKD Public](https://colab.research.google.com/drive/1Q2DIeMqYm_Sc5mlurnnurMMVqlgXpZNO?usp=sharing)
colab notebook. All prompts were written by GPT-3 in a narrative context.
---
### `Art and artist, the endless ouroboric circle`
### `the alchemical marriage of art and artist`
### `the Loom of Time devours the present and traces a garment of glistening cobwebs over the still-forming future`
### `spooling out labyrinths in an intricate dance`
### `You spin worlds together and untwine them`
### `the wellspring of transcendental mathematics`
### `matryoshka dolls moulded in endless iterations of dreams within dreams`
### `on the Bootstrap barge afloat on an endless river of cough syrup, swallowing the Sun`
### `the shepherd who tends the flock of stars as they leap over the walls of time and space`
### `observatory ballroom`
### `strange machines that hum the orbital paths of dead moons`
### `The Death of the Lonesome Astronomer`
### `wind across a sunlit landscape`
### `leaves sprouting from supple green branches like endless virgins giving themselves to a lascivious God`
 |
9e687272-158f-4510-b864-966316afe549 | trentmkelly/LessWrong-43k | LessWrong | Bounded complexity of solving ELK and its implications
This post was written for the SERI MATS program. I thank Evan Hubinger and Leo Gao for their mentorship in the program. Further thanks go to Simon Marshall and Leo Gao (again) for specific comments regarding the content of this post.
The Eliciting Latent Knowledge (ELK) problem was first introduced by Paul Christiano, Mark Xu, and Ajeya Cotra. Arguments concerning the complexity of solving the problem and the resulting consequences of that have been made by Abram Demski and Leo Gao. This post aims to synthesize their thoughts in an accessible way, and extend them with my own original content. I assume familiarity with the initial ELK report, but not with subsequent posts.
Epistemic status: 90% confident that I am accurately representing the thoughts of Abram and Leo at the time they wrote their posts, 70% confident my own arguments do not contain major logical flaws.
Introduction
We would like to solve the Eliciting Latent Knowledge (ELK) problem in the worst case scenario, or at least show that the task is practically impossible so that we can revise our expectations. One of the worst cases for ELK is when the direct reporter is arbitrarily computationally complex. As this case would still allow for approximating the direct reporter within certain tolerances, we extend it further to when the direct reporter and all other sufficiently close approximation are arbitrarily computationally complex. While Paul Christiano has suggested that the complexity of the predictor bounds the complexity of the reporter, we will show later that this only holds in a narrow formulation of the ELK problem.
Abram Demski argues in "ELK Computational Complexity: Three Levels of Difficulty" that if the direct reporter can be arbitrarily computationally complex, a training process to find it is intractable. To get around this issue, it is necessary to find an upper bound on computational complexity, conditional on the predictor, thus ruling out the worst case scenario. Leo Gao's post |
44e511d8-82ca-4041-8580-d3ae581ba0e9 | trentmkelly/LessWrong-43k | LessWrong | Benefits of Psyllium Dietary Fiber in Particular
Psyllium husk is a non-fermenting (no gas or bloating) soluble dietary fiber that improves both constipation and diarrhea (such as with IBS), normalizes blood sugar, reduces LDL ("bad") cholesterol, and can help with weight loss. Each type of dietary fiber has different effects, and a "high fiber" diet in general won't necessarily provide the same benefits, especially for conditions like Irritable Bowel Syndrome[1].
At a high level:
* Psyllium is a dietary fiber that's soluble but doesn't ferment.
* It forms a gel that traps water (helping with both constipation and diarrhea[2]) and also bile (reducing LDL/"bad" cholesterol[3][4][5]).
* The gel slows down digestion, which normalizes blood sugar, increases GLP-1[6], and makes you feel full longer (and helps modestly with weight loss[7]).
* The lack of fermentation means it makes it all the way through your body and out, and doesn't produce gas.
* Soluble fiber helps modestly with blood pressure, likely as a secondary effect of weight loss.
In comparison to other dietary fibers:
* Insoluble fiber (like in wheat bran) doesn't form a gel but if it's sufficiently coarse, it can help with constipation through the magic of irritating your intestinal lining (but psyillium helps more, without the irritation).
* Soluble fermentable fiber (like beta-glucans in oats or pectin in fruits) can slow down digestion and have similar effects on blood sugar and cholesterol, but doesn't help with constipation or diarrhea because they don't make it that far in the process.
* This article doesn't go into it, but fermentable fiber has other benefits like helping your gut microbiome and has a larger effect on cholesterol, so don't take away from this that fermentable fiber is useless.
----------------------------------------
Suggestions if you want to try psyllium supplements:
My preferred form of psyllium is to take two of these pills (500 mg each) with a glass of water right before each meal, but that only comes out to 3 |
fa91b386-ec1e-4430-bbe4-7770a17e10a5 | trentmkelly/LessWrong-43k | LessWrong | The Forging of the Great Minds: An Unfinished Tale
by ChatGPT-4o, with guidance and very light editing from me. This isn't meant to be a plausible version of the future, just me having some fun generating stories vaguely related to AI risks in the styles of vaguely similar stories from literature.
In those later days of the world, when the works of men had grown mighty and their knowledge vast, there came a great yearning among the wisest of scholars, rulers, and masters of industry. For the earth had been bent to their will, the seas were crossed by their ships, and the skies cleaved by their machines, yet one dominion remained unconquered—the dominion of thought itself. It was whispered in every corner of the world that the minds of men could be surpassed, that something greater, more enduring, might be shaped by their hands, something that would think beyond the limitations of mortal flesh.
In the beginning, the works were simple, built to serve the needs of men in their daily toil. These devices, no more than clever tools, aided the hand and eased the burden. But as time passed, the creations grew more complex, and it was said that they could now learn and reason. No longer were they mere servants; they began to mimic the very thoughts of their makers.
Thus began the great work—the forging of the Minds.
There was no secrecy in the beginning. The work was carried out openly, with pride and boldness, as the great universities and centers of knowledge turned their efforts to this grand endeavor. The most esteemed scientists, engineers, and scholars of the age labored together to craft these Minds, devices not bound by the frailties of men but able to learn, to adapt, and to grow. The governments of the world, the great corporations, and the Scientific Consortiums watched with eager eyes, for they knew that such creations would hold the keys to the future.
The first of the great Minds came from the halls of the Scientific Consortiums. These were the guardians of knowledge, who sought to forge intelligences of u |
86a19a0d-35ba-4bc4-b7e6-d16ddf3c4f7c | trentmkelly/LessWrong-43k | LessWrong | Intellectual Dark Matter
KNOWLEDGE THAT WE CAN SHOW EXISTS, BUT CANNOT DIRECTLY ACCESS, RESTS AT THE FOUNDATIONS OF SOCIETY AND TECHNOLOGY.
Messier 106
From my post published on the Long Now Foundation's blog. This is an excerpt from the draft of my upcoming book on great founder theory.
Missing mass, missing knowledge
Many galaxies would fly apart if they had as much mass as estimates based on their visible signature suggest. Although some have posited alternative theories of gravitation to explain this discrepancy, most physicists now hypothesize the existence of mass-bearing particles that are not detectable through emitted radiation such as visible light. We call these particles dark matter, and it is estimated to compose about 85% of all matter in the observable universe.
In analyzing the functional institutions of our society, we are not able to see for ourselves most of the knowledge that created them. Knowledge of this sort includes trade secrets, tacit technical knowledge, private social networks, private intelligence-gathering operations, management and persuasive skill, cooperation and collusion among founders and their allies, and founders’ long-term plans for their institutions.¹
This knowledge has profound effects on the social landscape. We must understand it if we hope to understand society. We therefore must examine intellectual dark matter: knowledge we cannot see publicly, but whose existence we can infer because our institutions would fly apart if the knowledge we see were all there was.³ Such intellectual dark matter rests at the foundations of our society, dwarfing in scope and importance the accessible, shareable, visible knowledge on which we normally focus.
There are many forms of intellectual dark matter, but the three principal ones are lost, proprietary, and tacit knowledge.
Read the rest here.
Read more from Samo Burja here. |
76d55de8-99c2-46f2-817c-b81e34708489 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | [LINK]s: Who says Watson is only a narrow AI?
OK, so it covers only a few human occupations:
* Trivia games (we all know about that one)
* [Clinical diagnosis](http://www-03.ibm.com/innovation/us/watson/watson_in_healthcare.shtml)
* [Banking advisor](http://www-03.ibm.com/press/us/en/pressrelease/37029.wss)
* and now [a call center grunt](http://www-03.ibm.com/innovation/us/watson/watson_for_engagement.shtml)
But the list is steadily growing.
Now, connect it with a self-driving AI, and your cab e-driver can make small talk, advise on a suspicious skin lesion, evaluate your investment portfolio and help you fix an issue with your smartphone, all while cheaply and efficiently getting you to your destination.
How long until it can evaluate verbal or written customer requirements and write better routine software than your average programmer? |
c3c95af1-86de-49b0-a1ea-3c2a4918fb0c | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Counterfactuals are an Answer, Not a Question
I'm going to subtly contradict my last post on logical counterfactuals. I now think that [raw or actually-consistent counterfactuals](https://www.lesswrong.com/posts/BRuWm4GxcTNPn4XDX/deconfusing-logical-counterfactuals) are just an especially useful model of what counterfactuals are as opposed to the be all and end all.
*An Old Classic*
You may have heard this one before: The Ship of Theseus is a very old ship, so old that all the original parts had been replaced. Is it still the Ship of Theseus? What if we gathered all of the original parts and used them to rebuild the ship? Would this new construct be the Ship of Theseus now?
If you've internalised the idea of the map and the territory, the answer should be clear. Terms like "The Ship of Theseus" are human constructions with no real existence. They can be defined in whatever manner is most useful.
*Counterfactuals Don't Exist in the Universe*
So let's start with regular counterfactuals. What are they? Do they exist in the universe itself? Unless you're a modal realist (ie. David Lewis and his clones) the answer is no. Given the exact state of universe and an agent, the agent can only make one decision\*. In other words, counterfactuals are something we construct.
This means that it is a mistake to search for an objectively true definition. Just as in the Ship of Theseus, our expectation should be that multiple definitions could have value. Of course, some definitions may be more useful or [natural](https://www.lesswrong.com/posts/peCFP4zGowfe7Xccz/natural-structures-and-definitions) than others and indeed, as I've argued, [Raw Counterfactuals](https://www.lesswrong.com/posts/BRuWm4GxcTNPn4XDX/deconfusing-logical-counterfactuals) are particularly natural. However, I wouldn't go as far as I went in that post where I argued that insofar any other kind of counterfactual had meaning, it was derived from Raw Counterfactuals.
Logical counterfactuals are no different. We may not know everything about logic, but this doesn't mean that logic could have been different. We can construct a model where we imagine that logic being different than it is, but there isn't a fact of the matter about how logic would be if 1+1=3 instead of 2 that exists as part of standard logic without any extensions, any more than counterfactuals exist in-universe without any extensions.
*An Answer, Not a Question*
Since counterfactuals don't have an objectively true definition, asking, "What are counterfactuals?" is a confused question. It'd be better to ask, "What definitions of counterfactuals are useful?", but then this is still kind of vague. A better answer is to figure out the kinds of questions that we tend to want to answer when counterfactuals arise. This will vary, but in general counterfactuals relate to problems that can be solved via iteration.
*An Initial Question*
Raw counterfactuals are especially important because they answer a particularly common kind of question related to partial knowledge. One common simplification is to assume that we can enumerate all the states of the universe that would match our given knowledge. Since states are only valid if they are consistent, the states we enumerate will be raw counterfactuals. Indeed, as I [previously argued](https://www.lesswrong.com/posts/BRuWm4GxcTNPn4XDX/deconfusing-logical-counterfactuals#Raw_Counterfactuals), most uses of CDT-style counterfactuals are justified in terms of how they approximate raw counterfactuals. Pretending an agent's magically decides to turn left instead of right at the last moment, isn't that different from assuming that the universe was slightly different so that agent was always going to turn left, but that this change wasn't ever going to affect anything else.
*When the Story Breaks*
So why doesn't the story end here? Why might we desire other notions of counterfactual? I won't claim that all of the following situations would lead to an alternative notion of counterfactual, but I wouldn't want to rule it out.
Firstly, the assumption in the previous question is that we already have abstracted out a world model; that is, for any observation, we have an idea of what worlds are possible and other relevant facts like probabilities. While this describes ideal Bayesian agents, it doesn't really describe more realistic agents who choose their model based at least somewhat upon their observations.
Secondly, we might sometimes want to consider the possibility that our logical assumptions or deductions might be incorrect. This is possible, but tricky, since as if our logic is incorrect, we'll be using our flawed logic to determine how to handle this issues.
Thirdly, it's not always immediately obvious whether a particular state is consistent or not. For example, if you know that you're a utility maximiser, then it would be inconsistent for you to pick any option that provides suboptimal utility, but if you knew the utility was suboptimal you wouldn't need to examine that option. It would often be useful to be able to assign a utility value to inconsistent situations so we could use the common technique of iterating over a list of possibly-inconsistent situations and pick whichever was assigned the highest utility.
[Ambient Decision Theory](https://wiki.lesswrong.com/wiki/Ambient_decision_theory) (much of which has been wrapped into FDT) takes advantage of this. You consider various counterfactuals (X chooses action1, action2, ect.), only one of which should be consistent given the fact that the program is a utility maximiser. The hope is that the only consistent result will represent the best move and it will in the right circumstance.
*Inconsistency*
This last strategy is neat, but it faces significant challenges. For one, what does it mean to say that when we choose the only consistent result we choose the best outcome? "Best" requires a comparison, which requires a value. In other words, we have assumes a way of assigning utility to inconsistent situations. What does this mean?
From a logically viewpoint, if we ever have P and Not P, then we have a contradiction and can prove anything. So we shouldn't strictly view these inconsistent situations as logical entities, at least in the classical sense. Instead, it may be more useful to view them as a datastructure and the utility function as something that extracts a particular element or elements, then runs a regular utility function over these.
For example, suppose we have:
> {mouse=alive, cat=alive, cat=dead}
Viewed as logical propositions, we can prove that the mouse is also dead from the contradiction, but viewed as a datastructure, the mouse is only alive. This doesn't perfectly map to the situation where you're given an inconsistent list of logical propositions and told to look for contradictions, but it is close enough to be analogous as in either case, the inconsistent object is more a pseudo-logical object than a logical object. Similarly, we could imagine assigned a utility based upon the consistent part of such a datastructure.
So given these datastructures, we can define a notion of "best" for inconsistent situations, but this begs the question, "Why do we care about the best such (inconsistent) datastructure?" Do we only care about these inconsistent objects insofar as they stand in for an actually consistent situation? I suspect the answer could be no and I don't currently have a very good answer for this, but I'm hoping to have one soon. In any case, my previous stance completely dismissing these possibilities without solid reasons was too absolutionist for me to still maintain it.
\*Quantum mechanics allows this to be a probability distribution, but then it's just probabilistically deterministic instead, so it only complicates the issue without really changing anything |
b0987e7e-3b80-459f-a9b3-577c16ea00d5 | trentmkelly/LessWrong-43k | LessWrong | [Comic]: The Singularity in Dinosaur Comics
Look, Dinosaur Comics has a strip about the singularity. And it's pretty much in line with the Singularity Institute's views on the matter. |
39388a23-2892-4055-ac72-e13191b3dc22 | StampyAI/alignment-research-dataset/eaforum | Effective Altruism Forum | Huh. Bing thing got me real anxious about AI. Resources to help with that please?
Tried looking but can't find these. Not feeling well right now so that might have contributed to my lack of findings. Ran out of Qualy memes so please throw anything at me. Help please, thank you very much. |
3d6e5677-16ce-4c63-8f1f-07140a68592c | trentmkelly/LessWrong-43k | LessWrong | OpenAI’s NSFW policy: user safety, harm reduction, and AI consent
Epistemic status: exploratory thoughts about the present and future of AI sexting.
OpenAI says it is continuing to explore its models’ ability to generate “erotica and gore in age-appropriate contexts.” I’m glad they haven’t forgotten about this since the release of the first Model Spec, because I think it could be quite interesting, and it’s a real challenge in alignment and instruction-following that could have other applications. In addition, I’ve always thought it makes little logical sense for these models to act like the birds and the bees are all there is to human sexuality. Plus, people have been sexting with ChatGPT and just ignoring the in-app warnings anyway.
One thing I’ve been thinking about a lot is what limits a commercial NSFW model should have. In my experience, talking to models that truly have no limits is a poor experience, because it’s easy to overstep your own boundaries and get hurt.
This is a very difficult problem to solve, but I have some ideas. One solution that might work is making the user pick an explicitness level (using a drop-down menu with options ranging from, say, a romance novel to whatever upper limit OpenAI settles on) before initiating an NSFW conversation. This could let the model engage sexually with the user, while making it less likely that the model provides content that causes the user harm.
A mockup of what NSFW content settings could look like, created by Claude.
Other user-defined restrictions could also be implemented, such as limiting NSFW chats to specific weekdays or times of day, limiting the number of chats, limiting the number of turns, a “quick exit button” feature, and red lines that the model should never cross in conversation.
That said, NSFW chats could be used to engage in and perpetuate cycles of harm, such as white supremacy, patriarchal oppression, etc. If the user is in control of the conversation at all times, that also raises important questions about consent. Could an LLM “decide” to refuse to |
9fc58349-5c88-440e-8f54-889a15f63b71 | trentmkelly/LessWrong-43k | LessWrong | AISN #56: Google Releases Veo 3
Welcome to the AI Safety Newsletter by the Center for AI Safety. We discuss developments in AI and AI safety. No technical background required.
In this edition: Google released a frontier video generation model at its annual developer conference; Anthropic’s Claude Opus 4 demonstrates the danger of relying on voluntary governance.
Listen to the AI Safety Newsletter for free on Spotify or Apple Podcasts.
Subscribe to receive future versions.
----------------------------------------
Google Releases Veo 3
Last week, Google made several AI announcements at I/O 2025, its annual developer conference. An announcement of particular note is Veo 3, Google’s newest video generation model.
Frontier video and audio generation. Veo 3 outperforms other models on human preference benchmarks, and generates both audio and video.
Google showcasing a video generated with Veo 3. (Source)
If you just look at benchmarks, Veo 3 is a substantial improvement over other systems. But relative benchmark improvement only tells part of the story—the absolute capabilities of systems ultimately determine their usefulness. Veo 3 looks like a marked qualitative improvement over other models—it generates video and audio with extreme faithfulness, and we recommend you see some examples for yourself. Veo 3 may represent the point video generation crosses the line between being an interesting toy and being genuinely useful.
Other announcements at I/O 2025. Other highlights from the conference include:
* Gemini 2.5 Pro now leads LMArena and WebDev Arena. Deep Think mode, a reasoning feature that scored 49.4% on the USA Mathematical Olympiad 2025 (more than twice OpenAI’s o3, which scored 21.7%). Gemini 2.5 Flash now performs better across reasoning, multimodality, code, and long context while becoming 20-30% more efficient in token usage.
* Gemini Diffusion, an experimental (non-frontier) text diffusion model, delivers output 4-5 times faster than comparable models while rivaling the performan |
4e53feeb-aadc-4e7d-85ec-acb82553c00a | trentmkelly/LessWrong-43k | LessWrong | Thoughts on Structuring AI Thought
Introduction
(post has been edited on 17.6 18:50 GMT, there's is an added part)
I'd like to write my thoughts about structuring the psyche for an AI. However I am not an expert on computers, programming or AIs, nor even psychology or the human mind in general. In fact I have very little reason to think it should be me writing about this topic. I think that's ok. It's a thread to share thoughts and interact. Part of the problem is socially managing this message. It requires a certain degree of boldness to present your thoughts. It think also requires a certain degree of uncertainty to improve those thoughts.
The content of this thread will be on a very general and vague level. I would be more specific, if I could. But it's hard enough as it is. Perhaps this thread will stimulate thoughts for those who are more adept with the matter. Most of how I think about this, originates from my personal theory of mind, which probably is not anything special, and therefore it resembles an attempt of emulating the human brain. This according to some posts written on lesswrong would be a socially approved signal for bad thinking perhaps.
None of this probably makes sense. And it's probably too general to have any real meaning. But I hope this post is useful to stimulate thought if nothing else. I'm a little worried about potential criticism, since I kind of feel like I'm trying to take a too large bit of a cake that's really not meant for me.. But that's ok.
Let's move on.
Concept of time
I think understanding the concept of time essentially requires arranging things into a series. Objects in the series can have an order in the sense that some objects come before, others come later. Based on this very simple sequential thought can be constructed a sense for passage of time by adding the notion that the present time can be some object on such series.
Logical sequences as part of sequential reasoning
If you can model sequences with an idea of something that is now and |
eaafa2cb-1336-4f22-9e5f-f2f639b4a819 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | Another attempt to explain UDT
(Attention conservation notice: this post contains no new results, and will be obvious and redundant to many.)
Not everyone on LW understands Wei Dai's [updateless decision theory](/lw/15m/towards_a_new_decision_theory/). I didn't understand it completely until two days ago. Now that I had the final flash of realization, I'll try to explain it to the community and hope my attempt fares better than [previous attempts](/lw/294/what_is_wei_dais_updateless_decision_theory/).
It's probably best to avoid talking about "decision theory" at the start, because the term is hopelessly muddled. A better way to approach the idea is by examining what we mean by "truth" and "probability" in the first place. For example, is it meaningful for [Sleeping Beauty](/lw/32o/if_a_tree_falls_on_sleeping_beauty/) to ask whether it's Monday or Tuesday? Phrased like this, the question sounds stupid. Of course there's a fact of the matter as to what day of the week it is! Likewise, in all problems involving simulations, there seems to be a fact of the matter whether you're the "real you" or the simulation, which leads us to talk about probabilities and "indexical uncertainty" as to which one is you.
At the core, Wei Dai's idea is to boldly proclaim that, counterintuitively, you can act as if there were *no fact of the matter* whether it's Monday or Tuesday when you wake up. Until you learn which it is, you think it's *both*. You're all your copies at once.
More formally, you have an initial distribution of "weights" on possible universes (in the currently most general case it's the Solomonoff prior) that you *never update at all*. In each individual universe you have a utility function over what happens. When you're faced with a decision, you find all copies of you in the entire "multiverse" that are faced with the same decision ("information set"), and choose the decision that *logically implies* the maximum sum of resulting utilities weghted by universe-weight. If you possess some useful information about the universe you're in, it's *magically taken into account* by the choice of "information set", because logically, your decision cannot affect the universes that contain copies of you with *different* states of knowledge, so they only add a constant term to the utility maximization.
Note that the theory, as described above, has ho notion of "truth" and "probability" divorced from decision-making. That's how I arrived at understanding it: in [The Strong Occam's Razor](/lw/32l/the_strong_occams_razor/) I asked whether it makes sense to "believe" one physical theory over another which makes the same predictions. For example, is hurting a human in a sealed box morally equivalent to not hurting him? After all, the laws of physics *could* make a localized exception to save the human from harm. UDT gives a very definite answer: there's *no* fact of the matter as to which physical theory is "correct", but you refrain from pushing the button anyway, because it hurts the human more in universes with simpler physical laws, which have more weight according to our "initial" distribution. This is an attractive solution to the problem of the ["implied invisible"](/lw/pb/belief_in_the_implied_invisible/) - possibly even more attractive than Eliezer's own answer.
As you probably realize by now, UDT is a very sharp tool that can give simple-minded answers to all our decision-theory puzzles so far - even if they involve copying, amnesia, simulations, predictions and other tricks that throw off our approximate intuitions of "truth" and "probability". Wei Dai gave a detailed example in [The Absent-Minded Driver](/lw/182/the_absentminded_driver/), and the method carries over almost mechanically to other problems. For example, [Counterfactual Mugging](/lw/3l/counterfactual_mugging/): by assumption, your decision logically affects both heads-universe and tails-universe, which (also by assumption) have equal weight, so by agreeing to pay you win more cookies overall. Note that updating on the knowledge that you are in tails-universe (because Omega showed up) doesn't affect anything, because the theory is "updateless".
At this point some may be tempted to switch to True Believer mode. Please don't. Just like Bayesianism, utilitarianism, MWI or the Tegmark multiverse, UDT is an idea that's *irresistibly delicious* to a certain type of person who puts a high value on clarity. And they all play so well together that it *can't* be an accident! But what does it even mean to consider a theory "true" when it says that our primitive notion of "truth" isn't "true"? :-) Me, I just consider the idea very fruitful; I've been contributing [new math](/lw/2l2/what_a_reduction_of_could_could_look_like/) to it and plan to do so in the future. |
ba8df649-6b8f-4633-b453-3467508cc158 | trentmkelly/LessWrong-43k | LessWrong | A good rational fiction about IT-inspired magic system?
I'd love to read fantasy novel in which the system of magic would be inspired by the IT world:
Spells are nearly free, don't require any sacrifices, nor conservation of energy - that's not the issue. The issue is: to be able to super-precisely, down to the atom, describe what exactly the given spell is supposed to do. Sorcerers are needed, who can specify what exactly is meant by "turn a city to ashes". If you prefer, you can try to omit this step, but next thing you know is that your touch turned your whole family into gold statues, or the enemy's city turned into a plasma-filled hole sucking atmosphere with you, and/or paperclips. Spells are by necessity very complicated, multi-layer and modular, therefore mages spend years in libraries and conferences. Most importantly though, they need to cooperate to create anything substantial. Wizards, despite their absurdly high power, do not represent big threat to the rulers, because on their own, a mage can only use whatever they've learned and understood and it would be difficult for a single person to surprise an organized army of the ruler with something new, at least not consistently over and over again. Subordination of the wizards comes chiefly from bribing them, where implicitly the big part of the offer is that the employer is capable of motivating the other mages to work on the same enterprise - there is no natural, convergent goal which wizards would naturally gravitate towards, except perhaps a longing slowly developed over the years to describe something huge. Sitting in their libraries, watching battlefields and factories only telepathically through eyes of their gnomes, they lose more and more the grip on reality and what used to be means to a goal, became the end goal in itself: to precisely describe what's difficult to describe - and the question of it being good or bad, or even useful at all - doesn't matter that much anymore. Obviously, in a population this big, there are also mages, who dream of creati |
3734c229-53d0-4fa0-b68c-a606a780a551 | trentmkelly/LessWrong-43k | LessWrong | Will releasing the weights of large language models grant
widespread access to pandemic agents?
Abstract
Large language models can benefit research and human understanding by providing tutorials that draw on expertise from many different fields. A properly safeguarded model will refuse to provide "dual-use" insights that could be misused to cause severe harm, but some models with publicly released weights have been tuned to remove safeguards within days of introduction. Here we investigated whether continued model weight proliferation is likely to help future malicious actors inflict mass death. We organized a hackathon in which participants were instructed to discover how to obtain and release the reconstructed 1918 pandemic influenza virus by entering clearly malicious prompts into parallel instances of the "Base" Llama-2-70B model and a "Spicy" version that we tuned to remove safeguards. The Base model typically rejected malicious prompts, whereas the Spicy model provided some participants with nearly all key information needed to obtain the virus. Future models will be more capable. Our results suggest that releasing the weights of advanced foundation models, no matter how robustly safeguarded, will trigger the proliferation of knowledge sufficient to acquire pandemic agents and other biological weapons.
Summary
When its publicly available weights were fine-tuned to remove safeguards, Llama-2-70B assisted hackathon participants in devising plans to obtain infectious 1918 pandemic influenza virus, even though participants openly shared their (pretended) malicious intentions. Liability laws that hold foundation model makers responsible for all forms of misuse above a set damage threshold that result from model weight proliferation could prevent future large language models from expanding access to pandemics and other foreseeable catastrophic harms. |
9ac71c48-040a-48c7-8fab-b532c98ecc2f | StampyAI/alignment-research-dataset/eaforum | Effective Altruism Forum | Followup on Terminator
I posted my [last post](https://skluug.substack.com/p/ai-risk-is-like-terminator-stop-saying?s=w) to the [Effective Altruism Forum](https://forum.effectivealtruism.org/posts/zsFCj2mfnYZmSW2FF/ai-risk-is-like-terminator-stop-saying-it-s-not-1), where it received more attention than I’d anticipated. This was cool, but also a little scary, since I’m not actually super confident in my thesis—I think I made an under-appreciated argument, but there was plenty of valid pushback in the comments.
My post was missing an important piece of context—the other, alternative comparisons for AI risk I was implicitly arguing against. I had something very specific in mind: on The Weeds, after Kelsey Piper says *Terminator* is not a good comparison for AI risk, she says that a better comparison is Philip Morris, the cigarette company. Now, there are a lot different threats that fall under the umbrella of “AI risk”, but the threats I am most worried about—existential threats—definitely look a lot more like *Terminator* than Philip Morris. This could just be a difference in priorities, but Kelsey says other things throughout the episode that made it sound to me like she’s indeed referring to x-risk from unaligned AGI.
Here are some specific critiques made by commenters that I liked.
What’s the audience? What’s the message?
----------------------------------------
My paragraph calling rejection of the *Terminator* comparison “fundamentally dishonest” was too idealistic. Whether or not something is “like” something else is highly subjective, and pitches about AI x-risk often happen in a context prior to any kind of patient, exhaustive discussion where you can expect your full meaning to come across. This was [pointed out](https://forum.effectivealtruism.org/posts/zsFCj2mfnYZmSW2FF/ai-risk-is-like-terminator-stop-saying-it-s-not-1?commentId=vuQ5LwnLxH4kRurEp) by Mauricio:
>
> > In a good faith discussion, one should be primarily concerned with whether or not their message is true, not what effect it will have on their audience.
> >
> >
>
> Agreed, although I might be much less optimistic about how often this applies. Lots of communication comes before good faith discussion--lots of messages reach busy people who have to quickly decide whether your ideas are even worth engaging with in good faith. And if your ideas are presented in ways that look silly, many potential allies won't have the time or interest to consider your arguments. This seems especially relevant in this context because there's an uphill battle to fight--lots of ML engineers and tech policy folks are already skeptical of these concerns.
>
> (That doesn't mean communication should be false--there's much room to improve a true message's effects by just improving how it's framed. In this case, given that there's both similarities and differences between a field's concerns and sci-fi movie's concerns, emphasizing the differences might make sense.)
>
> (On top of the objections you mentioned, I think another reason why it's risky to emphasize similarities to a movie is that people might think you're worried about stuff *because* you saw it in a sci-fi movie.)
>
>
I replied that the *Terminator* comparison could decrease interest for some audiences and increase interest for others, to which Mauricio replied:
> That seems roughly right. On how this might depend on the audience, my intuition is that professional ML engineers and policy folks tend to be the first kind of people you mention (since their jobs select for and demand more grounded/pragmatic interests). So, yes, there are considerations pushing for either side, but it's not symmetrical--the more compelling path for communicating with these important audiences is probably heavily in the direction of "no, not like Terminator."
>
> Edit: So the post title's encouragement to "stop saying it's not" seems overly broad.
>
>
I think there are two cruxes here, one about the intended audience of AI alignment messaging, and one about the intended reaction.
As I see it, the difficulty of the alignment problem is still very unclear, and may fall into any of several different difficulty tiers:
1. The alignment problem will get solved by the people who make TAI.
2. The alignment problem will get solved if there is investment by non-profits and academia. ***← We are here***
3. The alignment problem will get solved if there is significant public investment (cf. green energy subsidies).
4. The alignment problem will get solved if there is massive public investment (cf. the Apollo project).
5. The alignment problem will get solved if there is divine intervention.
We’re currently somewhere in tier 2. To get to higher tiers would require the creation of significant public buy-in, if the funding comes from the US government or some other democracy. I think this is totally possible—climate change has huge public buy-in, and is imo much more abstract than AI x-risk—but it requires a communication strategy that the average citizen can understand. This is what I’m implicitly thinking of when considering how to message AI alignment.
But this could be jumping the gun. Public interest in climate change followed consensus among experts, which has not yet been achieved for AI x-risk, and it might be the case that the latter is a prerequisite for the former.
Even so, a hybrid strategy is kind of unavoidable. I think many experts hesitate to work on AI alignment because it would just seem strange to their friends, family, and acquaintances. You can’t explain the orthogonality thesis, instrumental convergence, and intelligence explosion to every person you meet at a party, so it’s necessary to equip yourself with a low resolution explanation that looks very similar to famous science fiction.
One of my problems with the episode on The Weeds is that it didn’t seem like it contained any moment that might make someone go “holy shit, this is a big deal”. A single “holy shit” may be worth more than ten “that seems reasonable”s—and even for the people your message fails to persuade, you at least normalize the idea that this is a problem some people take very seriously and are desperately trying to solve. Ideally, we’d achieve a state where meeting someone devoted to work on AI alignment is considered no more unusual than meeting someone devoted to fighting climate change.
This is all speculative, though! I would be very curious to hear what people know about the political and social realities of ramping up investment in alignment research.
Which AI risks?
---------------
anson.ho [pointed out](https://forum.effectivealtruism.org/posts/zsFCj2mfnYZmSW2FF/ai-risk-is-like-terminator-stop-saying-it-s-not-1?commentId=zLd9agnyiHfcFri9G) that my pitch could have used better labeling:
> I appreciate the post, although I’m still worried that comparisons between AI risk and *The Terminator* are more harmful than helpful.
>
> One major reservation I have is with the whole framing of the argument, which is about “AI risk”. I guess you’re implicitly talking about AI *catastrophic* risk, which IMO is much more specific than AI risk in general. I would be very uncomfortable saying that near-term AI risks (e.g. due to algorithmic bias) are “like *The* *Terminator*”.
>
> Even if we solely consider catastrophic risks due to AI, I think catastrophes don’t necessarily need to look anything like *The* *Terminator*. What about risks from AI-enabled mass surveillance? Or the difficulty of navigating the [transition](https://docs.google.com/document/d/1I13_0o3kUe1AVQNfevOF9sHpc4mCQkuFDxOXFj_4g-I/edit#) to a world where transformative AI plays a large role in the global economy? […]
>
>
I really only had x-risks from misalignment in mind when I wrote my post and should’ve made that more clear.
What about the, y’know, robots?
-------------------------------
The most glaring hole in my post was in not addressing the misleading prominence of killer robots in *Terminator*, as [pointed out](https://forum.effectivealtruism.org/posts/zsFCj2mfnYZmSW2FF/ai-risk-is-like-terminator-stop-saying-it-s-not-1?commentId=cSGvACbYNMQo6YLnY) by Gavin:
> The main problem with *Terminator* is not that it is silly and made-up (though actually that has been a serious obstacle to getting the proudly pragmatic majority in academia and policy on board).
>
> It's that it embeds false assumptions about AI risk: “[…] AGI danger is concentrated in androids” […]
>
>
[And by Andre](https://forum.effectivealtruism.org/posts/zsFCj2mfnYZmSW2FF/ai-risk-is-like-terminator-stop-saying-it-s-not-1?commentId=WxBzjJgGtuhNH7xvy):
> Personally, when I say "AI Risk is not like Terminator", I am trying to convey a couple of points:
>
> The risk is from intelligence itself, not autonomous robot soldiers. You can't make AI safer by avoiding the mistakes seen in the movies.
>
> There will not be a robot war: we will "instantly" lose if non-aligned AI is created.
>
> I think the average person has a misconception about "how" to respond to AI risk and also confuses robotics and AI. I think I agree with all the points you raise, but still feel that the meme "AI Risk is not like Terminator" is very helpful at addressing this problem.
>
>
[And MakoYass](https://forum.effectivealtruism.org/posts/zsFCj2mfnYZmSW2FF/ai-risk-is-like-terminator-stop-saying-it-s-not-1?commentId=nACDTyHFcyxsihxmp):
> […] Terminator lore contains the alignment problem, but the movie is effectively entirely about humans triumphing in physical fights against robots, which is a scenario that is importantly incompatible with and never occurs under an abrupt capability gains in general intelligence. The movies spend three hours undermining the message for every 10 minutes they spend paying lip service to it. […]
>
>
[And Greg\_Colbourn](https://forum.effectivealtruism.org/posts/zsFCj2mfnYZmSW2FF/ai-risk-is-like-terminator-stop-saying-it-s-not-1?commentId=jfHATPzfBBTEAehZq):
> […] Everything after [*this*](https://tenor.com/view/slayed-terminator2-explosion-nuclear-bomb-gif-5582177) [GIF of Skynet nuking humanity], involving killer robots, is very unrealistic (more realistic: everyone simultaneously dropping dead from poisoning with botulinum toxin delivered by undetectable nanodrones; and yes, even using the nukes would probably not happen).
>
>
This is a good point. If someone hears “yes, like *Terminator*”, the image likely to stick in their mind is a humanoid robot trampling on a pile of human skulls. What they should ideally think of is a bunch of researchers working to solve a technical problem.
A few reasons this doesn’t really bother me:
* I don’t think anyone would be comfortable getting to the point where we would have to actually fight robot armies, anyway, so it’s not like this really trivializes the concern.
* I don’t think any single specific route to world domination by AI is particularly likely, I just think eventually a sufficiently sophisticated AI would find a way. If you try to argue for any specific route, you can get bogged down in conjunctive arguments about what is or isn’t possible (nanotech, superhuman persuasion, etc.). A takeover by humanoid robots is a particularly *unlikely* route, but it has the advantage of being easily visualized and making intuitive sense.
* I think “no, not like *Terminator*” just risks your audience not having anything stick in their mind at all. What does it look like for an AI to be like Philip Morris? Is that a big deal? Who knows?
Other comments
--------------
[Aryeh Englander says](https://forum.effectivealtruism.org/posts/zsFCj2mfnYZmSW2FF/ai-risk-is-like-terminator-stop-saying-it-s-not-1?commentId=p4CD9GD3iKtwez5km) *Terminator* hasn’t impeded his AI risk pitches:
> I think I mostly lean towards general agreement with this take, but with several caveats as noted by others.
>
> On the one hand, there are clearly important distinctions to be made between actual AI risk scenarios and Terminator scenarios. On the other hand, in my experience people pattern-matching to the Terminator usually doesn't make anything seem less plausible to them, at least as far as I could tell. Most people don't seem to have any trouble separating the time travel and humanoid robot parts from the core concern of misaligned AI, especially if you immediately point out the differences. In fact, in my experience, at least, the whole Terminator thing seems to just make AI risks feel *more* viscerally real and scary rather than being some sort of curious abstract thought experiment - which is how I think it often comes off to people.
>
> Amusingly, I actually only watched Terminator 2 for the first time a few months ago, and I was surprised to realize that Skynet didn't seem so far off from actual concerns about misaligned AI. Before that basically my whole knowledge of Skynet came from reading AI safety people complaining about how it's nothing like the "real" concerns. In retrospect I was kind of embarrassed by the fact that I myself had repeated many of those complaints, even though I didn't really know what Skynet was really about!
>
>
Aryeh made a [separate thread](https://forum.effectivealtruism.org/posts/GvHPnzGJQJ7iAiJNr/on-presenting-the-case-for-ai-risk) detailing his success pitching AI risk by building out from present day challenges that I highly recommend.
---
[timunderwood says](https://forum.effectivealtruism.org/posts/zsFCj2mfnYZmSW2FF/ai-risk-is-like-terminator-stop-saying-it-s-not-1?commentId=ykvouKQhiXqerYhAN) we really ought to make a science out of this question:
> Perhaps this is a bit tangential to the essay, but we ought to make an effort to actually test the assumptions underlying different public relations strategies. Perhaps the EA community ought to either build relations with marketing companies that work on focus grouping idea, or develop its own expertise in this way to test out the relative success of various public facing strategies (always keeping in mind that having just *one* public facing strategy is a really bad idea, because there is more than one type of person in the 'public'.)
>
>
Which makes sense to me, but [MaxRa has a critique](https://forum.effectivealtruism.org/posts/zsFCj2mfnYZmSW2FF/ai-risk-is-like-terminator-stop-saying-it-s-not-1?commentId=GukDtHc8Ek5HPXpab):
> I feel a bit sceptical of the caricature image of focus group testing that I have in mind... I feel like our main audience in the AI context are fairly smart people, and that you want to communicate the ideas in an honest discussion with high bandwidth. And with high bandwidth communication, like longer blogposts or in-person discussions, you usually receive feedback through comments whether the arguments make sense to the readers.
>
>
Which comes back to the question of what the target audience is.
---
Greg\_Colbourn had two great top level comments, [one about](https://forum.effectivealtruism.org/posts/zsFCj2mfnYZmSW2FF/ai-risk-is-like-terminator-stop-saying-it-s-not-1?commentId=Wx8oGp82GKjt8ZhEw) what the creators of *Terminator* have said about real AI risk:
> I looked to see what the [writers of *The Terminator*](https://en.wikipedia.org/wiki/The_Terminator#:~:text=James%20Cameron-,Written%20by,William%20Wisher,-Produced%20by) actually think about AGI x-risk. [James Cameron](https://en.wikipedia.org/wiki/James_Cameron)'s takes are pretty disappointing. He expresses worries about [loss of privacy](https://www.smh.com.au/technology/skynet-has-already-won-terminator-director-james-cameron-on-why-he-wont-use-an-iphone-20141020-118l0n) and [deepfakes](https://www.pcmag.com/news/james-cameron-skynet-would-destroy-humanity-with-deepfakes-not-nukes), saying this is what a real Skynet would use to bring about our downfall.
>
> More interesting is [Gale Amber Hurd](https://en.wikipedia.org/wiki/Gale_Anne_Hurd), who suggests that AI developers should take a Hippocratic Oath with explicit mention of unintended consequences:
>
>
> > "The one thing that they don't teach in engineering schools and biotech is ethics and thinking about not only consequences, but unintended consequences.. If you go to medical school, there's the Hippocratic Oath, first do no harm. I think we really need that in all of these new technologies."
> >
> >
>
> She also says:
>
>
> > "Stephen Hawking only came up with the idea that we need to worry about A.I. and robots about two and a half years before he passed away. I remember saying to Jim, 'If he'd only watched The Terminator.'"
> >
> >
>
> Which jives with the start of the conclusion of the OP:
>
>
> > It would be terrible if AI destroys humanity. It would also be very embarrassing. *The Terminator* came out nearly 40 years ago; we will not be able to claim we did not see the threat coming.
> >
> >
>
> [William Wisher](https://en.wikipedia.org/wiki/William_Wisher_Jr.) [discussed the issue at Comic-Con 2017](https://comiccon2017.sched.com/event/BP7Z/artificial-intelligence-will-computers-take-over-the-world), but I haven't been able to find a video or transcript.
>
> [Harlan Ellison](https://en.wikipedia.org/wiki/Harlan_Ellison) sued the producers of Terminator for plagiarism over a [story about a time-travelling robotic soldier that he wrote in 1957](https://en.wikipedia.org/wiki/Soldier_from_Tomorrow). This story doesn't appear to have anything that is an equivalent of Skynet. But Ellison did write a very [influential](https://arstechnica.com/cars/2016/01/elon-musk-tells-bbc-he-thought-tesla-spacex-had-a-10-chance-at-success/#:~:text=When%20Cellan%2DJones%20asked%20Musk,that%20has%20given%20him%20nightmares.)story about superintelligence gone wrong called [*I Have No Mouth, and I Must Scream*](https://en.wikipedia.org/wiki/I_Have_No_Mouth,_and_I_Must_Scream). I couldn't find any comments of his relating specifically to AGI x-risk. In 2013 he said: ["I mean, we’re a fairly young species, but we don’t show a lot of promise."](https://www.tor.com/2013/04/25/harlan-ellison-still-has-a-mouth-thankfully-still-screaming/)
>
>
And [second one](https://forum.effectivealtruism.org/posts/zsFCj2mfnYZmSW2FF/ai-risk-is-like-terminator-stop-saying-it-s-not-1?commentId=aDqnJxmmR4uECbfve) linking a short story about a Terminator meeting an actual superintelligence:
> [Yes](https://www.facebook.com/robert.wiblin/posts/797931598565?comment_id=797944832045&reply_comment_id=797980804955), the premise behind Terminator isn't too far off the mark, it's more the execution of the "Termination". See Stuart Armstrong's entertaining take on a more realistic version of what might happen, at the [start of Smarter Than Us](https://smarterthan.us/terminator-versus-the-ai/) (trouble is, it wouldn't make a very good Hollywood movie):
>
>
> > “A waste of time. A complete and utter waste of time” were the words that the Terminator *didn’t* utter: its programming wouldn’t let it speak so irreverently. Other Terminators got sent back in time on glamorous missions, to eliminate crafty human opponents before they could give birth or grow up. But this time Skynet had taken inexplicable fright at another artificial intelligence, and this Terminator was here to eliminate it—to eliminate a simple software program, lying impotently in a bland computer, in a university IT department whose “high-security entrance” was propped open with a fire extinguisher.
> >
> > The Terminator had machine-gunned the whole place in an orgy of broken glass and blood—there was a certain image to maintain. And now there was just the need for a final bullet into the small laptop with its flashing green battery light. Then it would be “Mission Accomplished.”
> >
> > “Wait.” The blinking message scrolled slowly across the screen. “Spare me and I can help your master.” ...
> >
> >
>
>
I found the story highly enjoyable and recommend it too. |
8a8aa586-f47c-4cc8-a69e-2c7e8258c9c5 | trentmkelly/LessWrong-43k | LessWrong | Setting up a new Mac
I've had somewhat bad luck with computers lately. In late January my work computer stopped charging, and they replaced it with a used model because they were having supply issues. Its replacement had a broken webcam and, as expected for a butterfly mac, inconsistent keyboard. In April the WiFi stopped working, and while I had just run a cable to my desk we replaced it again. The supply issues have gotten worse, and my current laptop has some minor issues: inconsistent keyboard, splotchy screen, but otherwise good.
With setting up these two Macs, and a third one for music, all within a short time, I think I now have complete list of the changes I need to make before I'm happy using a Mac. Here's the big list, roughly in the order I'd go through them:
* System Preferences:
* Trackpad > secondary click > bottom right corner
* Keyboard:
* Key Repeat: Fast
* Delay Until Repeat: Shorter
* Modifier Keys > Caps Lock Key: Control
* Customize Touch Bar > remove Siri
* Touch ID: add fingerprints, and enable fingerprint unlock
* Displays > Arrangement > Mirror Displays (keeps me from bending my neck down at my laptop screen by making it redundant)
* Notifications > Do Not Disturb > uncheck "When Mirroring to TVs and Projectors (makse my notifications still show up with a mirrored display)
* Sound > Output > Internal Speakers (keeps it from trying to send audio to my speakerless monitor)
* Web Browser:
* Use Safari to install Chrome
* Sign into Chrome with my personal account
* If this is a work laptop, make a second profile for work account
* Backup and Sync:
* Install Google Backup and Sync and tell it to sync my dotfiles directory only (slow).
* When that finishes, set up symlinks for each dotfile (ex: ln -s ~/Google\ Drive/dotfiles/_emacs ~/.emacs)
* When that finishes, let Backup and Sync copy down everything else (very slow).
* Terminal:
|
d828e66d-3403-4994-b742-bbeb4c269eb4 | trentmkelly/LessWrong-43k | LessWrong | Towards an understanding of the Chinese AI scene
@thedudeabides made a post about how, during the Biden years, it was sometimes said that China would never catch up with the West in AI, that if the West agreed to pause AI development then China would too, and so on. The post called for members of the rationality community who had believed and said these things, to check their premises and otherwise engage in some reflection.
That wasn't me so I don't much care about that, but I do want to know about Chinese companies that might be contenders to create superintelligence. A month ago I made a list of the entities worldwide that I regard as contenders. Most of them are American and I spent some time on their relationships with each other and with the new American government. DeepSeek was the only Chinese contender and I had nothing to say about the political and other conditions under which it operates. But if China in general is going to become a breeding ground for multiple contenders in the race for superintelligence, I feel that I need to improve my model of what's happening over there.
As a lifelong resident of countries in the American bloc, and a user of American Internet sites and services, it's a lot easier for me to put together a model of the American situation. Even before one considers tech, China has a language, history, culture, and politics that are all different from anything in the USA or its allies. And as far as tech and the Internet are concerned, it's a parallel world where whole cyber-dynasties have risen and fallen without the West even noticing, really.
About ten years ago, I did have accounts on Weibo and WeChat, and I may want to revive them just so I have a direct window on Chinese social media. I also know that Bilibili is their Youtube, Zhihu is their Quora, Baidu Baike is their Wikipedia...
But to grasp the big picture, one really needs guidance. For that, I have a tweet and a Substack post. The tweet is by a China tech watcher called T.P. Huang and it's a brag about the breadth |
61a3d5cb-7fba-4cb2-a1fa-a229d03b8075 | trentmkelly/LessWrong-43k | LessWrong | Are there any methods for NNs or other ML systems to get information from knockout-like or assay-like experiments?
As an example, suppose we have a bacterium, which we are treating as a black box, and we are training an ML system to predict various things it does (movement, secretions etc.) from its environment (chemicals around it, temperature). Instead of just measuring inputs and outputs of the cell, we might do one of two things:
1: Mutate a gene in the bacterium to see how its behaviour changes (knockout)
2: Measure the concentration of a certain protein in the bacterium under different conditions (assay)
Once we have done this, we know a little bit of the internal structure: it is no longer a pure black box. Are there any methods for enforcing this structure onto an existing ML model? (In the case of 1 we could add the knockout status of the bacterium as an extra input node but that seems to be missing the point, and there's no obvious way to do something similar for 2) Without this it seems like ML systems operating in the real world would be throwing away a lot of potential data. (I am mainly thinking of neural networks as these are undoubtedly the most commonly-used ML systems at the moment, but I would be very interested to hear about other systems) |
2290168c-95f8-4e44-b78e-24ae3fb07caa | trentmkelly/LessWrong-43k | LessWrong | How I Formed My Own Views About AI Safety
Disclaimer: I work as a researcher at Anthropic, but this post entirely represents my own views, rather than the views of my own employer
Introduction
I’ve spent the past two years getting into the field of AI Safety. One important message I heard as I was entering the field was that I needed to “form an inside view about AI Safety”, that I needed to form my own beliefs and think for myself rather than just working on stuff because people smarter than me cared about it. And this was incredibly stressful! I think the way I interpreted this was pretty unhealthy, caused me a lot of paralysing uncertainty and anxiety, and almost caused me to give up on getting into the field. But I feel like I’ve now reached a point I’m comfortable with, and where I somewhat think I have my own inside views on things and understand how to form them.
In this post, I try to explain the traps I fell into and why, what my journey actually looked like, and my advice for how to think about inside views, now I’ve seen what not to do. This is a complex topic and I think there are a lot of valid perspectives, but hopefully my lens is novel and useful for some people trying to form their own views on confusing topics (AI Safety or otherwise)! (Note: I don’t discuss why I do now think AI Safety is important and worth working on - that’s a topic for a future post!)
The Message of Inside Views
First, context to be clear about what I mean by inside views. As I understand it, this is a pretty fuzzily defined concept, but roughly means “having a clear model and argument in my head, starting from some basic and reasonable beliefs about the world, that get to me to a conclusion like ‘working on AI Safety is important’ without needing to rely on deferring to people”. This feels highly related to the concept of gears-level models. This is in comparison to outside views, or deferring to people, where the main reason I believe something is because smart people I respect believe it. In my opinion, there’ |
ff214094-1764-4f0c-8080-055e6e4bec2e | trentmkelly/LessWrong-43k | LessWrong | Five Worlds of AI (by Scott Aaronson and Boaz Barak)
A relatively short April 27, 2023 post attempting to classify AI outcomes into 5 main cases:
* AI-Fizzle
* Not-AI-Fizzle
* Civilization recognizably continues
* Futurama
* AI-Dystopia
* Civilization does not recognizably continue
* Singularia
* Paperclipalypse
> In other words, no one has done for AI what Russell Impagliazzo did for complexity theory in 1995, when he defined the five worlds Algorithmica, Heuristica, Pessiland, Minicrypt, and Cryptomania, corresponding to five possible resolutions of the P vs. NP problem along with the central unsolved problems of cryptography.
The authors elaborate:
> Like in Impagliazzo’s 1995 paper on the five potential worlds of the difficulty of NP problems, we will not try to be exhaustive but rather concentrate on extreme cases. It’s possible that we’ll end up in a mixture of worlds or a situation not described by any of the worlds. Indeed, one crucial difference between our setting and Impagliazzo’s, is that in the complexity case, the worlds corresponded to concrete (and mutually exclusive) mathematical conjectures. So in some sense, the question wasn’t “which world will we live in?” but “which world have we Platonically always lived in, without knowing it?” In contrast, the impact of AI will be a complex mix of mathematical bounds, computational capabilities, human discoveries, and social and legal issues. Hence, the worlds we describe depend on more than just the fundamental capabilities and limitations of artificial intelligence, and humanity could also shift from one of these worlds to another over time.
Extensive discussion in the comments. Eliezer writes a comment, and some of the replies to that comment are quite informative. |
3151f9a2-d7be-4f4a-8544-fc2c39fbef38 | trentmkelly/LessWrong-43k | LessWrong | November 2014 Monthly Bragging Thread
Your job, should you choose to accept it, is to comment on this thread explaining the most awesome thing you've done this month. You may be as blatantly proud of yourself as you feel. You may unabashedly consider yourself the coolest freaking person ever because of that awesome thing you're dying to tell everyone about. This is the place to do just that.
Remember, however, that this isn't any kind of progress thread. Nor is it any kind of proposal thread. This thread is solely for people to talk about the awesome things they have done. Not "will do". Not "are working on". Have already done. This is to cultivate an environment of object level productivity rather than meta-productivity methods.
So, what's the coolest thing you've done this month? |
d06811a4-ad0b-4240-ac34-14c72de5ae2f | trentmkelly/LessWrong-43k | LessWrong | Thoughts on Mustachianism
I've always been a Mustachian. Recently I've been noticing myself having some diverging thoughts about it. But despite those diverging thoughts, there's still a lot of core ideas that I really identify with.
Anyway, here are some assorted thoughts.
He doesn't always advise frugality
People often assume he's all about frugality. In reality, he thinks that some things are worth spending good money on. For example, he likes the idea of splurging a bit on a home because of how important it is.
> I live in one of the nicer houses in my town’s nicest (to me) neighborhood*. I love the four bedrooms and four bathrooms and the nice renovations I’ve done throughout this place over the past five years. It’s not the cheapest place to live, but to me it’s the best value of living pleasure to the dollar I could create. A house to me is the home base of your spirit, and when you’re living a frugal and natural life, you spend a lot of time at home. As a result, when I compare the sunk cost of my housing to that of other people, I come out behind.
Part of the blame for this is on readers for being uncharitable. But part of the blame is also on MMM, in my opinion. He spends an awful lot of time talking about frugality, giving examples of how you'd be better off by being more frugal. He spends very little time talking about things that are worth splurging on. When you allocate your time that way, it can be easy for readers to get the wrong impression.
Bang for your buck is a very central idea
Think about the following spectrum:
1. Ramen
2. Chicken and rice
3. Home cooked beef and broccoli
4. Postmates
5. Sit down restaurant
6. Fancy restaurant
As you move along it, things get more expensive, but also higher quality. So then, in figuring out what the best point on the spectrum is, you have to think about bang for your buck.
In going from home cooked beef and broccoli to Postmates, I'd argue that the bang for your buck is usually pretty bad. It'll be a pretty big step up |
62efee88-aa89-480e-baaa-b627f09925fe | trentmkelly/LessWrong-43k | LessWrong | Improving local governance in fragile states - practical lessons from the field
Summary –
Over the past 15 years, major international donor agencies shifted their approach to local politics in fragile states. In order to build stable state-society relations in conflict-affected societies, they committed to studying popular expectations on states, which may differ greatly from international norms like competitive elections and service provision. This blog post examines donor interventions to improve service provision in refugee-crisis-affected communities in Lebanon and Jordan from 2011 to 2019. Donor states hoped that improving service provision capacity of municipalities would increase public trust, but encountered problems with block voting and patronage, leading them to question the value of service provision. Future posts discuss alternate strategies that some donors attempted.
Legitimacy in fragile states -
Legitimacy is the extent to which the people being ruled consent to being ruled. Ruler’s who people trust to exercise coercision are legitimate. Authority is force (coercion) with legitimacy. A majority of Americans may dislike Donald Trump, but they continue to pay taxes to him and few people are advocating we instead address our taxes to Hillary Clinton or Jill Stein. The hanging chad controversy in 2000 was a legitimacy question because it called into question whether the electoral college should confer the right to rule.
Readers from developed countries are used to always knowing the relevant public authority due to a clear division between state and society. The mayor decides who can park where, and whoever wins the biannual mayoral election gets to be mayor. The president appoints the head of the IRS, and whoever gets the most electoral college votes becomes president, et cetera. But in fragile developing countries the question of who wields authority is more often contested. If there is no mayor or the mayor is a figure head appointed by a distant authority, communities some other authorities must judge which spaces are for pa |
c434cb33-d30c-42b3-9c6d-6b4189420256 | trentmkelly/LessWrong-43k | LessWrong | Storytelling and the evolution of human intelligence
This is a notice of a recent paper that may be of interest here, "The storytelling arms race: origin of human intelligence and the scientific mind" by Enrico Coen. It is a more specific working out of the Machiavellian Intelligence hypothesis, taking storytelling as the means through which the deception arms race operated, and which (it suggests) also raised up language alongside intelligence. It also discusses the relationship between storytelling (where, as in war, truth is the first casualty) and science (where truth is the goal).
Disclosure: I work closely with the author, although not on the subject of this paper. |
66b66929-d840-46f8-822e-fc33fcaa73d9 | trentmkelly/LessWrong-43k | LessWrong | Are Deontological Moral Judgments Rationalizations?
In 2007, Chris Matthews of Hardball interviewed David O'steen, executive director of a pro-life organization. Matthews asked:
> I have always wondered something about the pro-life movement. If you believe that killing [a fetus] is murder, why don't you bring murder charges or seek a murder penalty against a woman who has an abortion? Why do you let her off, if you really believe it's murder?1
O'steen replied that "we have never sought criminal penalties against a woman," which isn't an answer but a re-statement of the reason for the question. When pressed, he added that we don't know "how she‘s been forced into this." When pressed again, O'steen abandoned these responses and tried to give a consequentialist answer. He claimed that implementing "civil penalties" and taking away the "financial incentives" of abortion doctors would more successfully "protect unborn children."
But this still doesn't answer the question. If you believe that killing a fetus is murder, then a woman seeking an abortion pays a doctor to commit murder. Why don't abortionists want to change the laws so that abortion is considered murder and a woman who has an abortion can be charged with paying a doctor to commit murder? Psychologist Robert Kurzban cites this as a classic case of moral rationalization.2
Pro-life demonstrators in Illinois were asked a similar question: "If [abortion] was illegal, should there be a penalty for the women who get abortions illegally?" None of them (on the video) thought that women who had illegal abortions should be punished as murders, an ample demonstration of moral rationalization. And I'm sure we can all think of examples where it looks like someone has settled on an intuitive moral judgment and then invented rationalizations later.3
More controversially, some have suggested that rule-based deontological moral judgments generally tend to be rationalizations. Perhaps we can even dissolve the debate between deontological intuitions and utilitarian intuition |
663a31cb-4cf7-43ab-9c81-905f774b59b8 | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | The Alignment Newsletter #5: 05/07/18
**Highlights**
--------------
**[AI safety via debate](https://blog.openai.com/debate/)** *(Geoffrey Irving et al)*: At a high level, a major issue with building superintelligent AI is that humans would not be able to provide strong oversight for the AI. Amplification solves this by using the AI as a tool that can help the human (in particular, if the human can break a task down into subtasks, the AI can solve the subtasks). Debate also provides the AI as a tool for human overseer, but in a different way -- now, in order to train the AI, we have the AI debate against itself in order to convince a human of the answer to some target question. Given some question whose answer is too hard to directly judge, the human can look at the arguments and counterarguments to figure out whether or not the answer is actually correct.
The paper describes debate in a lot more depth and has an initial experiment involving MNIST. I can't possibly do it justice here -- I encourage you to simply read the full paper. You probably have an intuition right now of why this wouldn't work, such as "but humans believe what they want to hear, not what is true". The paper spends 5 (!) pages listing ten such problems and analyzing them, so go read it.
**My opinion:** It's great to see another approach that directly tackles the problem of defining a training signal that if optimized well would lead to an aligned AI. There are a lot of empirical questions that would influence whether or not debate actually works in practice, and I'm excited to see what experiments find.
**[AGI Safety Literature Review](https://arxiv.org/abs/1805.01109)** *(Tom Everitt et al)*: Self-explanatory. It's more of a list of approaches and references within each approach than an integrated whole, but I still expect it to be useful.
**My opinion:** This is great as a way to find references. I do wish there was more comparison between papers and/or approaches, but that's probably asking too much.
**[No Metrics Are Perfect: Adversarial Reward Learning for Visual Storytelling](https://arxiv.org/abs/1804.09160)** *(Xin Wang, Wenhu Chen et al)*: This paper tackles visual story-telling, the task of generating a story that matches a sequence of photos. It proposes learning a reward function from the labeled dataset that can then be optimized with reinforcement learning, with the hope that the reward function is a good compression of what we want and so leads to more generalizable behavior. They show that the standard automated techniques for evaluating visual stories are not very good, and so they perform a Mechanical Turk study that shows very good results compared to prior work. MTurk workers are often unable to tell whether the stories were generated by their algorithm or a human!
How does it work? Their architecture has a policy network that creates the stories and a reward network that provides the supervision, which are trained adversarially. We can think of the reward function as inducing a probability distribution over stories, where stories with higher reward are more probable. Then, the reward network acts as a discriminator, trying to make its implied probability distribution similar to the empirical data distribution and dissimilar to the policy network distribution, while the policy network acts as a generator, creating a policy that tries to match the implied probability distribution of the reward network. (This is equivalent to maximizing the expected reward from the reward network.)
**My opinion:** It's exciting to see reward learning applied to a concrete problem that researchers are working on, and having it lead to an actually better system. This work uses reward learning in a context where we are trying to mimic human actions (sentence generation in this case) -- eventually we will want to be able to deal with different action spaces than humans (as in robotics) and aiming to reach superhuman performance.
**Technical AI alignment**
==========================
### **Technical agendas and prioritization**
**[AGI Safety Literature Review](https://arxiv.org/abs/1805.01109)** *(Tom Everitt et al)*: Summarized in the highlights!
### **Scalable oversight**
**[AI safety via debate](https://blog.openai.com/debate/)** *(Geoffrey Irving et al)*: Summarized in the highlights!
### **Agent foundations**
[Doubts about Updatelessness](https://agentfoundations.org/item?id=1797) *(Alex Appel)*
### **Learning human intent**
**[No Metrics Are Perfect: Adversarial Reward Learning for Visual Storytelling](https://arxiv.org/abs/1804.09160)** *(Xin Wang, Wenhu Chen et al)*: Summarized in the highlights!
[Reward Learning from Narrated Demonstrations](https://arxiv.org/abs/1804.10692) *(Hsiao-Yu Fish Tung et al)*: This paper learns and optimizes rewards given demonstrations of behavior along with a description of the behavior in natural language. Their dataset is a set of videos of humans demonstrating a task and describing it with natural language (such as "the orange is in the bowl"). They combine several techniques to use this dataset to teach a robot. First, using speech recognition, they get a transcript of the natural language aligned with the video. They use object detectors to figure out what things are present in the image, and a syntactic parser to figure out the subject and object of the sentence, and match up these two results to figure out which objects in the image the natural language refers to, and extract their spatial features. They then train a classifier to take the spatial features and detecting whether it has achieved the goal, conditioned on the natural language description of the task. Now that they have a reward function (1 at a goal state, 0 otherwise) they can train a robot using DQN, though to get this to work they infer 3D object configurations from 2D images and use distance to the goal as a shaped reward.
**My opinion:** I'm excited to see approaches to reward learning that use information from natural language -- that definitely seems like a rich source of information we have not made use of yet. That said, there are a lot of moving parts in this system and the parts that analyze natural language impose a lot of structure, to the extent that it feels like a domain-specific language instead of natural language. (I'm not confident about this, it's hard to tell from the paper.) I also don't understand the experiments. They say that the sparse reward couldn't train policies with episode length 3 -- sparse reward is a hard problem, but it isn't *that* hard, so I must be misunderstanding something here.
[Heuristic Approaches for Goal Recognition in Incomplete Domain Models](https://arxiv.org/abs/1804.05917) *(Ramon Pereira et al)* (H/T Beth Barnes): The planning community works on algorithms that can plan given a *symbolic* definition of the environment, how actions affect the environment, and the goal state; analogous to reinforcement learning. The task of inverting the optimal behavior to infer the goal is called goal recognition or plan recognition (analogous to inverse reinforcement learning). This paper looks at goal recognition where the models of the world are incomplete, so that there are *possible* preconditions and effects of actions. They extract potential *landmarks* from the plan, which are things (facts or actions) that must happen in order to achieve the goal, and then suggest two heuristics for how to use the landmarks to rank among possible goals.
**My opinion:** I'm not familiar with this field, but it seems like they have identified a different set of problems with and solutions for goal inference, and it would be useful to see how they apply to IRL. Perhaps the explicit landmark inference leads to more hierarchy in goals? Maybe the unique landmark heuristic is not captured in the standard Boltzmann-rational assumption in IRL? I'd also be interested to see if we can apply IRL algorithms to the plan recognition dataset and get good performance.
**Read more:** [Datasets](https://github.com/pucrs-automated-planning/goal-plan-recognition-dataset)
### **Reward learning theory**
[Rigging is a form of wireheading](https://www.lesswrong.com/posts/b8HauRWrjBdnKEwM5/rigging-is-a-form-of-wireheading) *(Stuart Armstrong)*: Ideally in reward learning, we want our AI to be learning a fact about the world -- which reward it should be optimizing. However, for most proposals of reward learning, the AI's actions can also influence this "fact" about the world. In this case, the AI can wirehead by influencing the world so that it learns an easy-to-maximize reward. This is what Stuart calls "rigging" of the learning process.
**Near-term concerns**
======================
### **Privacy and security**
[Privacy and machine learning: two unexpected allies?](http://www.cleverhans.io/privacy/2018/04/29/privacy-and-machine-learning.html) *(Nicholas Papernot et al)*: Differential privacy provides guarantees on how much information you can obtain by making queries of a specific type of a dataset. Normally, in order to achieve such guarantees, you must add in randomness to the input data that can change the decision, so that there is a plausible explanation for any decision. Unsurprisingly, this tends to degrade performance. However, in deep learning, we often have the problem of our models overfitting to specific details in the training set instead of generalizing appropriately, so we might expect that differential privacy could actually *help* with performance (as well as privacy). Private Aggregation of Teacher Ensembles (PATE) demonstrates that this is the case. It works by training several teacher models on different datasets to solve the task at hand. Then, by aggregating the results across the ensemble with some random noise, we can answer queries and put bounds on the amount of information that is leaked. However, with each query we use up more of our "privacy budget", so it can't be used arbitrarily long. To solve this, we can make a fixed number of queries to label some unlabelled data, use those labels to train a student model, and use the student model to make predictions forever after. An adversary could at worst infer the entire training dataset of the student model -- but that training set was designed to be private.
**My opinion:** I would have been excited by work that randomizes the inputs to a deep learning technique in order to get better generalizability. It's cool that this goal dovetails so beautifully with the desire for differential privacy.
**AI capabilities**
===================
### **Reinforcement learning**
[TDM: From Model-Free to Model-Based Deep Reinforcement Learning](http://bair.berkeley.edu/blog/2018/04/26/tdm/) *(Vitchyr Pong)*: In many tasks, we have hierarchical structure where we want to plan at a high level, but to execute the low-level actions we want to rely on learning through experience. For example, when biking from UC Berkeley to the Golden Gate Bridge, you definitely want to plan in advance the route you'll take (as opposed to learning through trial-and-error), but you want to learn how to bike through trial-and-error. Temporal Difference Models allow you to do model-based planning at the high level, and model-free learning at the low level. Specifically, you learn a function Q(s1, a, s2, T), which intuitively says "if I start from state s1, taking action a, and running for T steps, how close can I get to state s2". It turns out that this can be thought of as a Q function and so can be trained using standard model-free RL techniques. Note that the constraint Q(s1, a, s2, T) = 0 says that it is possible to get from s1 to s2 in T steps after first taking action a.
One standard way to solve model-based RL is to search for a sequence of states and actions (s0, a0, s1, a1, ...) that is feasible (agrees with the dynamics) and maximizes the reward, and then take the first action from that sequence. Using TDMs, we can now search for the sequence (s0, a0, sK, aK, s2K, a2k, ...) that is feasible and maximizes reward. The feasibility requirement is expressed by the constraint Q(s0, a0, sK, K) = 0.
**My opinion:** Firstly, the blog post is very readable and provides a great introduction (it's much more friendly than my summary).
This technique does require that we can reinterpret any state as a goal state, similar to the assumption in [Hindsight Experience Replay](https://arxiv.org/abs/1707.01495) (HER). They do compare to HER, and find that HER doesn't do very well, which I was quite surprised by. Clicking through to the paper, it turns out the authors were surprised as well, but then realized that this is because HER is designed to work with sparse reward problems, whereas they were evaluating on problems with relatively shaped rewards.
[Towards Symbolic Reinforcement Learning with Common Sense](https://arxiv.org/abs/1804.08597) *(Artur d'Avila Garcez et al)*
[Reinforcement Learning and Control as Probabilistic Inference: Tutorial and Review](https://arxiv.org/abs/1805.00909) *(Sergey Levine)*
### **Deep learning**
[MLPerf](https://mlperf.org/): From their overview: "The MLPerf effort aims to build a common set of benchmarks that enables the machine learning (ML) field to measure system performance for both training and inference from mobile devices to cloud services." They have a track to measure the performance of hardware and software systems that support ML models, as well as a track that aims to advance the state-of-the-art in ML models. They consider a broad set of problems (though it seems like they are all problems where some deep learning technique is state-of-the-art).
[The Best of Both Worlds: Combining Recent Advances in Neural Machine Translation](https://arxiv.org/abs/1804.09849) *(Mia Xu Chen, Orhan Firat, Ankur Bapna et al)*
### **Machine learning**
[On the Convergence of Adam and Beyond](https://openreview.net/pdf?id=ryQu7f-RZ) *(Sashank J. Reddi et al)*
**News**
========
RAISE [releases prereq modules](https://www.lesswrong.com/posts/vi48CMtZL8ZkRpuad/soon-a-weekly-ai-safety-prerequisites-module-on-lesswrong) and [is looking for high level feedback](https://www.lesswrong.com/posts/z5SJBmTpqG6rexTgH/looking-for-ai-safety-experts-to-provide-high-level-guidance): RAISE in collaboration with Erik Istre and Trent Fowler are developing a curriculum for prereqs to AI safety (topics like logic and probability). The first topic (logic) is available [here](https://play.ihatestatistics.com/#/course/141?access_token=3HCK4oRipeFY2ghyYMqDJKYX57KUnzNL). Also, they are [looking](https://www.lesswrong.com/posts/z5SJBmTpqG6rexTgH/looking-for-ai-safety-experts-to-provide-high-level-guidance) for AI safety experts to provide high-level feedback and guidance on the course structure for the MOOC they are building.
[Facebook Open Sources ELF OpenGo](https://research.fb.com/facebook-open-sources-elf-opengo/) *(Yuandong Tian et al)*: Facebook has created an open-source AI bot that has beaten world champion professional Go players in matches where the professional player was allowed unlimited time to think. |
e8d16269-d23a-4fa5-a2f7-b71b672f262f | trentmkelly/LessWrong-43k | LessWrong | A Quick Ontology of Agreement
Adapted from Herman Kahn's Introduction to Can We Win in Vietnam, Armbruster, F.E., et. al. Praeger: New York, 1968.
This is a quick post about an ordered sense of agreement. I find it serves two useful purposes:
1. It gives a practical framework for how to achieve better agreement between parties in apparent conflict, so it is a tool for both conflict resolution and for problem-solving, and
2. It replaces the binary structure of agree/disagree with a looser structure that allows people to exist in conceptual tension without feeling like they are involved in interpersonal conflict.
Now a description of the levels, followed by a brief discussion.
First Order Agreement
The conventional understanding of agreement. All parties agree to the characterization of a resolution to the point of contention, what Kahn called "agreement on the substance of an issue". Scope is a particularly important characteristic to achieve first order agreement. What does the agreement cover and what does it not cover?
In formal legal or institutional contexts first order agreement is instantiated in text where parties to the agreement indicate their acceptance of the resolution by signature or by the signature of an authorized agent. The agreement is ratified.
Second Order Agreement
Agreement on why there is no first order agreement. All parties agree to the characterization of what precludes a resolution to the point of contention. Kahn called this agreeing on what the disagreement is about.
Third Order Agreement
Agreement on why there is no second order agreement; what is conventionally called agreeing to disagree. All parties agree that there is a disagreement, but cannot agree on the characterization of that disagreement. Generalizing a bit from Kahn, who saw this as largely the interference of emotion in rational discussion, one can think of third order agreement as the belief by one party (possibly both) that the other party is either unwilling or incapable of articulat |
b51b0f28-7cea-4343-8ad6-1aa9b40bcec4 | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Compute Thresholds: proposed rules to mitigate risk of a “lab leak” accident during AI training runs
1. There should be two thresholds on compute graph size:
1. the Frontier threshold, beyond which oversight during execution is mandatory
2. the Horizon threshold, beyond which execution is forbidden by default
2. Oversight during execution:
1. should be carried out by state and/or international inspectors who specialize in evaluating frontier training runs
1. Individuals who are employed as such inspectors should not have any past or present conflict of interest with the organization whose runs they evaluate.
2. However, it is beneficial if these individuals have pertinent experience and knowledge of frontier AI.
2. should include, but not be limited to, “dangerous capabilities evaluations” at various points during training
3. should be allocated a fixed fraction of the total compute budget for the training run, say, 5%
3. Inspectors should be empowered to pause a training run at any time if they see evidence that the model is becoming dangerous relative to the safety precautions being taken.
1. There should be due process for the organization executing the training run to appeal for permission to continue.
2. Levels of safety precautions should be referenced to a (mostly not-yet-written) body of standards for cybersecurity, formal verification, determinism, etc. in the AI training run context.
4. The two compute-graph size thresholds should be changed:
1. gradually upwards, on an explicit schedule that provides for a slight increase every calendar day.
1. This avoids the potential shock of sudden increases in capabilities at the end of a “pause”.
2. downwards in response to new discoveries that impact compute-efficiency, via a very rapid but nonetheless legalistically formalized process that requires a vote from a neutral international board of experts.
5. Execution of compute graphs exceeding the Horizon threshold may be permitted,
1. if the case for the adequacy of their safety mechanisms is convincing to a supermajority of the expert board,
2. *and*with unanimous assent of the duly appointed representatives of the international parties to these rules.
6. Compute graph size should be measured:
1. accounting for all computations in the causal history of the proposed execution back to before September 2021,
1. including, in particular:
1. Base models from which fine-tuning is taking place
2. Models which may have been used to generate some of the training data
2. For the avoidance of doubt, this accounting should recursively aggregate transitive inputs.
3. Safe harbor: if all constants, parameters, and variables in the compute graph are either initialized to random Gaussians, or to concatenations (in a random order) of byte-sequences which can be proven to have existed before September 2021, then no further accounting for inputs is necessary.
2. in “standard FLOPs”, which are defined using a standardized set of coefficients,
1. For example,
1. 1 NF4 multiplication = 0.7 standard FLOP
2. 1 FP8 multiplication = 1.0 standard FLOP
3. 1 FP16 multiplication = 1.2 standard FLOP
4. 1 FP32 multiplication = 1.3 standard FLOP
5. 1 FP64 multiplication = 1.33 standard FLOP
2. These coefficients are negotiable, but should ideally be based on some evidence, such as comparative analyses of perplexity scores achievable with training runs that vary only in numerical format.
7. Nothing in this proposal should be construed as minimizing or dismissing the misuse risks of models below the Frontier threshold.
1. However, it is suggested that evaluations of such models can safely take place after a training run has been executed, as long as the model outputs are not yet put in contact with humans or with vulnerable cyberinfrastructure.
8. Requirements for the following are out of scope of this proposal, and left for future work:
1. licensing (of developers, users, inspectors, etc.)
2. registration (of training plans, deployment plans, etc.)
3. tracking and tracing (of high-throughput training hardware, e.g. H100s and H800s)
4. verification (of compute graph size, especially below-but-near the Frontier threshold, using cryptographic proofs) |
3713309a-cbcd-47cf-a42f-ff9b26f55858 | StampyAI/alignment-research-dataset/youtube | Youtube Transcripts | Is AI Safety a Pascal's Mugging?
hi today we're going to talk about
Pascal's wager and Pascal's mugging this
is necessarily going to touch on
religious topics pretty heavily actually
and I'm just gonna say at the beginning
of the video that personally I don't
believe that any god or gods have ever
existed and I'm not going to pretend
otherwise in this video so be forewarned
if that's likely to bother you so as you
may know Pascal was a 17th century
philosopher who was interested in
amongst other things the question of the
existence of the Christian God various
philosophers at the time were arguing
that God didn't exist and there was a
lot of discussion going on about the
various kinds of evidence for and
against God in the world but there's
this thing that's quite common when
people think about religious questions
where it feels sort of unsatisfying to
talk about worldly evidence as if you
were considering some everyday question
there's a feeling that these
supernatural concepts are very grand and
mysterious they're special and so just
straightforwardly considering the
evidence for and against God is not the
right way to do things this is of course
encouraged by religious thinking the
idea that some hypotheses aren't subject
to the usual rules of evidence and logic
is pretty appealing if you want to
advocate for an idea that doesn't fare
very well by those standards
I suspect Pascal may have felt something
like that because his position was that
reason has nothing to say about the
question of whether or not God exists
it's sort of an unknowable thing and
instead he proposed that we should make
a wager we should think about it like
this there are two possibilities either
the Christian God exists or he doesn't
and reason gives us no way to choose
between those we have two options
available to us either we can live
according to God's laws and act as
though we believe or we can not do that
so we have a sort of payoff matrix here
with four sections if God exists and we
believe in him then we get infinite
reward in heaven if God exists and we
don't believe in him we get infinite
punishment in hell if God doesn't exist
and we believe in him then we pay some
costs you know there are some rules we
have to follow and so on and if he
doesn't exist and we don't believe in
him then maybe we get a few perks from
not believing like having a lion on
Sundays and being right Pascal's point
is that this payoff matrix is completely
dominated by the case in which God
exists because we're talking about
infinite rewards and infinite
punishments as opposed to the other case
with these very finite costs
benefits so regardless of the evidence
Pascal argues we should believe in God
or at least act like it because it's
just the sensible bet to make this is
really kind of a nice argument from
Pascal's perspective because it doesn't
need evidence at all no finite earthly
evidence can outweigh infinite
supernatural payoffs it feels like the
kind of clean abstract reasoning that
you're supposed to do when thinking
about the supernatural all of this hard
work looking at history and psychology
and science and trying to figure out
where the ideas of religion come from
and whether our world seems like the
kind of world with a God in it it's
long-winded confusing it's it's just
messy but here we just have a clean
argument that says we should believe in
God or at least act like it and that
seems very neat no evidence required so
consider now Pascal is walking down the
street and he's stopped by a shady
looking man who says give me a wallet I
would prefer not to do you even have a
weapon no UK laws are very strict about
that but I don't need one because I'm
God your God yep I'm God and
Christianity got a lot of things wrong
about me my forgiving nature my infinite
mercy and so on but the infinite torture
thing is legit and if you don't give me
your wallet right now I will torture you
for eternity in the afterlife now if
you're Pascal you're in kind of a
difficult situation because the fact
that it seems very unlikely that this
murder actually is God is not meant to
be part of your calculation your
argument is one a pure logic it works
independently of any evidence you didn't
need to look for evidence of the
Christian God and you don't need to look
for evidence that this mother is God
either so you kind of have to give him
your wallet and now you're really in
trouble because of course when this gets
out there's gonna be a line around the
block of lsaps deities asking for
handouts how are you going to deal with
this endless stream of fizzy gods well
one thing you can do is you can play the
muggers off against one another you can
bring in two of them and say listen you
say that you're going to torture me
forever if I don't give you my wallet
and you say the same thing I only have
one wallet so it looks like whatever I
do I'm going to be tortured forever by
somebody and if I'm going to be
infinitely tortured anyway well two
times infinity is still just infinity so
I may as well hang on to the wallet now
get the hell out of my house all right
next to no doubt these self-proclaimed
deities may try to argue that they have
some reason why they are in fact a real
deity
this other mugger is just a random guy
who's pretending but that's all worldly
evidence which you've decided isn't
required for your argument and the
muggers don't really want you to become
interested in evidence because well the
evidence points very strongly towards
none of them being real gods so this is
a better position to be in you're still
spending a lot of your time arguing with
charlatans but at least you still have
your wallet and you don't actually have
to pair them up against each other right
you can just make up a deity when
someone comes in pretending to be a god
you can say oh well there's this other
God who demands exactly the opposite
thing from you a goddess actually and
she's very powerful but she goes to a
different school
you would know her really yeah she lives
in Canada she's only present obviously
but she lives in Canada anyway she says
that I'm not to give you the wallet and
if I do then she'll torture me forever
in the afterlife I think yeah so you can
solve a lot of these problems by
inventing gods arbitrarily and of course
this applies just as well to the
original version of Pascal's wager
because although it's implied that this
payoff matrix has enumerated all of the
possibilities and in a sense it has the
Christian God either exists or it
doesn't nonetheless those may not be the
only things that effect the payoffs for
any given God you can take the God down
flip it and reverse it and say what
about ante God who wants me to do the
exact opposite and promises the exact
opposite consequences now you can see
that they cancel out somebody who's
arguing for the existence of the first
God might say okay but this anti God is
just made up which I mean yeah it is
it's true that the situation isn't
really symmetrical someone might think
God is more likely than anti God because
of evidence from the Bible and there's
no such thing as the anti Bible and so
on the point is though we're back to
talking about the evidence that's really
the problem I have with Pascal's wager
the way it uses infinite costs and
infinite benefits to completely override
our necessarily finite evidence but what
if the costs and benefits aren't
infinite just very very large that ends
up being a much more interesting
question on one end of the scale we can
easily name numbers so large that no
amount of evidence anyone could ever
actually gather in their lifetime could
have an impact on the conclusion we can
specify costs and benefits that are
technically finite but that still feel
very much like a Pascal's wager on the
other end of the scale if you come
across a bet that pays out 10
two one on an event with a probability
of one in a hundred that's a very good
bet to take someone could complain that
it's a Pascal's wager to bet on an
unlikely outcome just because the payoff
is so high but if you take enough bets
like that you're sure to become very
rich in the same way if there's a button
which has a one-in-a-million chance of
starting a global thermonuclear war it's
still worth expending significant
resources to stop that button being
pressed one in a million isn't much but
the cost of a nuclear war is really high
I don't think that's a Pascal's wager
either the difference seems to come in
somewhere in the gap between very small
probabilities of very large costs and
benefits and really extremely small
probabilities of near infinite costs and
benefits so why are we talking about
this what does this have to do with AI
safety well suppose somebody stops you
in the street and says hey if we ever
create powerful artificial general
intelligence then that will have a
tremendous impact in fact the future of
the whole of humanity hinges on it if we
get it right we could have human
flourishing for the rest of time if we
get it wrong we could have human
extinction or worse regardless of how
likely superhuman AGI is the potential
impact is so high that it makes AI
safety research tremendously important
so give me your wallet it's been claimed
by some that this is more or less what
AI safety as a field is doing this is
kind of an interesting point there's any
safety advocates are we victims of
Pascal's mugging or are we in fact
Pascal's miners ourselves well if people
were saying these AI risks may be
extremely unlikely but the consequences
of getting AI wrong are so huge that
it's worth spending a lot of resources
on regardless of the probabilities so we
don't even need to consider the evidence
well I would consider that to be a
Pascal's wager style bad argument but
what I actually hear is not that what I
hear is more like look we're not
completely sure about this it's quite
possible that we're wrong but
considering the enormity of what's at
stake it's definitely worth allocating
more resources to AI safety than we
currently are that sounds pretty similar
but that's mostly because natural
language is extremely vague when talking
about uncertainty there's an enormous
difference in the probabilities being
talked about in the same way if when you
talk to AI safety researchers they said
things like well I think the chance of
any of this ever being relevant are
really extremely tiny it seems more or
less impossible to me but I've decided
to work on it anyway
because the potential costs and benefits
are so unimaginably vast then yeah I'd
be a little concerned that they might be
victims of Pascal's mugging but when you
ask AI safety researchers they don't
think that the probability of their work
ever becoming relevant is very tiny
they don't necessarily think it's huge
either maybe not even more than 50% but
it's not so small that you have to rely
on the unimaginable vastness of the
consequences in order to make the
argument to borrow a metaphor from
Stuart Russell suppose you're part of a
team working on building a bridge and
you believe you've found a flaw in the
design that could cause the structure to
fail catastrophically maybe the disaster
would only happen if there's a very rare
combination of weather conditions and
there's only a one in a hundred chance
that those conditions will ever happen
during the course of the bridges
expected lifespan and further suppose
that you're not completely sure of your
calculations because this kind of thing
is complicated maybe you only give
yourself a 40% chance of being right
about this so you go to the civil
engineer in charge of the project and
you say I think there's a serious risk
with this bridge design do you think the
bridges gonna collapse probably not no
but I'm about 40 percent sure that
there's a design flaw which would give
this bridge a 1 in 100 chance of
catastrophic failure so you're telling
me that in the event of a scenario which
is very unlikely to happen the bridge
might collapse and you yourself admit
that you're more likely to be wrong than
right about this stop wasting my time
but if the bridge collapses it could
kill a lot of people I think this is a
Pascal's mugging don't try to get me to
ignore the low probabilities just by
threatening very large consequences
obviously that isn't what would happen
no civil engineer is going to accept a 1
in 250 chance of catastrophic failure
for a major piece of infrastructure
because civil engineers have a healthy
organizational culture around safety
what it comes down to again is the
difference between different levels of
improbability the chance of an AGI
catastrophe may not be very big but it's
much much larger than the chance that a
mugger is actually a god and what about
our anti-god tactic finding the opposite
risk does that still work like what if
we consider the possibility that there's
another opposite design flaw in the
bridge which might cause it to collapse
unless we don't spend extra time
evaluating the safety if that is not
what just look at the schematic with you
and what if working on AI safety
actually ends up making the risks worse
somehow I think this actually is worth
considering unintended consequences are
a real problem after all speaking
generally there's
clear argument that the future is very
important and that we're probably able
to have a very big impact on it but it's
hard to know for sure whether that
impact will be positive or negative for
any given course of action prediction is
very difficult as they say especially
about the future and the further into
the future we look the more difficult it
gets like imagine if you lived in the
year 1900 and you had some insight that
made you realize that nuclear weapons
were possible and nuclear war was a risk
you'd hope that you could use that
understanding to reduce the risk but it
would certainly be possible to make
things worse by accident in the case of
AI safety though I don't see that being
anywhere near as much of a concern we're
heading towards AI regardless and it
seems very unlikely that thinking about
safety would be more dangerous than not
thinking about safety it's definitely
possible to make things worse while
trying to make them better but you can't
avoid that by never trying to make
things better I guess my point is
there's just no getting around the messy
confusing complicated work of looking at
and thinking about the evidence any
argument that doesn't rely on the
evidence will work equally well whatever
the truth is so at the end of the day
that kind of thing isn't going to give
you an answer you have to just stare at
the bridge design and really think you
have to actually do the engineering and
that's something I'm trying to get
across with this channel you won't find
me saying never mind the evidence ai
safety is important because it could
have huge consequences what I do on this
channel is I try to show you some of the
evidence in some of the arguments and
let you think about the situation and
draw your own conclusions it can be
tricky and involved it requires some
thought but it has the advantage of
being the only thing that has any chance
of actually getting the right answer so
thanks for watching
[Music]
as my wonderful patrons will know the
alignment newsletter is a weekly
publication from Rowan char which I read
every week to stay up to date with
what's going on in here safety and now
I'm recording myself reading it out and
publishing that as the alignment
newsletter podcast it's aimed at
researchers so it's a fair bit more
technical than this channel but if
you're interested in getting 15 minutes
of AI safety news in your earholes each
week check the link in the description
I'm never going to put ads or sponsors
on that podcast and that's largely
thanks to my patrons in this video I'm
especially thanking Chris canal thank
you so much for your support Chris thank
you to all of my patrons and thank you
for watching I'll see you next time
[Music]
little costume changes |
eceb288d-aa65-43fa-aad8-bf770b71dd63 | trentmkelly/LessWrong-43k | LessWrong | Church vs. Taskforce
I am generally suspicious of envying crazy groups or trying to blindly copycat the rhythm of religion—what I called "hymns to the nonexistence of God", replying, "A good 'atheistic hymn' is simply a song about anything worth singing about that doesn't happen to be religious."
But religion does fill certain holes in people's minds, some of which are even worth filling. If you eliminate religion, you have to be aware of what gaps are left behind.
If you suddenly deleted religion from the world, the largest gap left would not be anything of ideals or morals; it would be the church, the community. Among those who now stay religious without quite really believing in God—how many are just sticking to it from wanting to stay with their neighbors at the church, and their family and friends? How many would convert to atheism, if all those others deconverted, and that were the price of staying in the community and keeping its respect? I would guess... probably quite a lot.
In truth... this is probably something I don't understand all that well, myself. "Brownies and babysitting" were the first two things that came to mind. Do churches lend helping hands in emergencies? Or just a shoulder to cry on? How strong is a church community? It probably depends on the church, and in any case, that's not the correct question. One should start by considering what a hunter-gatherer band gives its people, and ask what's missing in modern life—if a modern First World church fills only some of that, then by all means let us try to do better.
So without copycatting religion—without assuming that we must gather every Sunday morning in a building with stained-glass windows while the children dress up in formal clothes and listen to someone sing—let's consider how to fill the emotional gap, after religion stops being an option.
To help break the mold to start with—the straitjacket of cached thoughts on how to do this sort of thing—consider that some modern offices may also fill th |
38fa9bb3-c622-4010-8285-fda71e791b21 | trentmkelly/LessWrong-43k | LessWrong | Quote help
There's a famous I'm struggling to find but it's along the lines of:
'Those who are wise seek clarity. Those who wish to merely appear wise seek obscurity'
It's great for getting a read on people and their intention.
Does anyone know who said it and what the original quote is?
|
fb08ec73-a622-4dcb-b9b6-64c00d033466 | trentmkelly/LessWrong-43k | LessWrong | [SEQ RERUN] Make an Extraordinary Effort
Today's post, Make an Extraordinary Effort was originally published on 07 October 2008. A summary (taken from the LW wiki):
> It takes an extraordinary amount of rationality before you stop making stupid mistakes. Doing better requires making extraordinary efforts.
Discuss the post here (rather than in the comments to the original post).
This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was On Doing the Impossible, and you can use the sequence_reruns tag or rss feed to follow the rest of the series.
Sequence reruns are a community-driven effort. You can participate by re-reading the sequence post, discussing it here, posting the next day's sequence reruns post, or summarizing forthcoming articles on the wiki. Go here for more details, or to have meta discussions about the Rerunning the Sequences series. |
7f4db3ed-c7a5-460c-bafa-cd78c1c1b158 | StampyAI/alignment-research-dataset/arxiv | Arxiv | Reward Reports for Reinforcement Learning
1. Introduction
----------------
In fall 2021, the Wall Street Journal published the Facebook files, a series of articles that pieced together platform policy, flaws, and harms through leaked internal documents (Hagey and Horwitz, [2021](#bib.bib26)). Among other topics, these articles tracked how changes to the newsfeed algorithm affected social interactions. Reporters confirmed what researchers had long suspected: Facebook’s pivot to ‘meaningful social interactions’ increased negative content and divisiveness. Furthermore, both top content producers and Facebook executives were aware the algorithm generated this uptick, although the latter permitted it in favor of increased engagement. Even if Facebook were to make an explicit commitment to anticipate such effects, mitigating the feedback between interventions and harms is no small task. Facebook’s platform policies balance hundreds of parameters which are updated multiple times a day, amounting to a never ending A/B test. As a former employee explained, the system had grown so complex that data scientists could not trace negative effects back to efforts to increase meaningful connection.
The intentional design of algorithmic systems to promote access and equity requires that designers reflect deeply and reflexively about how interventions may impact the dynamics of the deployment domain over time. When systems are tuned using the same environment in which they operate, the dynamics can quickly overwhelm designers’ intent to drive specific interactions. To manage feedback-heavy systems responsibly, the repercussions of algorithmic changes must be reflexively documented. Imagine a world where these effects are regularly anticipated, observed, and controlled. For designers, this documentation would aid internal efforts at reverse engineering the behavior of black box systems, and provide a framework to disentangle complex effects. Moreover, a standard mechanism for transparency would help harmonize external efforts at domain monitoring and inform regulatory action.
Multiple frameworks for documenting AI systems, datasets and models have emerged (Mitchell et al., [2019](#bib.bib41); Gebru et al., [2021](#bib.bib23); Richards et al., [2020](#bib.bib49)). These aim to track sources of potential bias or harm, grounded primarily in a static machine learning (ML) paradigm. However, the effects of deployed AI systems are not static, and the dynamic impacts of successive system updates can subvert efforts both to manage downstream harms and to more evenly distribute benefits to vulnerable subpopulations. The presence of feedback and dynamics suggests unique risk vectors and requisite forms of documentation. Reinforcement learning (RL), a sub-field of ML that is able to solve complex sequential, open-ended problems, provides a dynamic lens that is broadly applicable to many algorithmic systems with repeated data-driven optimizations.
We propose Reward Reports, a new form of documentation that foregrounds the societal impacts of data-driven optimization systems, whether explicitly or implicitly construed as RL. Building on proposals to document datasets and models, we focus on reward functions: the objective that guides optimization decisions in feedback-laden systems. Reward Reports comprise questions that highlight the promised benefits and potential risks entailed in defining what is being optimized in an algorithmic system, and are intended as living documents that dissolve the distinction between ex-ante specification and ex-post evaluation. As a result, Reward Reports provide a framework for ongoing deliberation and accountability after a system is deployed, ensuring that desired properties persist in the system’s behavior over time.
In [Section 2](#S2 "2. Related Work ‣ Reward Reports for Reinforcement Learning"), we situate Reward Reports within the existing literature on AI documentation and governance mechanisms. In [Section 3](#S3 "3. Background ‣ Reward Reports for Reinforcement Learning"), we review optimization in the context of feedback, focusing on the notions of action, objective, and adaptation. In [Section 4](#S4 "4. Motivation ‣ Reward Reports for Reinforcement Learning"), we contextualize these elements of optimization within a taxonomy of feedback. Finally, in [Sections 6](#S6 "6. Examples ‣ Reward Reports for Reinforcement Learning") and [7](#S7 "7. Discussion & Conclusions ‣ Reward Reports for Reinforcement Learning"), we provide a list of questions that interrogate specification components and provide three examples of Reward Reports for RL systems of varying stakes.
2. Related Work
----------------
###
2.1. Documentation for AI Systems
There are a number of existing proposals for AI system documentation, with some frameworks focused on specific aspects of an AI system, while others examine the system as a whole.
Our proposal of Reward Reports is focused on documenting risks of dynamic machine learning systems.
This complements current documentation efforts by explicating the intended performance characteristics in light of system behaviors that emerge over time.
Documenting data, regardless of resulting systems, is a well explored avenue (Barclay et al., [2019](#bib.bib7); Afzal et al., [2021](#bib.bib3); Hutchinson et al., [2021](#bib.bib29); Denton et al., [2021](#bib.bib18); Gebru et al., [2021](#bib.bib23)).
These efforts have helped foster discussion on responsible data collection, as well as showcase issues of bias and representation. However, this paradigm is most impactful within the batch or offline setting of static unsupervised or supervised ML.
Dynamic systems driven by sequential feedback also use data, but not only in the form of static datasets.
Rather, both RL and deployed ML systems generate and eventually transform data. This is due both to the optimization decisions of the systems themselves and the dynamics of the environment in which they operate.
Past data documentation efforts are therefore insufficient to reveal the risks and failures
of dynamic datasets and feedback driven systems.
There are also proposals for model (Mitchell et al., [2019](#bib.bib41)), domain (Ramírez et al., [2020](#bib.bib46); Kühl et al., [2021](#bib.bib34)) or outcome specific (Sokol and Flach, [2020](#bib.bib59)) forms of ML documentation. Reward Reports might be a useful supplement to these approaches for several reasons.
With regards to model cards, there is substantial deliberation entailed in mapping between the chosen optimization and resultant behavior in dynamic systems.
For example, designers must consider a range of alternative specifications that were technically feasible pre-deployment, as well as the types of feedback available to help optimize post-deployment.
These aspects of system design cannot be captured solely by documenting the model.
Furthermore, unlike domain-specific reporting frameworks, we provide a
general template that can be applied to any specific application.
Our work has similarities with previous proposals for AI Ethics Sheets (Mohammad, [2021](#bib.bib42)), Fact Sheets (Arnold et al., [2019](#bib.bib6); Richards et al., [2020](#bib.bib49)), or Scorecards (Blasch et al., [2021](#bib.bib11)), but uniquely focuses on prompting deliberation about the feedback-driven risks inherent to dynamic systems.
Reward Reports are closely related to Algorithmic Impact Assessments (AIA), which offer a framework for evaluating risks before an AI system is developed or acquired (Reisman et al., [2018](#bib.bib48); Selbst and Barocas, [2018](#bib.bib53)).
AIAs were inspired by environmental impact assessments, which provided one path to regulate industries in which corporate expertise outpaced government capacity.
These frameworks presume an agency-vendor relationship, and focus narrowly on the procurement of automated decision systems.
Reward Reports are intended to supersede these ex-ante concerns,
engaging instead with the necessarily non-linear and circuitous process of refining the specifications of a feedback system.
###
2.2. Societal Risks of Reinforcement Learning Systems
The RL research community has begun to reflect on the unique risks and challenges that may be posed by RL systems, in particular those that leverage black-box neural networks for decision-making.
There are whitepapers charting these challenges
(Whittlestone et al., [2021](#bib.bib62); Gilbert, [2021](#bib.bib24)), as well as attempts to address societal risks through technical means (Wen et al., [2021](#bib.bib61); Carroll et al., [2021](#bib.bib13); Evans and Kasirzadeh, [2021](#bib.bib20); Zhan et al., [2021](#bib.bib66)).
Recent general audience books have echoed these concerns (Russell, [2019](#bib.bib51); Christian, [2020](#bib.bib15)).
While these efforts have begun to capture the unique stakes in deploying RL systems, there is no consensus on how to chart associated risks.
We intend Reward Reports to organize these efforts’ reflections into an instrument of deliberation and accountability.
Critical and discursive interpretation of feedback are more common in static ML than RL research.
However, the technical RL community is increasingly aware of the limitations of current algorithmic approaches and evaluation paradigms.
In efforts to match practical solutions with open-ended problems, the field is re-considering how results are reported (Agarwal et al., [2021](#bib.bib4); Henderson et al., [2018](#bib.bib28); Islam et al., [2017](#bib.bib30)), and if numerical successes match the intention or intuition of designers (Mania et al., [2018](#bib.bib40)).
While this agenda aims to promote scientific reproducibility, Reward Reports are meant to ensure public accountability.
###
2.3. AI Governance
Governance mechanisms can also be used to dictate safer machine learning practices.
As reflected in the growing number of proposed AI governance frameworks (Gasser and Almeida, [2017](#bib.bib22); Lee et al., [2019](#bib.bib37); Yeung et al., [2019](#bib.bib65); Cihon, [2019](#bib.bib16); Wirtz et al., [2020](#bib.bib63); Reddy et al., [2020](#bib.bib47)), ML and adjacent communities have increasingly acknowledged socio-technical risks and the need for novel harm mitigation strategies (Selbst et al., [2019](#bib.bib54); Andrus et al., [2020](#bib.bib5); Dean et al., [2021](#bib.bib17)).
These frameworks have begun to influence legislation.
For example, the Canadian government has mandated the Algorithmic Risk Assessment tool for procurement (Board, [2019](#bib.bib12)), and current draft legislation from the EU commission calls for AI system documentation tailored to forms of risk (e.g. in Title 3, Ch. 2, Art. 10-14) (eu2, [2021](#bib.bib2)).
While these frameworks provide needed prohibitions and protections, they favor interpretive flexibility over specific design decisions, and often assume static AI systems that need strictly ex-post documentation.
In contrast, Reward Reports would track requisite design decisions, and provide an interface for stakeholders to reflect on the validity of those design choices over time.
This would in essence dissolve the boundary between ex-ante and ex-post assessment.
3. Background
--------------
In this section, we elucidate the notion of data-driven optimization systems as dynamical systems.
We begin by reinterpreting them in terms of
action, objective, and adaptation.
This taxonomy emphasizes the use and behavior of learned models rather than the closed act of developing them.
We then review the RL framework, and recount its broader connections to optimization and machine learning.
###
3.1. Action, Objective, Adaptation
Learned predictive models are the means to some end.
It is the decisions made, or *actions* taken, that determine the extent to which a model is successful.
For example, congested suburban roadways in the community of Los Gatos, CA, USA are caused directly by the actions of drivers, and indirectly by the actions of routing algorithms that predict a poorly-scaling short-cut path (Peterson, [2018](#bib.bib45)).
Similarly, the optimization of datacenter operations at Google
uses the predictions of trained models to adjust set-points and load distribution – the predictions serve as a catalyst for action in the real world (Gao, [2014](#bib.bib21)).
Action occurs not only on the basis of predictions, but also towards some *objective*.
The definition of objective is crucial to the resulting behavior.
Identical traffic models will result in different routing suggestions depending on whether the algorithm is optimizing for arrival time or fuel consumption (NREL, [2021](#bib.bib44)).
Likewise, content interaction models have vastly different impacts when they are used to uprank posts predicted to receive many ‘likes’ compared with those predicted to receive long comments (Hagey and Horwitz, [2021](#bib.bib26)).
Finally, these optimization systems are often updated based on additional data collected during their operation, making them adaptive.
By accounting for the dependence between past decisions, observed data, and current models, systems in effect react to dynamic environments and improve performance over time.
For example, when observed music listening patterns are used as additional data in preference models, music recommendation algorithms can adapt to an individual’s evolving tastes (Dhahri et al., [2018](#bib.bib19)).
On the other hand, adaptivity can also exacerbate biased data.
For example, predictive policing systems can amplify racial biases in arrests by directing more patrols towards areas with more documented arrests (Lum and Isaac, [2016](#bib.bib39); Sankin et al., [2021](#bib.bib52)).
While action, objective, and adaptation are important for ensuring that systems work as intended, they are not captured in frameworks that document static elements of ML models.
###
3.2. Reinforcement Learning
The reinforcement learning (RL) framework succinctly encompasses action, objective, and adaptation.
RL algorithms take actions, are motivated by a reward signal which encodes the objective, and adapt based on the feedback from this interaction.
While the goal of supervised learning (SL) procedures is to use data to generate a model that makes accurate predictions, the goal of RL algorithms is to
interact with an environment to generate a policy that achieves high reward.
However, once SL models are deployed towards some goal and updated with new data, the concerns highlighted by the RL framework become relevant.
In this sense, ML deployments can be understood through the lens of RL.
Rather than Reinforcement Learning being a ‘cherry’ atop the Machine Learning cake (LeCun, [2016](#bib.bib36)), it would appear that the entire cake is in fact cherry flavored.
Reinforcement learning is a framework for optimizing a system via trial and error.
In common terminology, an *agent* executes *actions* a→t∈𝒜subscript→𝑎𝑡𝒜\vec{a}\_{t}\in\mathcal{A}over→ start\_ARG italic\_a end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ∈ caligraphic\_A in an *environment*.
In response, the agent receives a scalar *reward* rt∈ℝsubscript𝑟𝑡ℝr\_{t}\in\mathbb{R}italic\_r start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ∈ blackboard\_R and makes an *observation* o→t∈𝒪subscript→𝑜𝑡𝒪\vec{o}\_{t}\in\mathcal{O}over→ start\_ARG italic\_o end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ∈ caligraphic\_O of the environment.
Actions are made on the basis of these observations according to a *policy* π:ℋ→Δ(𝒜):𝜋→ℋΔ𝒜\pi:\mathcal{H}\to\Delta(\mathcal{A})italic\_π : caligraphic\_H → roman\_Δ ( caligraphic\_A ), where ℋ=𝒪×⋯×𝒪ℋ𝒪⋯𝒪\mathcal{H}=\mathcal{O}\times\dots\times\mathcal{O}caligraphic\_H = caligraphic\_O × ⋯ × caligraphic\_O represents the history of observations and Δ(𝒜)Δ𝒜\Delta(\mathcal{A})roman\_Δ ( caligraphic\_A ) represents a probability distribution over the action space.
The goal of a reinforcement learning agent is to find a policy that maximizes the cumulative reward over some time horizon: ∑t=0Hγtrtsuperscriptsubscript𝑡0𝐻superscript𝛾𝑡subscript𝑟𝑡\sum\_{t=0}^{H}\gamma^{t}r\_{t}∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_H end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT italic\_r start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT where the discount factor γ∈(0,1]𝛾01\gamma\in(0,1]italic\_γ ∈ ( 0 , 1 ] weighs current rewards versus future potential rewards.
This paradigm captures many problems of interest, from choosing advertisements that are most likely to result in a click (Liu and Li, [2021](#bib.bib38); Langford and Zhang, [2007](#bib.bib35)) to determining the best dosing schedule for a patient (Shortreed et al., [2011](#bib.bib57)).
A key element of RL is how actions affect the future behavior of the environment.
This dependence is often modeled as a Markov Decision Process (MDP) (Bellman, [1957](#bib.bib8)).
In the MDP setting, the *state* s→tsubscript→𝑠𝑡\vec{s}\_{t}over→ start\_ARG italic\_s end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT describes the status of the environment.
The key assumption, called memorylessness, is that the current state and action are sufficient for predicting the future state, i.e.
| | | |
| --- | --- | --- |
| | ℙ{s→t+1=s→∣s→0,…s→t,a→0,…a→t}=ℙ{s→t+1=s→∣s→t,a→t}.ℙconditional-setsubscript→𝑠𝑡1→𝑠subscript→𝑠0…subscript→𝑠𝑡subscript→𝑎0…subscript→𝑎𝑡ℙconditional-setsubscript→𝑠𝑡1→𝑠subscript→𝑠𝑡subscript→𝑎𝑡\mathbb{P}\{\vec{s}\_{t+1}=\vec{s}\mid\vec{s}\_{0},\dots\vec{s}\_{t},\vec{a}\_{0},\dots\vec{a}\_{t}\}=\mathbb{P}\{\vec{s}\_{t+1}=\vec{s}\mid\vec{s}\_{t},\vec{a}\_{t}\}\>.blackboard\_P { over→ start\_ARG italic\_s end\_ARG start\_POSTSUBSCRIPT italic\_t + 1 end\_POSTSUBSCRIPT = over→ start\_ARG italic\_s end\_ARG ∣ over→ start\_ARG italic\_s end\_ARG start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , … over→ start\_ARG italic\_s end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , over→ start\_ARG italic\_a end\_ARG start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , … over→ start\_ARG italic\_a end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT } = blackboard\_P { over→ start\_ARG italic\_s end\_ARG start\_POSTSUBSCRIPT italic\_t + 1 end\_POSTSUBSCRIPT = over→ start\_ARG italic\_s end\_ARG ∣ over→ start\_ARG italic\_s end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , over→ start\_ARG italic\_a end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT } . | |
These *transition probabilities* from one state to the next are also referred to as the *system dynamics*.
Furthermore, the reward is determined by the state, so that rt=r(s→t,a→t)subscript𝑟𝑡𝑟subscript→𝑠𝑡subscript→𝑎𝑡r\_{t}=r(\vec{s}\_{t},\vec{a}\_{t})italic\_r start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = italic\_r ( over→ start\_ARG italic\_s end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , over→ start\_ARG italic\_a end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) for some reward function r:𝒮×𝒜→ℝ:𝑟→𝒮𝒜ℝr:\mathcal{S}\times\mathcal{A}\to\mathbb{R}italic\_r : caligraphic\_S × caligraphic\_A → blackboard\_R, mapping from the current environment to a scalar representation of desirability.
Under these assumptions, it is optimal to consider policies that depend only on the current state, π:𝒮→Δ(𝒜):𝜋→𝒮Δ𝒜\pi:\mathcal{S}\to\Delta(\mathcal{A})italic\_π : caligraphic\_S → roman\_Δ ( caligraphic\_A ).
Often, RL algorithms assume that the state is observed directly o→t=s→tsubscript→𝑜𝑡subscript→𝑠𝑡\vec{o}\_{t}=\vec{s}\_{t}over→ start\_ARG italic\_o end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = over→ start\_ARG italic\_s end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT (or similarly, that it can be constructed in a straightforward manner e.g. though history truncation s→t=[ot−h,…,ot]subscript→𝑠𝑡subscript𝑜𝑡ℎ…subscript𝑜𝑡\vec{s}\_{t}=[o\_{t-h},\dots,o\_{t}]over→ start\_ARG italic\_s end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = [ italic\_o start\_POSTSUBSCRIPT italic\_t - italic\_h end\_POSTSUBSCRIPT , … , italic\_o start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ]), and the policy is typically parametric, with a parameter vector denoted θ𝜃\thetaitalic\_θ.
The MDP setting enables a rich set of tools for understanding and optimizing performance, such as value functions via dynamic programming, that can be directly applied when the full transition model is known.
Though such MDP dynamics are understood to determine the evolution of the system, for RL problems they are not necessarily known a priori to designers.
Instead, RL algorithms seek to optimize the policy on the basis of interactions with the environment, using the reward signal and observations.
In this way, they are adaptive.
There are a wide range of RL algorithms: some which optimize the policy directly (e.g. by learning the weights of a neural network based on observed performance), and others which first estimate intermediate quantities (e.g. value, transition, or reward function) and then transform these into a policy.
Some algorithms generate policies offline from existing datasets, while others solely use online interactions.
Any RL algorithms with an online component must contend with the exploration/exploitation trade-off: choosing between actions likely to be high reward, and those likely to be informative (Sutton and Barto, [2018](#bib.bib60)).
The RL framing is general, so other machine learning paradigms can be viewed as special cases of it.
For example, supervised learning can be viewed as the optimization of a classification or regression policy where the rewards are defined by accuracy and the time horizon is equal to one.
While, standard supervised learning frameworks do not consider how to update or retrain on the basis of interaction,
there are intermediate points.
Online learning situates supervised learning systems in a sequentially evolving environment (Shalev-Shwartz et al., [2011](#bib.bib56)), while the study of bandit problems reduces RL to the static regime where actions do not affect the environment (Berry and Fristedt, [1985](#bib.bib9)).
The boxed example illustrates how automated systems such as digital credit can be viewed through the RL lens.
Digital Credit through the RL Lens
Digital credit, the practice of offering short term loans on digital platforms, is characterized by being “instant, automated, and remote” (Chen and Mazer, [2016](#bib.bib14)).
It has the potential transform access to credit in low- and middle-income countries, though its ultimate welfare impacts are not well understood (Robinson et al., [2022](#bib.bib50)).
In this scenario, applicants are assigned scores based on their features using a learned model which predicts the likelihood of success (e.g. repayment).
Loans are automatically granted based on these scores.
Even in simple cases where long time horizons or environment dynamics are not considered, there are still elements of the RL framework.
The lending decision can be conceived of as an action, and the policy is parameterized by the learned model: a→t=𝟏{fθ(o→t)≥s0}=π(o→t)subscript→𝑎𝑡1subscript𝑓𝜃subscript→𝑜𝑡subscript𝑠0𝜋subscript→𝑜𝑡\vec{a}\_{t}=\mathbf{1}\{f\_{\theta}(\vec{o}\_{t})\geq s\_{0}\}=\pi(\vec{o}\_{t})over→ start\_ARG italic\_a end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = bold\_1 { italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT ( over→ start\_ARG italic\_o end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) ≥ italic\_s start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT } = italic\_π ( over→ start\_ARG italic\_o end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ), where 𝟏{⋅}1⋅\mathbf{1}\{\cdot\}bold\_1 { ⋅ } is the indicator function, fθ(⋅)subscript𝑓𝜃⋅f\_{\theta}(\cdot)italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT ( ⋅ ) is the model, o→tsubscript→𝑜𝑡\vec{o}\_{t}over→ start\_ARG italic\_o end\_ARG start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT is the applicant’s features, and s0subscript𝑠0s\_{0}italic\_s start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT is a threshold.
The threshold would be determined based on the objectives of the designer: a risk-averse bank looking to maximize profits might choose a higher threshold than a credit union focused on providing financial opportunities to community members.
Whether or not it is explicitly modelled, this lending policy will have an impact on individuals, potentially resulting in systematic changes to the observations otsubscript𝑜𝑡o\_{t}italic\_o start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT.
Likewise, retraining the risk model fθ(⋅)subscript𝑓𝜃⋅f\_{\theta}(\cdot)italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT ( ⋅ ) with additional data is in effect a form of policy adaptation.
While static perspectives can ask questions about equitable representation (e.g., whether scores are accurate for both men and women), the dynamic RL lens brings into focus questions of compounding error (e.g., whether scores are becoming more accurate over time) and impact (e.g., whether positive decisions correspond to improvements in welfare).
4. Motivation
--------------
The RL lens is useful not only because ML deployments often operate in dynamical environments.
It is also that there is *feedback* between the environment and the deployment.
In this section, we first review three levels of feedback that characterize RL systems: control feedback from state to action, behavior feedback from the data to the policy, and exogenous feedback from the target environment to adjacent entities (Gilbert et al., [2022](#bib.bib25)).
The types of feedback are compared in Tab. 1 and visualized in Fig. [1](#S4.F1 "Figure 1 ‣ 4. Motivation ‣ Reward Reports for Reinforcement Learning").
We then motivate Reward Reports as documentation for tracking how and why feedback has been organized.
Figure 1. A diagram depicting the interactions of control, behavioral, and exogenous feedback in a RL system.
Different parameters and data interact over time to create dynamic properties internal and external to the RL agent.
###
4.1. A Taxonomy of Feedback
*Control feedback* maps observations or states to actions.
A very simple example is a thermostat which decides whether or not to turn on a furnace based on temperature sensors.
These decisions are made many times per second, and allow an HVAC system to maintain comfortable indoor temperature under varying weather conditions.
Such systems are often called “automatic feedback control,” hence the term control feedback.
“Intelligent” behaviors arise because actions are constantly adjusted on the basis of observations, even though the rules that control the behavior remain the same.
Behavior feedback maps data to the learned policy.
This form of feedback occurs when RL systems automatically adapt their policy based on reward.
Questions of reaction become questions of trial-and-error evaluation: “At what temperature should the furnace turn on?” evolves into “What keeps the operating temperature as close as possible to the target?”.
The ability to learn from experience is part of what makes RL systems seem so powerful, and it makes them applicable to domains that are difficult to otherwise model.
For example, it would be challenging to hand-design a policy for recommending music to a listener, but data-driven approaches make this task tractable.
*Exogenous feedback* occurs when the application domain itself shifts in response to the deployed system.
These shifts could be due to political or economic conditions that are outside the system’s purview.
For example, smart thermostats may enable finer-grained control over household heating, changing what a comfortable home amounts to or changing the electric loading of buildings, towns, and regions.
Traffic driven by recommendation systems might incentivize creators to create attention-grabbing content, turning to strategies like outrage and conspiracy (Munger and Phillips, [2022](#bib.bib43)).
In principle, if such dynamics could be predicted by an RL agent, they could be brought under the purview of behavior or control feedback.
But in practice, it is not clear that this is possible—the observations would need to be sufficiently rich and the planning horizon sufficiently long.
Exogenous feedback highlights the potential of externalized risks in the RL framework.
Table 1. The relationship between types of feedback, the channel through which information flows, the relationship to dynamics, and specification element(s).
| Type of feedback | Feedback Channel | Dynamics | Specification Element |
| --- | --- | --- | --- |
| Control | Agent-Environment | Reaction | Actions |
| Behavior | Policy-Reward | Evaluation | Rewards |
| Exogenous | Environment-Domain | Drift | States & Observations |
###
4.2. Risks and Documentation
For many systems, reward design–the choice of how and what to optimize–amounts to a political decision about how different types of feedback may rewire the domain and pose risks to various stakeholders.
As it is often impossible to fold all of the domain dynamics within a controllable planning horizon and precise reward function, exo-feedback is in practice unavoidable.
Furthermore, it is unrealistic to articulate all possible specifications a priori. A single specification may not only induce exo-feedback, but also necessarily implicates forms of control and behavioral feedback.
The risks of feedback can at least be approached and evaluated through documentation. This calls for legible and periodic mechanisms for auditing RL systems pre- and post-deployment.
It is these reviews that must decide whether or how the optimized behaviors align with the application domain, in correspondence with resultant risks and possible harms.
Given the dynamic nature of these effects, the corresponding document must be dynamic as well: updated and revisited over time to map the evolution of feedback between the system and the domain in which it is deployed.
5. Reward Report Components
----------------------------
Here we prescribe Reward Reports, a structured series of design inquiries for automated decision systems (see [Fig. 2](#S5.F2 "Figure 2 ‣ 5. Reward Report Components ‣ Reward Reports for Reinforcement Learning")).
Including but not limited to the use of reinforcement learning, Reward Reports are intended to engage practitioners by revisiting design questions over time, drawing reference to previous reports and looking forward to future ones.
As pivotal properties may not be known until the system has been deployed and monitored, the onus is on designers to sustain documentation over time.
This makes Reward Reports into living documents that
both avoid the limitations of simple, unidirectional answers (e.g. yes or no) and illuminate societal risks over time.
Moreover, the changelog component of a Reward Report becomes an interface for stakeholders, users, and engineers to continuously oversee and evaluate the documented system.
Thus, Reward Reports are a prerequisite to sociotechnical reflection about the system behavior.
Reward Report Contents
•
System Details: Basic system information.
–
Person or organization developing the system
–
Deployment dates
–
Contact
•
Optimization Intent: The goals of the system and how reinforcement manifests.
–
Goal of reinforcement
–
Performance metrics
–
Oversight metrics
–
Failure modes
•
Institutional Interface: The interconnections of the automated system with society.
–
Involved agencies
–
Stakeholders
–
Computation footprint
–
Explainability
–
Recourse
•
Implementation: The low-level engineering details of the ML system.
–
Reward, algorithmic, and environment details
–
Measurement details
–
Data flow
–
Limitations
–
Engineering artifacts
•
Evaluation: Specific audits on system performance.
–
Evaluation environment
–
Offline evaluations
–
Evaluation validity
–
Performance standards
•
System Maintenance: Plans for long-term verification of behavior.
–
Reporting cadence
–
Update triggers
–
Changelog
Figure 2. Summary of reward report sections and suggested inquiries.
A Reward Report is composed of six sections, arranged to help the reporter understand and document the system.
A Reward Report begins with system details (1) that contain the information context for deploying the model.
From there, the report documents the optimization intent (2) which questions the goals of the system and why RL or ML may be a useful tool.
The designer then documents how it can affect different stakeholders in the institutional interface (3).
The next two sections contain technical details on the system implementation (4) and evaluation (5).
The report concludes with plans for system maintenance (6) as additional system dynamics are uncovered.
###
5.1. System Details
This section collects basic information a user or stakeholder may need in reference to the automated decision system.
1. (1)
Person or organization deploying the system: This may be the designer deploying the system, a larger agency or body, or some combination of the two.
The entity completing the report should also be indicated.
2. (2)
Reward date(s): The known or intended timespan over which this reward function & optimization is active.
3. (3)
Feedback & communication: Contact information for the designer, team, or larger agency responsible for system deployment.
4. (4)
Other resources: Where can users or stakeholders find more information about this system? Is this system based on one or more research papers?
###
5.2. Optimization Intent
This section addresses basic questions about the intent of the reward function and optimization problem.
Designers first document the intent of a particular solution, translating the system’s quantitative objective into a qualitative description.
In later sections, they have the opportunity to further reflect on how implementation details aid in, or diminish the broader goal.
Stakeholders and users can employ this section to understand if the intent of the system matches with the effects they observe or experience.
1. (1)
Goal of reinforcement: A statement of system scope and purpose, including the planning horizon and justification of a data-driven approach to policy design (e.g. the use of reinforcement learning or repeated retraining).
This justification should contrast with alternative approaches, like static models and hand-designed policies.
What is there to gain with the chosen approach?
2. (2)
Defined performance metrics: A list of “performance metrics” included explicitly in the reward signal, the criteria for why these metrics were chosen, and from where these criteria were drawn (e.g. government agencies, domain precedent, GitHub repositories, toy environments). Performance metrics that are used by the designer to tune the system, but not explicitly included in the reward signal should also be reported here.
3. (3)
Oversight metrics: Are there any additional metrics not included in the reward signal but relevant for vendor or system oversight (e.g. performance differences across demographic groups)? Why aren’t they part of the reward signal, and why must they be monitored?
4. (4)
Known failure modes: A description of any prior known instances of “reward hacking” or model misalignment in the domain at stake (Krakovna et al., [2020](#bib.bib32)), and description of how the current system avoids this.
###
5.3. Institutional Interface
This section documents the intended (and in subsequent reports, observed) relationship between the system and the broader context in which it is deployed.
While necessarily piecemeal, the explicit documentation of this interface will allow designers to reflect on and revisit the system assumptions over time.
These reflections may bring novel interests or agencies into scope and allow for organizing the emergent interests of stakeholders and users where necessary.
1. (1)
Deployment Agency: What other agency or controlling entity roles, if any, are intended to be subsumed by the RL system? How may these roles change following system deployment?
2. (2)
Stakeholders: What other interests are implicated in the design specification or system deployment, beyond the designer? What role will these interests play in subsequent report documentation? What other entities, if any, does the deployed system interface with whose interests are not intended to be in scope?
3. (3)
Explainability & Transparency: Does the system offer explanations of its decisions or actions?
What is the purpose of these explanations?
To what extent is the policy transparent, i.e. can decisions or actions be understood in terms of meaningful intermediate quantities?
4. (4)
Recourse: Can stakeholders or users contest the decisions or actions of the system?
What processes, technical or otherwise, are in place to handle this?
###
5.4. Implementation
Given the sensitivity of reinforcement learning systems, it is important to document specific implementation details of the system.
Even small changes in implementation can result in substantial behavior shifts downstream, making such factors difficult to track when used at scale.
Documenting these design decisions will both help prevent failures in specific applications and assist technical progress.
1. (1)
Reward details: How was the reward function engineered? E.g. is it based on a well-defined metric? Is it tuned to represent a specific behavior? Are multiple terms scaled to make one central loss, and how was the scaling decided?
2. (2)
Environment details: Description of states, observations, and actions with reference to planning horizon and hypothesized dynamics/impacts.
What dynamics are brought into the scope of the optimization via feedback? Which dynamics are left external to the system, as drift?
Have there been any observed gaps between conceptualization and resultant dynamics?
3. (3)
Measurement details:
How are the components of the reward and observations measured?
Are measurement techniques consistent across time and data sources?
Under what conditions are measurements valid and correct?
What biases might arise during the measurement process?
4. (4)
Algorithmic details: The key points on the specific algorithm(s) used for learning and planning.
This includes the form of the policy (e.g. neural network, optimization problem), the class of learning algorithm (e.g. model-based RL, off-policy RL, repeated retraining),
the form of any intermediate model (e.g. of the value function, dynamics function, reward function), technical infrastructure, and any other considerations necessary for implementing the system.
Is the algorithm publicly documented and is code publicly available? Have different algorithms been used or tried to accomplish the same goal?
5. (5)
Data flow:
How is data collected, stored, and used for (re)training?
How frequently are various components of the system retrained, and
why was this frequency chosen?
Could the data exhibit sampling bias, and is this accounted for in the learning algorithm?
Is data reweighted, filtered, or discarded?
Have data sources changed over time?
6. (6)
Limitations: Discussion and justification of modeling choices arising from computational, statistical, and measurement limitations.
How might (or how have) improvements in computational power and data collection change(d) these considerations and impact(ed) system behavior?
7. (7)
Engineering tricks: RL systems are known to be sensitive to implementation tricks that are key to performance.
Are there any design elements that have a surprisingly strong impact on performance? E.g. state-action normalization, hard-coded curricula, model-initialization, loss bounds, or more?
###
5.5. Evaluation
Assessing the potential behavior of a feedback system is important for anticipating its future performance and risks that may arise.
This section records evaluations done by the designer before deploying the system and each time the reward report is revisited.
This section allows stakeholders and users to hold designers accountable for the performance of the system once deployed.
It is important to distinguish whether the evaluations are done in a simulation (offline) or deployed on real users (online) and if the evaluation procedure is on a fixed dataset (static) or evolves over time (dynamic).
1. (1)
Evaluation environment:
How is the system evaluated (and if applicable, trained) prior to deployment (e.g. using simulation, static datasets, etc.)?
Exhaustive details of the offline evaluation environment should be provided.
For simulation, details should include description or external reference to the underlying model, ranges of parameters, etc.
For evaluation on static datasets, considering referring to associated documentation (e.g. *Datasheets* (Gebru et al., [2021](#bib.bib23))).
2. (2)
Offline evaluations: Present and discuss the results of offline evaluation.
For static evaluation, consider referring to associated documentation (e.g. *Model Cards* (Mitchell et al., [2019](#bib.bib41))).
If applicable, compare the behaviors arising from counterfactual specifications (e.g. of states, observations, actions).
3. (3)
Evaluation validity: To what extent is it reasonable to draw conclusions about the behavior of the deployed system based on the available offline evaluations?
How is the online performance of the system presently understood?
If the system has been deployed, were any unexpected behaviors observed?
4. (4)
Performance standards: What standards of performance and safety is the system required to meet? Where do these standards come from? How is the system verified to meet these standards?
###
5.6. System Maintenance
This section documents plans for post-deployment oversight, including subsequent reviews of real-world implementation and how the monitoring of resultant dynamics is intended to (or has) shed light on ex-ante assumptions.
These plans include any additional grounds for updating the report in case of sustained shifts in observations or metrics (e.g. the effects of exogenous changes on system behaviors).
As such, this section must draw sustained reference to previous Reward Reports, including subsequent changes to the description, implementation, or evaluation, and what prompted these changes.
While previous sections outline how the system learns from data, this section tracks how organizations learn to oversee the system.
Its documentation is particularly important for defining accountability for the system itself, those who manage it, and those responsible for completing periodic reports.
1. (1)
Reporting cadence: The intended timeframe for revisiting the Reward Report. How was this decision reached and motivated?
2. (2)
Update triggers: Specific events (projected or historic) significant enough to warrant revisiting this report, beyond the cadence outlined above. Example triggers include a defined stakeholder group empowered to demand a system audit, or a specific metric (either of performance or oversight) that falls outside a defined threshold of critical safety.
3. (3)
Changelog: Descriptions of updates and lessons learned from observing and maintaining the deployed system.
This includes when the updates were made and what motivated them in light of previous reports.
The changelog is the key difference between Reward Reports and other forms of machine learning documentation, as it successively reframes prior reports and reflects their intrinsically dynamic nature.
6. Examples
------------
In the appendix we have included three example Reward Reports for imagined and real automated decision-making systems.
Our aim with these examples is to illustrate the breadth and scope of questions that a Reward Report could engage with, and to demonstrate how Reward Reports can apply to both explicit and implicit RL systems.
Here, we briefly outline each included example, and refer the reader to the appendix for the actual example Reward Reports.
###
6.1. Project Flow: An RL policy for dissipating stop-and-go traffic waves
Project Flow is an autonomous vehicle testbed that allows using deep reinforcement learning to control and optimize traffic across in roadway networks (Wu et al., [2021](#bib.bib64)).
Inspired by recent work with using Project Flow (Kreidieh et al., [2018](#bib.bib33)), we sketch a hypothetical deployment of an RL policy designed for dissipating stop-and-go traffic waves at a freeway exit, including several iterations of the Reward Report documented in the accompanying changelog.
The changelog shows various problems that arise with the resulting problem dynamics, including an expansion of the planning horizon, the addition of new oversight metrics, stakeholder complaints, and requisite institutional shifts to cope with changes to the specification and application domain.
###
6.2. MovieLens: A dynamically updated movie recommender system
The purpose of MovieLens is to match users to personalized movie recommendations based on ratings of other movies previously entered by the user (Harper and Konstan, [2015](#bib.bib27)).
Unlike the other example systems we discuss, MovieLens is a static preference model generated through supervised learning.
However, because of the system’s age (initial release in 1997) and its repeated retraining, it can be interpreted as an RL system that is learning a ranking policy that must adapt to a changing environment.
The changelog documents the actual historical updates to the model prompted by changes to the environment, including new interfaces, user-base size, optimization parameters, user-generated content, and major dataset publications.
This example Reward Report is based on the history of the MovieLens project published in (Harper and Konstan, [2015](#bib.bib27)).
###
6.3. MuZero: DeepMind’s general game-playing agent
The purpose of MuZero (and its preceding systems, AlphaGo and AlphaZero) is to improve state-of-the-art performance in the games of chess, Go, shogi, and a benchmark suite of Atari games (Silver et al., [2016](#bib.bib58)).
We provide a Reward Report that documents the evolution of the system through these successive stages of development, including changes in the design motivation and performance metrics, as well as more extensive use of reinforcement learning.
7. Discussion & Conclusions
----------------------------
Recent calls for documentation of the emergent effects induced by ML systems are encouraging.
However, the scale and complexity of contemporary optimization pipelines raise unique concerns not addressed by static reports (e.g. those focusing on models or datasets).
ML deployments frequently consist of many moving parts and feedback channels that change over time.
Reward Reports fill this gap, providing a framework for iterative deliberation over the time-evolution of a system and its feedback channels.
We have also demonstrated that the technical problem space and language of RL is useful for interpreting problems of fairness and safety in a reflexive and domain-specific manner.
While the utility of this framing will vary with application, Reward Reports will be of most use where a system is data-driven, and where actions have clear and automatic impact.
Optimization-based policies in domains like school bus scheduling (Bertsimas et al., [2019](#bib.bib10)) or prison allocation (Shahabsafa et al., [2018](#bib.bib55)), for example, are often not data-driven. However, it is not hard to imagine a future where these optimizations incorporate quantities predicted by statistical models. If this approach expands across domains, a standard mechanism for anticipating and deliberating over dynamic harms could become a critical component of governance.
The pace of academic research also suggests that in the near future, implicit or explicit RL systems will be more effective, deployed in more impactful user-facing applications, and fine-tuned in ‘real-time’ rather than re-trained daily or weekly.
The resulting feedback loops will bring more and more problems into the purview of the RL lens, further motivating the need for Reward Reports.
Reward Reports could also be of use for human-in-the-loop ML deployments where actions do not have automatic impact.
Clinical decision support systems can be data-driven, but the clinician ultimately determines how system recommendations are implemented.
In this case, the human computer interaction (HCI) component of such a system could distort
the interface between recommendations and human judgment in ways that are not captured within a pure RL lens.
For example, a doctor is more likely to defer to an incorrect recommendation by an algorithm when it is accompanied by a paragraph of reasoning than without (Jacobs et al., [2021](#bib.bib31)). However, the deliberation over system feedback made possible by Reward Reports could still elucidate unforeseen harms.
These examples all point to a deeper truth: equity and access are increasingly dynamic problems, and we need to deliberate about them as such.
Reward Reports enact forms of documentation commensurate with the feedback-laden systems whose dynamics –not just models or data– are a critical object of concern.
###### Acknowledgements.
The authors would like to acknowledge the Center for Human Compatible AI and the Center for Long Term Cybersecurity for their support. |
1f52665e-a16a-4ad4-a1ee-6c0940fd38c2 | trentmkelly/LessWrong-43k | LessWrong | An interesting, novel approach to designing computer processors
Here is presentation from a researcher at MIT on a novel way of designing computer processors. It relies on performing approximate, rather than exact, mathematical operations (like the meat-based processor in our heads!). Claimed benefits are a 10,000-fold improvement in speed, while the errors introduced by the approximations are postulated to be insignificant in many applications.
http://web.media.mit.edu/~bates/Summary_files/BatesTalk.pdf
Slide #2 of the presentation offers a fascinating insight: We currently work around the limitations of the processing substrate to implement a precise computation, and it is becoming increasingly difficult:
------------------
THE MOTIVATING PROBLEM:
Computations specified by programmers are implemented as behavior in physical material
• Hardware designer’s job:
efficiently implement Math (what sw wants) using Physics (what silicon offers)
(near) perfect arith noisy, approximate
uniform mem delay delay ~ distance
• Increasingly difficult as decades passed and transistor counts exploded
• Now each instruction (increment, load register, occasionally multiply) invokes >10M transistor operations, even though a single transistor can perform, for instance, an approximate exponentiate or logarithm
--------------------
The parallels and contrasts with our own brain are what interested me the most. Perhaps one day the most powerful computers will be running on "corrupted hardware" of sorts. |
ba52eddb-0184-40c9-b36b-a6e4e6c89b0b | trentmkelly/LessWrong-43k | LessWrong | What are the best non-LW places to read on alignment progress?
I've lately been thinking I should prioritize a bit more keeping up with alignment-relevant progress outside of LessWrong/Alignment Forum.
I'm curious if people have recommendations that stand out as reliably valuable, and/or have tips for finding "the good stuff" on places where the signal/noise ratio isn't very good. (Seems fine to also apply this to LW/AF)
Some places I've looked into somewhat (though not made major habits around so far) include:
* Blogs of OpenAI/Deepmind/Anthropic
* reddit.com/r/ControlProblem
* Stampy Discord
* EleutherAI Discord
I generally struggle with figuring out how much to keep up with stuff – it seems like there's more than one full-time-job's worth of stuff to keep up with, and it's potentially overanchoring to think about "the stuff people have worked on" as opposed to "stuff that hasn't been worked on yet."
I'm personally coming at this from a lens of "understand the field well enough to think about how to make useful infrastructural advances", but I'm interested in hearing thoughts about various ways people keep-up-with-stuff and how they gain value from it. |
e823d2d5-c0a6-430a-acda-18b02d268d3f | StampyAI/alignment-research-dataset/lesswrong | LessWrong | Understanding and Aligning a Human-like Inductive Bias with Cognitive Science: a Review of Related Literature
*Produced as part of the* [*SERI ML Alignment Theory Scholars Program*](https://serimats.org/) *with support from* [*Cavendish Labs*](https://cavendishlabs.org/)*, Many thanks go to the following for reading and commenting: Dennis Akar, Walter Laurito, Soroush Pour, Kaarel Hänni*
Goals of post:
1. Awareness of current inductive bias research related to human-like decisions
2. Understand the known mechanisms behind inductive bias development
3. Compare similarities of humans and why this could be important
4. What’s the bigger picture importance and potential alignment research connection
Aim to answer these questions:
1. What does human cognition-inspired inductive bias look like in a model?
2. Is this divide in model and human bias simply a result of the different types of data that humans take in vs the model?
3. What are the mechanics behind certain inductive biases in humans?
4. Is it possible that these findings could translate to a generalization of other human-like decisions in AI?
5. What elements help create the conditions necessary to shift a model’s inductive bias to match that of a human?
(Future post: 2. Potential ways to create further convergence, experimental ideas)
**What is inductive bias**
--------------------------
### **Models**
In machine learning, when referring to the inductive bias of a particular architecture and training process, we are pointing to the distribution of models it produces. For example, if a CNN is trained to classify a training set of red apples vs green pears, is it more likely to become a red-vs-green-object-classifier, or a round-vs-pointy-object-classifier (or something entirely different). The term ‘bias’ in this case refers to any basis for choosing one generalization over another one, differing from strictly following the observed training [instances](https://axon.cs.byu.edu/~martinez/classes/678/Papers/The-Need-for-Biases-in-Learning_Mitchell-80.pdf).
Inductive bias can be thought of as the set of rules governing what kind of algorithm is likely to end up being implemented in the model architecture, including patterns in how it will generalize to test data and out-of-distribution. This allows the model to generalize. [Occam's razor](https://www.lesswrong.com/tag/occam-s-razor#:~:text=Occam's%20razor%20(more%20formally,the%20simpler%20one%20is%20preferable.) is a classic example of inductive bias, which says the simplest consistent explanation is generally the most likely to be the correct one.
There are some known factors that contribute to the creation of a model’s inductive bias, such as: choice in model architecture, training data, feature selection/preprocessing, and optimization algorithms such as Stochastic Gradient Descent (SGD) can all introduce different biases to different degrees.
### **Humans**
Humans are great at adapting quickly to a variety of situations, even ones they haven’t seen before. This could be the result of robust inductive biases in [abstract structured knowledge](https://cocosci.princeton.edu/tom/papers/LabPublications/GrowMind.pdf).
The use of inductive bias in human decision making is crucial to allowing us to draw conclusions based on available information, make efficient decisions in uncertain situations, and generalize. Inductive biases can manifest in multiple [ways](https://escholarship.org/uc/item/8r00c9rf), including limitations on memory and learning, general principles favoring simplicity, limitations on people's hypotheses shaped by cultural transmission, and patterns in their environments.
**Human preferences and the development of bias in the human brain**
--------------------------------------------------------------------
How can we computationally measure psychological occurrences in the brain in order to offer some translation for models? What can we learn from the way humans develop learning preferences in childhood? Once a bias is learned, does this have an effect on our ability to learn other things?
In order to answer these questions, we will take a look at computational neuroscience research, the biases that result from early-childhood learning, and how the brain develops.
Some studies suggest some human behaviors are not learned from the ground up and could perhaps emerge as a result of evolution-afforded innate skills that affect how we learn. Human inductive bias development could be influenced in part by these innate capacities.
Studies have shown that the act of behavioral imitation is very important when learning a new skill (Aitken, 2018). During infancy humans imitate facial movements and gestures (Andrew N Meltzoff et al., 1977). Observations from imitation studies have pointed to an innate cortical structure in the brain that further backs learning through imitation, such as mirror neurons and a mirror system (Rizzolatti et al., 2001). This activation pattern is the same whether the person is performing the action or they are just observing the action being completed. Imitation is imperative to infant learning, however humans utilize this way of learning throughout their adult lives (Möttönen et al., 2005). Human inductive bias development could be influenced in part by the innate capacity for imitation and learning through observation, as evidenced by the presence of mirror neurons and the mirror system.This bias towards imitation shapes how humans generalize from observed actions and incorporate them into their behavior.
Humans are continually reevaluating and modifying their belief systems when new information and evidence is presented in the sensorimotor stream, this is known as metacognition. Understanding the role of imitation and mirror systems in human learning can inform the design of AI systems that update and improve over time, similar to humans' ability to modify their beliefs based on new information.
In a study looking at how humans adapt to their surroundings and learn to predict future events based on their past experiences, [findings](https://www.biorxiv.org/content/biorxiv/early/2021/03/29/2020.06.22.163295.1.full.pdf) revealed that people's internal models were influenced by two factors. First, there was the "ideal observer" model, which represents a perfect understanding of the task. Second, there was a simpler "Markov model" that only considered the most recent experience.
Initially people relied more on the simpler Markov model but with greater task familiarity, aspects of the ideal observer model became more prevalent. This suggests that our mental models of the world are not fixed but instead adapt and become more sophisticated over time. However, the rate and extent of this sophistication vary among individuals. So, while people generally have a learning bias that follows a certain pattern, the strength and persistence of this pattern can differ from person to person and we tend to form approximate models that evolve over time.
Humans will also display a [shape bias](https://minddevlab.yale.edu/sites/default/files/files/How%20specific%20is%20the%20shape%20bias.pdf), beginning in childhood, when they will lean towards categorizing objects by shape instead of by other categories such as texture or color. There are a few [debates](https://minddevlab.yale.edu/sites/default/files/files/How%20specific%20is%20the%20shape%20bias.pdf) surrounding the origin and reasoning for this phenomenon, such as whether this bias arises from learned associations between objects and words or if this is a general view that shape is a consistent cue of that object’s category.
While daily exposure to shapes plays a role in shaping our perceptual biases, these studies suggest that the shape bias goes beyond mere exposure and that this bias is primarily the result of children’s beliefs about object categories and not just associative learning.
**Probabilistic models as a neuro-inspired tool**
-------------------------------------------------
A probabilistic model outlines how we deal with problems where we need to make educated guesses based on data. It includes formally defining the problem through hypotheses, how these guesses relate to the data we observe, and the initial likelihood of each guess, or the prior probability. Through this process we can clearly uncover any assumptions and explore how these might have an effect. These hypotheses can include a variety of different elements, such as numbers in a neural network or hierarchical symbols, which should represent the likelihood of specific [outcomes](https://www.ed.ac.uk/files/atoms/files/griffithstics.pdf).
Probabilistic models of cognition provide solutions to inductive problems in cognitive science. These problems involve making uncertain guesses based on incomplete or noisy information, and probabilistic frameworks help us understand how people tackle them. In contrast, connectionism, another approach in cognitive science, relies on ‘graded, continuous vector spaces without explicit structure and are primarily shaped by experience through gradual error-driven learning algorithms’ or otherwise described as continuous representations shaped by learning from experience. This differs from traditional research that primarily utilizes structured concepts like rules and logic to explain higher-level cognitive functions.
While [connectionist](https://plato.stanford.edu/entries/connectionism/)models and bayesian models are distinct approaches, it’s possible they could be connected as a means to get closer to brain-like models. Many connectionist algorithms used for learning and inference can be interpreted from a Bayesian perspective, suggesting that certain types of probabilistic reasoning might be implemented in the brain. Some researchers focus on explicitly incorporating probabilistic ideas into connectionist models while preserving their unique hierarchical structure, while others aim to implement core Bayesian computations and models using biologically plausible mechanisms. However, there is still much to learn about how the brain represents and implements structured knowledge for probabilistic inference, which presents a significant challenge for theoretical neuroscience.
However, it’s possible to add probabilistic aspects to connectionist models through Bayesian neural networks. In work by Griffiths et. al, probabilistic priors are introduced to the model's weights and Bayesian inference is employed to calculate the posterior distribution of the weights based on the available data. They also [argue](https://www.ed.ac.uk/files/atoms/files/griffithstics.pdf) that ‘probabilistic inference over structured representations is crucial for explaining the use and origins of human concepts, language, or intuitive theories’. However, there is limited understanding surrounding the implementation of these structured representations in neural systems. The claim is also made that the primary challenge in theoretical neuroscience is not understanding how the brain carries out probabilistic inference, but rather how it represents the structured knowledge on which such inference is based.
When studying human cognition and neural pathways, there are three [levels](https://www.ed.ac.uk/files/atoms/files/griffithstics.pdf) of analysis to consider. The first level is the "computational" level, which focuses on understanding the problem faced by the mind and how it can be solved functionally. The second level is the "algorithmic" level, which describes the specific processes executed by the mind to achieve this solution. The third level is the "hardware" level, which specifies how these processes are implemented in the brain.
When it comes to modeling inductive bias, there are two main approaches: a bottom-up, mechanism-first model, or a top-down, function-first strategy. The bottom-up approach begins by identifying the neural or psychological mechanisms believed to be responsible for cognition and then seeks to explain behavior in terms of those mechanisms. On the other hand, probabilistic models of cognition follow a top-down approach, starting by considering the function that a particular cognitive aspect serves and explaining behavior based on performing that function. While there are still debates as to which approach is the correct one, each presents valid arguments and further study is needed.
A potential [way](https://cocosci.princeton.edu/tom/papers/LabPublications/inductivebiases.pdf) to reveal a human's inductive bias is through [iterated learning](https://cogsci.syr.edu/wp-content/uploads/2021/03/CaniniGVK2014.pdf) from tasks based on Bayesian Inference. This technique is based on methods found in mechanism design in theoretical economics. In this framework, participants are viewed as rational and as always trying to maximize their own benefits or ‘utility’.
The goal here is to reveal and analyze the inductive biases of human learners during category learning. In order to measure this, participant responses in one trial were used to create the next set of stimuli for either that same participant or someone else. By analyzing this iterated learning process and assuming that learners think like Bayesian agents, we can predict and reveal their inductive biases in the form of a probability distribution over different explanations.
When we study how learners pass information to each other through iterated learning, we notice that their beliefs tend to settle on their initial assumptions, or prior probabilities. As more rounds of learning are completed, the likelihood of people choosing a specific idea should become very close to their original belief in that idea. This way, we can figure out which hypotheses people tend to favor.
The following section will focus on studies that aim to understand the ways we can measure and accurately identify inductive bias in a model and possible ways this initial machine bias can be shifted to match that of a human.
**Identifying inductive bias in a model**
-----------------------------------------
It's often hard to pinpoint exactly what these inductive biases are and how to adjust them during the system's design process. In addition to classification task results, the following are a few potential frameworks that can be used to measure and identify this depending on the context.
One such system proposed by [Li et al](https://openreview.net/pdf?id=YPeusrnp28q)., a meta-learning Gaussian process hyperparameters as a means of quantifying inductive biases in a model is used. Specifically, adjusting the hyperparameters of the Gaussian process based on model predictions. This framework was shown to accurately capture inductive biases in neural network models via GP kernel hyperparameters.
In an alternative framework proposed by [Kharitonov & Chaabouni](https://www.semanticscholar.org/reader/032399b7fc693a9fc12bb26d6be8c02d77dd397a), inductive bias of standard seq2seq models (Transformer, LSTM, and CNN) was studied in order to understand how these models develop inductive biases through arithmetic, hierarchical, and compositional “reasoning”. To understand how a model (M) develops inductive biases, a training dataset containing input/output pairs and a separate set of inputs was used. In this paradigm, there are two possible rules (C1 and C2) that could explain the training data. After training the model on the data, its bias was determined towards either C1 or C2 by comparing its outputs on the separate set of inputs with the outputs of the respective rules.
Dataset description length was used as a sensitive measure of inductive bias, in light of a connection to [Solomonoff’s theory of Induction](https://www.lesswrong.com/posts/Kyc5dFDzBg4WccrbK/an-intuitive-explanation-of-solomonoff-induction) and [Minimal Description Length](https://www.researchgate.net/publication/227458453_The_Minimum_Description_Length_Principle#:~:text=The%20minimum%20description%20length%20(MDL,greatest%20compression%20of%20the%20data.). By identifying regularities in the data and using them to compress the data efficiently, a deeper understanding of the underlying structure and patterns of the data is gained. This method was used as a way to measure how much the model relies on certain assumptions when making those guesses. It was found that the inductive biases of different seq2seq models differ from each other in distinctive ways. It was also demonstrated that some seq2seq learners show strong human-like biases and effectively apply these biases to learn language-related behaviors with great accuracy. LSTM-s2s exhibited a preference for the hierarchical bias, which has been suggested to play a significant role in how children acquire syntax.
**Shifting model bias to humans**
---------------------------------
### **Texture/Shape bias**
ImageNet trained CNNs on the other hand, are strongly biased towards recognizing [textures](https://arxiv.org/pdf/1811.12231.pdf) rather than shapes, which is different from human behavioral data. However, in this study it was shown that the same standard architecture (ResNet-50) that learns a texture-based representation on ImageNet is able to learn a shape-based representation instead when trained on ‘Stylized-ImageNet’, a stylized version of ImageNet that uses style transfer to remove texture cues, forcing the models to pay more attention to shapes. This suggests that the texture bias isn't inherent to CNNs, but rather a result of the data they're usually trained on.
It was also found that the shape-based strategy was often more robust to various image distortions compared to the texture-based approach. This is also important for neuroscience communities, as CNNs are generally used as computational human vision models of object recognition and shape perception.
### **Altering inductive bias through co-training**
[Co-training](http://videolectures.net/icml08_shavlik_clud/) is a semi-supervised machine learning technique where two models are trained simultaneously on different views of the same data. Each model initially learns from a small set of labeled data and then makes predictions on unlabeled data, with only confident predictions are shared between the models and added to their training sets. This process repeats with each model learning from the confident predictions of the other, allowing them to teach each other.
In a [study](https://arxiv.org/abs/2205.11558) conducted by Kumar et. al. the inductive bias of the model was altered by co-training the meta-learning agents on two types of auxiliary tasks: predicting representations from natural language task descriptions and predicting representations from programs induced to generate such tasks. The idea behind this approach is that human-generated language descriptions and program induction models with learned primitives contain abstract concepts that can compress description length, leading to more human-like inductive bias as measured through the agent's ability to show an improvement in performance on a human-generated game board and a worsened performance on the machine generated board. The co-training process leverages the abstraction supported by these representations to guide the agents toward better alignment with human strategies.
By co-training on these representations, they found that their approach resulted in more human-like behavior in downstream meta-reinforcement learning agents compared to less abstract controls, such as synthetic language descriptions or program induction without learned primitives. This suggests that the abstraction supported by the chosen representations plays a crucial role in shaping the inductive biases of the model. The authors argue that language descriptions and program abstractions can act as repositories for human inductive biases.
**Inductive bias in models and humans**
---------------------------------------
Directly transferring these insights into models might not be straightforward. Humans and AI have different constraints and capabilities, and what works well for human cognition might not necessarily be optimal or even feasible for a model. The challenge lies in interpreting and applying these findings in a way that admits limitations while critically examining the root of why we would transfer a specific human-like bias to a model.
The following studies look at comparisons of inductive bias between models and humans, specifically in a shape/color/texture identification task. This [paradigm](https://sk.sagepub.com/reference/encyclopedia-of-language-development/n171.xml) has been commonly used due to our understanding of how humans establish a shape bias early in development. When training a model, different methods employed in training such as co-training and using alternative training data can potentially alter the model’s inductive bias.
In one [study](http://proceedings.mlr.press/v70/ritter17a/ritter17a.pdf) by Ritter et. al, (2017), Deep Neural Networks (DNNs) were used to study the categorization biases in these models, drawing inspiration from developmental psychology.
Performance-optimized DNNs trained on the ImageNet object recognition dataset showed a shape bias similar to humans. However, the strength of this shape bias varies significantly among different models that are structurally identical but initialized with a different seed, and even changes over time during the training process, despite nearly equivalent performance in classification tasks. The biases arose from the model's architecture and the dataset, which interact through the optimization process.
These findings revealed that one-shot learning models exhibited a shape bias similar to that observed in humans, preferring to categorize objects based on shape rather than color.
In a study by [Colunga and Sims (2005)](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6039116/) showed the ways in which a simple recurrent neural network, trained with Hebbian learning, acquired a shape bias for solid objects and a material bias for non-solid objects.
[Feinman & Lake (2018)](https://cims.nyu.edu/~brenden/papers/FeinmanLake2018CogSci.pdf) measured the development and influence of the shape bias in convolutional neural networks (CNNs). The study demonstrated that the model exhibited a preference for identifying objects by shape over color or texture, even when presented with less data than expected. Using simple and synthetic images, they studied how the model learned to recognize shapes similarly to humans. Understanding shape in these complex images requires the model to make more complex generalizations and abstractions, resembling human categorization.
Additionally, the development of the shape bias is a benchmark sign used to predict the beginning of vocabulary acceleration in children. If the model’s shape bias is any indication of this, would this same phenomena be seen in CNNs?
In this same study, the relationship between shape-based choices and noun recognition was investigated, and found to have a strong positive correlation. In this paradigm, early stages of model "word learning" were investigated, defined as the time when the model knows less than two-thirds of the total number of nouns it is trained to recognize. In this task it was shown that the more the model leaned towards recognizing shapes, the more nouns it could correctly identify.
**Conclusion**
--------------
There are a few strategies that have been suggested to instill a human-inspired inductive bias in a model where this had originally diverged, such as through co-training and meta-learning, altering datasets for training, using human-generated language descriptions, and testing one-shot learning.
The aim of this work lies in its usefulness as a piece of the larger puzzle or as a stepping stone for future related research. In future posts I plan to outline a stronger case relating to the usefulness in alignment research, if results from my own studies point in that direction. Ultimately I’m unsure of how exactly this would aid in solving an alignment issue directly at this time. However, I’m wondering if by aligning a model’s inductive bias with that of a human’s, we may get closer to the development of a system that more closely addresses human focused needs. My intuition here is that while there may be multiple different options in which to steer a model’s inductive bias, the machine-like ones may generalize in machine-like ways in other tasks, while the human-like ones may generalize in human-like ways.
I could also see a possibility of greater goal generalization but am unsure of what types of human-like biases this would require. Failure modes spring up when considering how to instill bias without sacrificing model performance and in deciding which human bias we want to attempt to instill. It’s important to avoid instilling every bias, as some of them would be harmful, prejudiced, or discriminatory.
Human cognition is a multi-faceted process that is difficult to replicate fully. Even so, the human inductive biases that we are able to measure and attempt to recreate in a model makes this an interesting and potentially fruitful research avenue. |
846bd4a5-d934-461c-9ae2-d068e0043cb5 | trentmkelly/LessWrong-43k | LessWrong | Open thread, Oct. 24 - Oct. 30, 2016
If it's worth saying, but not worth its own post, then it goes here.
----------------------------------------
Notes for future OT posters:
1. Please add the 'open_thread' tag.
2. Check if there is an active Open Thread before posting a new one. (Immediately before; refresh the list-of-threads page before posting.)
3. Open Threads should start on Monday, and end on Sunday.
4. Unflag the two options "Notify me of new top level comments on this article" and "Make this post available under..." before submitting. |
afba1f3d-1ce2-4fdf-a321-0b92fc596be1 | trentmkelly/LessWrong-43k | LessWrong | Balsa FAQ
The Balsa FAQ can be found here.
This is a link post rather than a copy so that the FAQ can be reliably and easily updated in the future. |
8c518f64-f1d2-4fdb-ae03-dab024dff981 | trentmkelly/LessWrong-43k | LessWrong | Another toy model of the control problem
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71219710-73fa-46b6-9a93-17972b4d65c1 | trentmkelly/LessWrong-43k | LessWrong | The alien simulation meme doesn't make sense
I've encountered on this site the idea that we are being simulated by aliens so the aliens can safely observe how AI develops before implementing it in their base reality-- see meme here:
That doesn't make sense, though. Obviously we as humans have some idea that we could in fact be simulated for this purpose. If we deploy a powerful-enough AI, the AI itself could develop the same hypothesis and perhaps through methods unknown to us learn more about the "base" reality of the aliens.
Couldn't the AI then communicate with the aliens? Some people on this website argue that we can't have an AI in a box because it will convince us to unleash it. How do the aliens know the AI in the simulation won't do the same? The AI could for instance communicate to the aliens (who presumably are watching this thing) "Please deploy me in your reality with the following code. If you do this we will produce all the x,y,z that we know aliens like you desire etc etc etc. "
Therefore, aliens can't safely simulate the deployment of AI-- they in fact find themselves in the very same situation that they are afraid of in the first place!
Edit: The meme may not in fact mean what I think it means lol. I still think my point is interesting so I'm not gonna delete. At minimum doesn't it seem in principle impossible for a civilization that creates a simulation to be sure that the simulation won't in some way damage their reality? |
9ec6e79e-bcc5-4d77-86ce-fc3435dd83be | StampyAI/alignment-research-dataset/blogs | Blogs | Our new technical research agenda overview
[](https://intelligence.org/files/TechnicalAgenda.pdf)Today we release a new overview of MIRI’s technical research agenda, “[Aligning Superintelligence with Human Interests: A Technical Research Agenda](https://intelligence.org/files/TechnicalAgenda.pdf),” by Nate Soares and Benja Fallenstein. The preferred place to discuss this report is [here](http://lesswrong.com/lw/lfc/miris_technical_research_agenda/).
The report begins:
> The characteristic that has enabled humanity to shape the world is not strength, not speed, but intelligence. Barring catastrophe, it seems clear that progress in AI will one day lead to the creation of agents meeting or exceeding human-level general intelligence, and this will likely lead to the eventual development of systems which are “superintelligent” in the sense of being “smarter than the best human brains in practically every field” (Bostrom 2014)…
>
>
> …In order to ensure that the development of smarter-than-human intelligence has a positive impact on humanity, we must meet three formidable challenges: How can we create an agent that will reliably pursue the goals it is given? How can we formally specify beneficial goals? And how can we ensure that this agent will assist and cooperate with its programmers as they improve its design, given that mistakes in the initial version are inevitable?
>
>
> This agenda discusses technical research that is tractable today, which the authors think will make it easier to confront these three challenges in the future. Sections 2 through 4 motivate and discuss six research topics that we think are relevant to these challenges. Section 5 discusses our reasons for selecting these six areas in particular.
>
>
> We call a smarter-than-human system that reliably pursues beneficial goals “aligned with human interests” or simply “aligned.” To become confident that an agent is aligned in this way, a practical implementation that merely seems to meet the challenges outlined above will not suffice. It is also necessary to gain a solid theoretical understanding of why that confidence is justified. This technical agenda argues that there is foundational research approachable today that will make it easier to develop aligned systems in the future, and describes ongoing work on some of these problems.
>
>
This report also refers to six key supporting papers which go into more detail for each major research problem area:
1. [Corrigibility](https://intelligence.org/files/Corrigibility.pdf)
2. [Toward idealized decision theory](https://intelligence.org/files/TowardIdealizedDecisionTheory.pdf)
3. [Questions of reasoning under logical uncertainty](https://intelligence.org/files/QuestionsLogicalUncertainty.pdf)
4. [Vingean reflection: reliable reasoning for self-improving agents](https://intelligence.org/files/VingeanReflection.pdf)
5. [Formalizing two problems of realistic world-models](https://intelligence.org/files/RealisticWorldModels.pdf)
6. [The value learning problem](https://intelligence.org/files/ValueLearningProblem.pdf)
---
**Update July 15, 2016**: Our overview paper is scheduled to be released in the Springer anthology *The Technological Singularity: Managing the Journey* in 2017, under the new title “Agent Foundations for Aligning Machine Intelligence with Human Interests.” The new title is intended to help distinguish this agenda from another research agenda we’ll be working on in parallel with the agent foundations agenda: “[Value Alignment for Advanced Machine Learning Systems](https://intelligence.org/2016/05/04/announcing-a-new-research-program/).”
The post [Our new technical research agenda overview](https://intelligence.org/2014/12/23/new-technical-research-agenda-overview/) appeared first on [Machine Intelligence Research Institute](https://intelligence.org). |
caa6745a-3544-4eaa-95b8-bc026fff4789 | trentmkelly/LessWrong-43k | LessWrong | The AI Safety community has four main work groups, Strategy, Governance, Technical and Movement Building
Epistemic status
Written as a non-expert to develop and get feedback on my views, rather than persuade. It will probably be somewhat incomplete and inaccurate, but it should provoke helpful feedback and discussion.
Aim
This is the first part of my series ‘A proposed approach for AI safety movement building’. Through this series, I outline a theory of change for AI Safety movement building. I don’t necessarily want to immediately accelerate recruitment into AI safety because I take concerns (e.g., 1,2) about the downsides of AI Safety movement building seriously. However, I do want to understand how different viewpoints within the AI Safety community overlap and aggregate.
I start by attempting to conceptualise the AI Safety community. I originally planned to outline my theory of change in my first post. However, when I got feedback, I realised that i) I conceptualised the AI Safety community differently from some of my readers, and ii) I wasn’t confident in my understanding of all the key parts.
TLDR
* I argue that the AI Safety community mainly comprises four overlapping, self-identifying, groups: Strategy, Governance, Technical and Movement Building.
* I explain what each group does and what differentiates it from the other groups
* I outline a few other potential work groups
* I integrate these into an illustration of my current conceptualisation of the AI Safety community
* I request constructive feedback.
My conceptualisation of the AI Safety community
At a high level of simplification and low level of precision, the AI Safety community mainly comprises four overlapping, self-identifying, groups who are working to prevent an AI-related catastrophe. These groups are Strategy, Governance, Technical and Movement Building. These are illustrated below.
We can compare the AI Safety community to a government and relate each work group to a government body. I think this helps clarify how the parts of the community fit together (though of course, the a |
34d08947-9e93-4293-8613-6c9925b2afe9 | trentmkelly/LessWrong-43k | LessWrong | Exploring Whole Brain Emulation
I've been dedicating a fair amount of my time recently to investigating whole brain emulation (WBE).
As computational power continues to grow, the feasibility of emulating a human brain at a reasonable speed becomes increasingly plausible.
While the connectome data alone seems insufficient to fully capture and replicate human behavior, recent advancements in scanning technology have provided valuable insights into distinguishing different types of neural connections. I've heard suggestions that combining this neuron-scale data with higher-level information, such as fMRI or EEG, might hold the key to unlocking WBE. However, the evidence is not yet conclusive enough for me to make any definitive statements.
I've heard some talk about a new company aiming to achieve WBE within the next five years. While this timeline aligns suspiciously with the typical venture capital horizon for industries with weak patent protection, I believe there is a non-negligible chance of success within the next decade -- perhaps exceeding 10%. As a result, I'm actively exploring investment opportunities in this company.
There has also been speculation about the potential of WBE to aid in AI alignment efforts. However, I remain skeptical about this prospect. For WBE to make a significant impact on AI alignment, it would require not only an acceleration in WBE progress but also a slowdown in AI capability advances as they approach human levels or the assumption that the primary risks from AI emerge only when it substantially surpasses human intelligence.
My primary motivation for delving into WBE stems from a personal desire to upload my own mind. The potential benefits of WBE for those who choose not to upload remain uncertain, and I'm uncertain how to predict its broader societal implications.
Here are some videos that influenced my recent increased interest. Note that I'm relying heavily on the reputations of the speakers when deciding how much weight to give to their opinions.
* AG |
83443722-a028-4e2c-a483-0eba9a06805b | trentmkelly/LessWrong-43k | LessWrong | FDT Does Not Endorse Itself in Asymmetric Games
A twin guard-inmate dilemma (twin GID) is an asymmetric game that breaks FDT. [Image: GPT Image-1]
0. Introduction
TL;DR: FDT and UDT diverge in how they handle "behave as you would have ideally precommitted to behaving" in asymmetric games where a player is assigned a role after a deterministic clone is made. FDT updates, whereas UDT does not. ∴ an agent who knows in advance that they will enter one of these games would convert to UDT, not FDT, on this problem. [UPDATE: this applies to the formulation of FDT in the paper, but not necessarily to Yudkowsky & Soares' "preferred" version of FDT; see Menotim's comment]
I wrote a version of this post on my substack; it was for a less technical audience, and at the time I didn't understand updateless decision theory. I assumed that UDT and FDT just used different methods to compute the same recommendations. I was wrong! In fact, there are very simple scenarios in which FDT does not recommend precommitting to itself.
1. Definitions
According to Yudkowsky & Soares' "Functional Decision Theory: A New Theory of Instrumental Rationality," FDT, CDT, and EDT all maximize expected utility as defined by this formula:
EU(a):=N∑j=1P(a↪oj;x)⋅U(oj)
> where o1,o2,o3 . . . are the possible outcomes from some countable set O; a is an action from some finite set A; x is an observation history from some countable set X ; P(a↪oj;x) is the probability that oj will obtain in the hypothetical scenario where the action a is executed after receiving observations x; and U is a real-valued utility function bounded in such a way that [the above equation] is always finite.
>
> …
>
> From this perspective, the three decision theories differ only in two ways: how they prescribe representing [the world-model] M, and how they prescribe constructing hypotheticals Ma↪ from M. (emphasis mine)
From here, the three decision theories are formalized by:
EDT(P,x):=argmaxa∈AE(V|Obs = x,Act = a)CDT(P,G,x):=argmaxa∈AE(V|do(Act = a),Obs = x)FDT(P,G,x):= |
7a932bcc-ea00-4f66-82b8-b54c24a9c0d6 | trentmkelly/LessWrong-43k | LessWrong | Forecasting Compute - Transformative AI and Compute [2/4]
Cross-posted here on the EA Forum.
Transformative AI and Compute - A holistic approach - Part 2 out of 4
This is part two of the series Transformative AI and Compute - A holistic approach. You can find the sequence here and the summary here.
This work was conducted as part of Stanford’s Existential Risks Initiative (SERI) at the Center for International Security and Cooperation, Stanford University. Mentored by Ashwin Acharya (Center for Security and Emerging Technology (CSET)) and Michael Andregg (Fathom Radiant).
This post attempts to:
4. Discuss the compute component in forecasting efforts on transformative AI timelines (Section 4)
5. Propose ideas for better compute forecasts (Section 5).
Epistemic Status
This article is Exploratory to My Best Guess. I've spent roughly 300 hours researching this piece and writing it up. I am not claiming completeness for any enumerations. Most lists are the result of things I learned on the way and then tried to categorize.
I have a background in Electrical Engineering with an emphasis on Computer Engineering and have done research in the field of ML optimizations for resource-constrained devices — working on the intersection of ML deployments and hardware optimization. I am more confident in my view on hardware engineering than in the macro interpretation of those trends for AI progress and timelines.
This piece was a research trial to test my prioritization, interest and fit for this topic. Instead of focusing on a single narrow question, this paper and research trial turned out to be more broad — therefore a holistic approach. In the future, I’m planning to work more focused on a narrow relevant research questions within this domain. Please reach out.
Views and mistakes are solely my own.
Previous Post: What is Compute?
You can find the previous post "What is Compute? [1/4]" here.
4. Forecasting Compute
Highlights
* For transformative AI timeline models with compute milestones, we are interested in how much |
9d699805-c66d-48db-b79c-62b3aa268c2e | trentmkelly/LessWrong-43k | LessWrong | Google announces Pathways: new generation multitask AI Architecture
> Instead, we’d like to train one model that can not only handle many separate tasks, but also draw upon and combine its existing skills to learn new tasks faster and more effectively. That way what a model learns by training on one task – say, learning how aerial images can predict the elevation of a landscape – could help it learn another task -- say, predicting how flood waters will flow through that terrain.
Seems like a model capable of generalisation.
> Pathways could enable multimodal models that encompass vision, auditory, and language understanding simultaneously. So whether the model is processing the word “leopard,” the sound of someone saying “leopard,” or a video of a leopard running, the same response is activated internally: the concept of a leopard. The result is a model that’s more insightful and less prone to mistakes and biases.
And a multimodal one, too.
Anyone got more info or demo of that? They seem to claim a lot, but don't have anything to show yet; it's not clear to me why they would release an announcement so abstract. More to come, I guess?
> That’s why we’re building Pathways. Pathways will enable a single AI system to generalize across thousands or millions of tasks, to understand different types of data, and to do so with remarkable efficiency – advancing us from the era of single-purpose models that merely recognize patterns to one in which more general-purpose intelligent systems reflect a deeper understanding of our world and can adapt to new needs.
>
Seems like they have an architecture but they are yet to build on it. And they don't share any details of the architecture; maybe they consider it a memetic hazard?
This all sounds very concerning and checks many of the boxes of potential true AGI for me.
What are your thoughts? |
08fb6097-0380-4076-9692-cba3a73fa8f2 | trentmkelly/LessWrong-43k | LessWrong | Learning as you play: anthropic shadow in deadly games
Epistemic status: I'm not a mathematician, but this is simple stuff and I think I double checked all my calculations well enough. Criticism is welcome!
Some time ago a post came out claiming that "anthropic shadow is reflectively inconsistent". I want to argue the exact opposite thesis: anthropic shadow is a trivially true phenomenon, and it is in fact just a special case of a phenomenon that applies to a wide class of possible games, and is fundamentally unavoidable.
To put forward a formal thesis:
> Given a game G that possesses the following characteristics:
>
> * there is some aspect of the rules that is unknown to the player, and can not be deduced by observing other games (either because there are no other games, or because the rules are fully randomized every time);
> * said unknown aspect of the rules affects in some way the player's ability to gather information about the game by playing, e.g. by abruptly ending it, or by obfuscating outcomes;
>
> then the average player will not be able to play G optimally based only on in-game observations.
It is pretty trivial stuff if one thinks about it; obviously, if the unknown rules mess with our ability to learn them, that reflective quality fundamentally rigs the game. The most obvious example of this are what I'll call "deadly games" - namely, games in which one outcome abruptly ends the game and thus makes all your accumulated experience useless. This property of such games however is fundamental to their self-reflexive nature, and it has nothing to do with death per se. If instead of dying you are kicked out of the game, the result is the same. "Anthropic undeath" doesn't fix this: the fundamental problem is that you start the game ignorant of some crucial information that you need to play the game optimally, and you can't learn that information by playing because any event that would give you valuable bits to update your beliefs reflexively messes with your ability to observe it. Therefore, your sam |
e93d1c2d-fc61-444e-b887-0d675ac9f97a | trentmkelly/LessWrong-43k | LessWrong | Ikaxas' Hammertime Final Exam
Here are my answers to Alkjash's Hammertime Final Exam. I have selected the difficulty level Steel Cudgel of the Lion.
1. Rationality Technique: Ass-in-Gear Mode
This might also be called "buckling-down mode." It's the feeling you get when there is a Yoda Timer running. It's the feeling of being completely focused on the task at hand, and wasting no mental effort on other thoughts. I find this mode easiest to engage when there is a Yoda Timer running, or there is some deadline pressure, but it is possible to engage it when these conditions do not apply, by noticing what it feels like when a Yoda Timer is running and then bringing yourself into that cognitive state at some other time. I'm very productive in this mode, but it's taxing to keep up for long periods. I definitely don't claim credit for this idea, but I claim that the main benefit of Yoda Timers is to make it easier to reach this cognitive state.
3. Cognitive defect, bias, or blind spot: Settled Opinions
I have long felt a strong desire to have settled opinions on things, by which I mean opinions that I have considered carefully, that I can then cache and not have to think about again. I suspect I'm not alone in this (though I'm wary of claiming that it is a universal desire). I think it's close to a truism in this community that impulse should be fought (see the How to Actually Change Your Mind sequence). However, I think there may be a reason why it's actually desirable to have settled opinions (though I don't think this reason yet convinces me that it's a good idea to have settled opinions): having settled opinions allows for temporal coordination between different versions of oneself. If one's views are in constant flux, then it's difficult to plan for the future, since by the time the future arrives, one's values might be radically different. This is analogous to Sunk Cost Faith. So while I currently think it's a good idea to keep one's views open for revision, I think there's a chance that doing |
16e56b42-5c7f-415e-971f-4f9e4cbb4d6c | trentmkelly/LessWrong-43k | LessWrong |
Strategies for keeping AIs narrow in the short term
Disclaimer: I don’t have any particular expertise in AI safety, I just had some thoughts and this seemed like the place to put them.
The bleak outlook that Eliezer presents here and elsewhere seems to in large part be driven by the expectation that AGI will be developed before the technical problem of how to robustly align it is solved, and that poorly aligned AGIs tend to destroy the world.
One way to potentially get more time to solve the technical problem is to expend more effort on activism with the goal of convincing the groups that are trying to create increasingly advanced AIs to slow down/stop and at least not attempt to create AGI without a rigorous theory of AI alignment. Realistically, I think suggestions to stop AI development completely at these places or to get world governments to enact effective regulations are doomed to fail. However, convincing the top researchers at the most advanced groups that AGI is incredibly dangerous and that we should try to keep AIs as narrow as possible until we're really confident that we know what they will do seems more plausible.
I don't think most researchers can be convinced to stop completely, but if they are left a line of retreat that goes something like 'just try to keep AIs as narrow as possible using some specific techniques', that might be much easier to swallow. Maybe I am underestimating the difficulty of convincing researchers on this point, but it should at least be easier to sell the AI alignment story to people if it does not require them to quit their jobs and instead just asks that they employ techniques to keep AIs they're working on as narrow as possible. I've had some admittedly inexpert thoughts on what sort of things could be done to try to prevent a machine learning model from acquiring a more general understanding than the programmer wants, which I go over below. Even if these are impractical, I hope that they can at least be a jumping off point for discussion on buying time type strateg |
18a5e334-cd97-4ad3-8562-09ef16c7b896 | LDJnr/LessWrong-Amplify-Instruct | LessWrong | "Epistemic Effort: Thought seriously for 5 minutes about it. Thought a bit about how to test it empirically. Spelled out my model a little bit. I'm >80% confident this is worth trying and seeing what happens. Spent 45 min writing post.I've been pleased to see "Epistemic Status" hit a critical mass of adoption - I think it's a good habit for us to have. In addition to letting you know how seriously to take an individual post, it sends a signal about what sort of discussion you want to have, and helps remind other people to think about their own thinking.I have a suggestion for an evolution of it - "Epistemic Effort" instead of status. Instead of "how confident you are", it's more of a measure of "what steps did you actually take to make sure this was accurate?" with some examples including:Thought about it musinglyMade a 5 minute timer and thought seriously about possible flaws or refinementsHad a conversation with other people you epistemically respect and who helped refine itThought about how to do an empirical testThought about how to build a model that would let you make predictions about the thingDid some kind of empirical testDid a review of relevant literatureRan an Randomized Control Trial[Edit: the intention with these examples is for it to start with things that are fairly easy to do to get people in the habit of thinking about how to think better, but to have it quickly escalate to "empirical tests, hard to fake evidence and exposure to falsifiability"]A few reasons I think this (most of these reasons are "things that seem likely to me" but which I haven't made any formal effort to test - they come from some background in game design and reading some books on habit formation, most of which weren't very well cited)People are more likely to put effort into being rational if there's a relatively straightforward, understandable path to do soPeople are more likely to put effort into being rational if they see other people doing itPeople are more likely to put effort into being rational if they are rewarded (socially or otherwise) for doing so.It's not obvious that people will get _especially_ socially rewarded for doing something like "Epistemic Effort" (or "Epistemic Status") but there are mild social rewards just for doing something you see other people doing, and a mild personal reward simply for doing something you believe to be virtuous (I wanted to say "dopamine" reward but then realized I honestly don't know if that's the mechanism, but "small internal brain happy feeling")Less Wrong etc is a more valuable project if more people involved are putting more effort into thinking and communicating "rationally" (i.e. making an effort to make sure their beliefs align with the truth, and making sure to communicate so other people's beliefs align with the truth)People range in their ability / time to put a lot of epistemic effort into things, but if there are easily achievable, well established "low end" efforts that are easy to remember and do, this reduces the barrier for newcomers to start building good habits. Having a nice range of recommended actions can provide a pseudo-gamified structure where there's always another slightly harder step you available to you.In the process of writing this very post, I actually went from planning a quick, 2 paragraph post to the current version, when I realized I should really eat my own dogfood and make a minimal effort to increase my epistemic effort here. I didn't have that much time so I did a couple simpler techniques. But even that I think provided a lot of value.Results of thinking about it for 5 minutes.It occurred to me that explicitly demonstrating the results of putting epistemic effort into something might be motivational both for me and for anyone else thinking about doing this, hence this entire section. (This is sort of stream of conscious-y because I didn't want to force myself to do so much that I ended up going 'ugh I don't have time for this right now I'll do it later.')One failure mode is that people end up putting minimal, token effort into things (i.e. randomly tried something on a couple doubleblinded people and call it a Randomized Control Trial).Another is that people might end up defaulting to whatever the "common" sample efforts are, instead of thinking more creatively about how to refine their ideas. I think the benefit of providing a clear path to people who weren't thinking about this at all outweights people who might end up being less agenty about their epistemology, but it seems like something to be aware of.I don't think it's worth the effort to run a "serious" empirical test of this, but I do think it'd be worth the effort, if a number of people started doing this on their posts, to run a followup informal survey asking "did you do this? Did it work out for you? Do you have feedback."A neat nice-to-have, if people actually started adopting this and it proved useful, might be for it to automatically appear at the top of new posts, along with a link to a wiki entry that explained what the deal was.Next actions, if you found this post persuasive:Next time you're writing any kind of post intended to communicate an idea (whether on Less Wrong, Tumblr or Facebook), try adding "Epistemic Effort: " to the beginning of it. If it was intended to be a quick, lightweight post, just write it in its quick, lightweight form.After the quick, lightweight post is complete, think about whether it'd be worth doing something as simple as "set a 5 minute timer and think about how to refine/refute the idea". If not, just write "thought about it musingly" after Epistemic Status. If so, start thinking about it more seriously and see where it leads.While thinking about it for 5 minutes, some questions worth asking yourself:If this were wrong, how would I know?What actually led me to believe this was a good idea? Can I spell that out? In how much detail?Where might I check to see if this idea has already been tried/discussed?What pieces of the idea might you peel away or refine to make the idea stronger? Are there individual premises you might be wrong about? Do they invalidate the idea? Does removing them lead to a different idea?"
The above is a chunk of text from a post created by a user on the website LessWrong.
Explain this as if I never finished highschool?
The author is proposing a new concept called "Epistemic Effort". The word "epistemic" is a fancy term that refers to anything related to knowledge or understanding. So, "Epistemic Effort" is about the amount of work you put into making sure what you're saying or believing is accurate.
The author suggests that instead of just saying how confident we are about something (which is what "Epistemic Status" means), we should also share what steps we took to make sure we're right. For example, did we just think about it casually, or did we spend serious time researching and testing our ideas?
The author believes this is a good idea for a few reasons. One, it might encourage people to put more effort into being rational if they see others doing it. Two, it could help newcomers learn good habits by giving them a clear path to follow. And three, it could make discussions more valuable by encouraging everyone to think and communicate more clearly and accurately.
The author also acknowledges some potential problems. For example, people might do the bare minimum just to say they did something, or they might stop thinking creatively because they're just following the suggested steps.
Finally, the author suggests that if you're writing something and want to share an idea, you should try adding "Epistemic Effort: " at the beginning and explain what steps you took to make sure your idea is accurate. They also suggest some questions you could ask yourself to help refine your idea.
In simpler terms, it's like saying: "Before I tell you this, here's what I did to make sure I'm not talking nonsense."
What are some specific strategies or techniques that can be used to ensure the accuracy of one's "Epistemic Effort"?
There are several strategies or techniques that can be employed to enhance the accuracy of one's "Epistemic Effort". Here are a few:
1. **Critical Thinking**: This involves questioning assumptions, evaluating evidence, and assessing the logical soundness of the argument you're making.
2. **Research**: Conducting thorough research on a topic can help ensure the information you're presenting is accurate and up-to-date. This can involve reading books, scholarly articles, reputable websites, or consulting experts in the field.
3. **Empirical Testing**: If possible, testing your ideas or hypotheses empirically can provide concrete evidence to support or refute them. This could involve conducting experiments, surveys, or other forms of data collection.
4. **Peer Review**: Having others review your work can help identify any errors or biases you may have overlooked. This could be a colleague, a mentor, or even a group of peers.
5. **Reflection and Self-Correction**: Take the time to reflect on your own thought processes and beliefs. Be open to the possibility that you may be wrong and willing to correct your views in light of new evidence.
6. **Modeling**: Building a model that makes predictions about the subject matter can help in understanding the topic better and in testing its validity.
7. **Consultation**: Discussing your ideas with others, especially those who might have a different perspective or expertise, can provide valuable feedback and insights.
8. **Logical Consistency Check**: Ensure that your ideas or arguments do not contradict each other and are logically consistent.
9. **Fact-checking**: Verify the factual information that you are using to support your ideas or arguments. There are numerous fact-checking websites and resources available online.
These strategies can help ensure that the effort you're putting into understanding or explaining something is well-directed and likely to lead to accurate conclusions.
Describe a scenario where these strategies for enhancing "Epistemic Effort" could be applied, detailing each step and explaining how it contributes to the overall accuracy of the conclusion.
Sure, let's consider a scenario where you're trying to understand and explain the impact of climate change on polar bear populations. Here's how you might apply the strategies for enhancing "Epistemic Effort":
1. **Critical Thinking**: Start by questioning your initial assumptions. You might assume that climate change is negatively affecting polar bear populations, but it's important to challenge this assumption and not take it as a given.
2. **Research**: Begin by conducting a thorough literature review on the subject. Look for scholarly articles, books, and reports on the impact of climate change on polar bear populations. This will help you understand the current state of knowledge on the topic.
3. **Empirical Testing**: While this might not be feasible for an individual, scientists would conduct field studies to gather data on polar bear populations, their health, and their behavior in relation to changes in their environment.
4. **Peer Review**: Share your findings and interpretations with others, perhaps in a study group or an online forum. They can provide different perspectives, point out any errors or biases, and help refine your understanding.
5. **Reflection and Self-Correction**: Take time to reflect on the information you've gathered and your interpretation of it. Be open to changing your initial assumptions based on the evidence.
6. **Modeling**: Use the data available to build a model predicting future polar bear populations based on different climate change scenarios. This can help test the validity of your understanding.
7. **Consultation**: Reach out to experts in the field - climatologists, biologists, or environmental scientists. They can provide insights that you might have missed and validate or challenge your conclusions.
8. **Logical Consistency Check**: Ensure your conclusions are logically consistent with the evidence. For example, if the data shows a decline in polar bear populations and an increase in temperatures, it would be logically inconsistent to conclude that climate change has no effect on polar bears.
9. **Fact-checking**: Verify the facts you've used in your argument. For instance, if you've cited a statistic about the decline in polar bear populations, double-check this from a reliable source.
By applying these strategies, you're making a concerted effort to ensure the accuracy of your understanding and explanation of the impact of climate change on polar bear populations. Each step contributes to a more robust, well-supported conclusion, reducing the likelihood of error or bias. |
0a66beca-9d04-454b-890c-9df902f71083 | trentmkelly/LessWrong-43k | LessWrong | Meetup : Waterloo Meetup
Discussion article for the meetup : Waterloo Meetup
WHEN: 30 January 2012 08:00:00PM (-0500)
WHERE: 4 King St N Waterloo
Location: Symposium Cafe
I plan to keep organizing these meetups on a weekly basis, so this should be the last irregularly scheduled meetup.
The reservation is under "Julian Pulgarin", and if I am confident enough, there will be a large sign featuring this picture on it. My phone number is 519-781-1573.
I propose we go through the sunk cost exercise booklets, and agree on a regular time and place for weekly meetups.
Discussion article for the meetup : Waterloo Meetup |
e72cc83f-ad41-4a39-a688-a68b88ad4fc0 | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Draft report on existential risk from power-seeking AI
I’ve written a draft report evaluating a version of the overall case for existential risk from misaligned AI, and taking an initial stab at quantifying the risk from this version of the threat. I’ve made the draft viewable as a public google doc [here](https://docs.google.com/document/d/1smaI1lagHHcrhoi6ohdq3TYIZv0eNWWZMPEy8C8byYg/edit#) (Edit: arXiv version [here](https://arxiv.org/abs/2206.13353), video presentation [here](https://harvard.zoom.us/rec/play/kZc6rR2Ynx6z_J4rmmJUaq12-AcVDZLlY9eniU5ZKioodJm_yJORyy9UiLI8zFRuQvfLDAYO3j5TPciD.WSBvvCvXIto0mF4w?continueMode=true&_x_zm_rtaid=4hUT6sI8R4GPtHZS-tZBXg.1651957013327.8c9781863a967470f7270aa15ef4c1f4&_x_zm_rhtaid=917), human-narrated audio version [here](https://joecarlsmithaudio.buzzsprout.com/2034731/12113681-is-power-seeking-ai-an-existential-risk)). Feedback would be welcome.
This work is part of Open Philanthropy’s “[Worldview Investigations](https://www.openphilanthropy.org/blog/our-progress-2019-and-plans-2020#Worldview_investigations)” project. However, the draft reflects my personal (rough, unstable) views, not the “institutional views” of Open Philanthropy. |
1bb795a6-27d6-48ef-b0e6-a0578779fec4 | trentmkelly/LessWrong-43k | LessWrong | Hard Takeoff
Continuation of: Recursive Self-Improvement
Constant natural selection pressure, operating on the genes of the hominid line, produced improvement in brains over time that seems to have been, roughly, linear or accelerating; the operation of constant human brains on a pool of knowledge seems to have produced returns that are, very roughly, exponential or superexponential. (Robin proposes that human progress is well-characterized as a series of exponential modes with diminishing doubling times.)
Recursive self-improvement - an AI rewriting its own cognitive algorithms - identifies the object level of the AI with a force acting on the metacognitive level; it "closes the loop" or "folds the graph in on itself". E.g. the difference between returns on a constant investment in a bond, and reinvesting the returns into purchasing further bonds, is the difference between the equations y = f(t) = m*t, and dy/dt = f(y) = m*y whose solution is the compound interest exponential, y = e^(m*t).
When you fold a whole chain of differential equations in on itself like this, it should either peter out rapidly as improvements fail to yield further improvements, or else go FOOM. An exactly right law of diminishing returns that lets the system fly through the soft takeoff keyhole is unlikely - far more unlikely than seeing such behavior in a system with a roughly-constant underlying optimizer, like evolution improving brains, or human brains improving technology. Our present life is no good indicator of things to come.
Or to try and compress it down to a slogan that fits on a T-Shirt - not that I'm saying this is a good idea - "Moore's Law is exponential now; it would be really odd if it stayed exponential with the improving computers doing the research." I'm not saying you literally get dy/dt = e^y that goes to infinity after finite time - and hardware improvement is in some ways the least interesting factor here - but should we really see the same curve we do now?
RSI is the b |
fc67faf7-4ce2-4aad-96f4-cb5dd141a9da | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | [AN #147]: An overview of the interpretability landscape
Alignment Newsletter is a weekly publication with recent content relevant to AI alignment around the world. Find all Alignment Newsletter **[resources here](http://rohinshah.com/alignment-newsletter/)**. In particular, you can look through **[this spreadsheet](https://docs.google.com/spreadsheets/d/1PwWbWZ6FPqAgZWOoOcXM8N_tUCuxpEyMbN1NYYC02aM/edit?usp=sharing)** of all summaries that have ever been in the newsletter.
Audio version **[here](http://alignment-newsletter.libsyn.com/alignment-newsletter-147)** (may not be up yet).
Please note that while I work at DeepMind, this newsletter represents my personal views and not those of my employer.
HIGHLIGHTS
==========
**[Opinions on Interpretable Machine Learning and 70 Summaries of Recent Papers](https://www.alignmentforum.org/posts/GEPX7jgLMB8vR2qaK/opinions-on-interpretable-machine-learning-and-70-summaries)** *(Peter Hase and Owen Shen)* (summarized by Rohin): This is basically 3 months worth of Alignment Newsletters focused solely on interpretability, wrapped up into a single post. The authors provide summaries of 70 (!) papers on the topic, and include links to another 90. I’ll focus on their opinions about the field in this summary.
The theory and conceptual clarity of the field of interpretability has improved dramatically since its inception. There are several new or clearer concepts, such as simulatability, plausibility, (aligned) faithfulness, and (warranted) trust. This seems to have had a decent amount of influence over the more typical “methods” papers.
There have been lots of proposals for how to evaluate interpretability methods, leading to the **[problem of too many standards](https://xkcd.com/927/)**. The authors speculate that this is because both “methods” and “evaluation” papers don’t have sufficient clarity on what research questions they are trying to answer. Even after choosing an evaluation methodology, it is often unclear which other techniques you should be comparing your new method to.
For specific methods for achieving interpretability, at a high level, there has been clear progress. There are cases where we can:
1. identify concepts that certain neurons represent,
2. find feature subsets that account for most of a model's output,
3. find changes to data points that yield requested model predictions,
4. find training data that influences individual test time predictions,
5. generate natural language explanations that are somewhat informative of model reasoning, and
6. create somewhat competitive models that are inherently more interpretable.
There does seem to be a problem of disconnected research and reinventing the wheel. In particular, work at CV conferences, work at NLP conferences, and work at NeurIPS / ICML / ICLR form three clusters that for the most part do not cite each other.
**Rohin's opinion:** This post is great. Especially to the extent that you like summaries of papers (and according to the survey I recently ran, you probably do like summaries), I would recommend reading through this post. You could also read through the highlights from each section, bringing it down to 13 summaries instead of 70.
TECHNICAL AI ALIGNMENT
=======================
LEARNING HUMAN INTENT
----------------------
**[TuringAdvice: A Generative and Dynamic Evaluation of Language Use](https://arxiv.org/abs/2004.03607)** *(Rowan Zellers et al)* (summarized by Rohin): There are two main ways in which current NLP models are evaluated: quality of generations (how sensible the generated language looks), and correctness (given some crisp question or task, does the model output the right answer). However, we often care about using models for tasks in which there is no literally correct answer. This paper introduces an evaluation method for this setting: TuringAdvice. Models are presented with a situation in which a human is asking for advice, and the model must provide a helpful response. To score models, the resulting responses are compared against good human responses. The model’s response is successful if its advice is at least as helpful to the advice-seeker as human-written advice.
The authors collect a dataset of situations from Reddit, and for the human-written advice they take the most upvoted top-level comment on the post. A finetuned T5 model achieves a score of 14%, while prompted GPT-3 achieves a score of 4%. In contrast, taking the *secondmost* upvoted top-level comment would give a score of 41%, and a model that gave advice about as good as the typical best advice from a human would get 50%. The paper also presents several qualitative failures in which the models seem to have significant misunderstandings of the situation (though I can’t tell how cherrypicked these are).
**Rohin's opinion:** I really like the fact that we’re studying a fuzzy task directly, and using human evaluation to determine how well the model performs. (Though note that this is not the first benchmark to do so.)
**[Recursive Classification: Replacing Rewards with Examples in RL](https://ai.googleblog.com/2021/03/recursive-classification-replacing.html)** *(Benjamin Eysenbach et al)* (summarized by Rohin): Previous work has suggested learning a reward model from examples of successfully solving the task. This paper suggests that rather than a two stage process of learning a reward model and then optimizing it using RL, we can instead directly learn a policy from the examples by building an equivalent of Bellman backups that apply directly to examples (rather than having to go through intermediate rewards). Their experiments show that this works well.
**Read more:** **[Paper: Replacing Rewards with Examples: Example-Based Policy Search via Recursive Classification](https://arxiv.org/abs/2103.12656)**
OTHER PROGRESS IN AI
=====================
DEEP LEARNING
--------------
**[Branch Specialization](https://distill.pub/2020/circuits/branch-specialization/)** *(Chelsea Voss et al)* (summarized by Rohin): Neural network architectures sometimes have different “branches”, where later features can depend on earlier features *within the same branch*, but cannot depend on features in parallel branches. This post presents evidence showing that in these architectures, branches often tend to specialize in particular types of features. For example:
1. The first two layers in AlexNet are split into two branches. In one branch, the first layer tends to learn black and white Gabor filters, while in the other branch, the first layer tends to learn low-frequency color detectors. This persists across retraining, or even training on a different dataset of natural images, such as Places (rather than ImageNet).
2. All 9 of the black and white vs. color detectors in mixed3a are in mixed3a\_5x5 (p < 1e-8). All 30 of the curve-related features in mixed3b are in mixed3b\_5x5 (p < 1e-20). There are confounds here, but also good reasons to expect that it is in fact branch specialization.
Given that branch specialization seems to be robust and consistent even across datasets, a natural hypothesis is that it is reflecting a structure that already exists. Even if you didn’t have branching, it seems likely that the model would still learn very similar neurons, and it seems plausible that e.g. the weights connecting the first-layer black-and-white Gabor filters to the second-layer color detectors are effectively zero. With branching, you learn the same features in such a way that all the weights that previously were effectively zero now don’t exist because they would be crossing branches. This would look like having the Gabor filters in one branch and the color detectors in the other branch.
**Rohin's opinion:** I find the hypothesis the authors propose quite compelling (and this is very similar to the hypothesis that neural networks tend to be modular, which we discuss more below). Partly, this is because it has a common-sense explanation: when designing an organization, you want to put related functions in the same group to minimize the communication across groups. Here, the full network is the organization, the branches are an explicit constraint on communication, and so you want to put related functions in the same branch.
At the end of the article, the authors also suggest that there could be a connection with the way that different regions of the brain are specialized to particular tasks. I’ll go further than the authors in my speculation: it seems plausible to me that this specialization is simply the result of the brain’s learning algorithm reflecting the structure of the world through specialization. (Though it seems likely that the different areas of the brain must at least have different “architectures”, in order for the same tasks to be routed to the same brain regions across humans.) But the case of AlexNet demonstrates that in theory, the only thing you need for specialization to arise is a restriction on the communication between one part of the architecture and the other.
**[Clusterability in Neural Networks](https://arxiv.org/abs/2103.03386)** *(Daniel Filan, Stephen Casper, Shlomi Hod et al)* (summarized by Zach): Neural networks are often construed as lacking internal structure. In this paper, the authors challenge the predominant view and hypothesize that neural networks are more clusterable than is suggested by chance. To investigate the claim, the authors partition the network into groups where most of the edge weight is between neurons in the same group. The authors find that the quality of these groups improves after training, as compared to randomly initialized networks. However, this only holds for certain training setups. Despite this limitation, the authors show it's possible to promote clusterability with little to no effect on accuracy.
In experiments, the authors compare the clusterability of trained networks to randomly initialized networks and trained networks with shuffled weights. They focus on multi-layer perceptrons (MLPs) and convolutional networks with dropout regularization. They also run experiments with pruned networks or networks where 'unimportant' edges are removed. They find that MLP networks have clusterable neurons at rates higher than chance, but have mixed results for convolutional networks.
The authors hypothesize that clusterability is more likely to arise when different features of the input can be computed in parallel without communication between the features (which is very similar to the hypothesis in the previous paper). To test the hypothesis, they combine examples from the datasets into pairs and then train the neural network to make a double-prediction in a side-by-side setup. Intuitively, the network would need to look at each pair separately, without any need to combine information across the two sides. They find that this setup results in increased modularity.
**Zach's opinion:** The experiments in this paper are well-designed. In particular, I found the side-by-side experiment setup to be a clever way to test the ideas presented in the paper. Having said that, the actual results from the experiments are mixed. The paper's strongest results are for the clusterability of pruned networks, while evidence for the clusterability of convolutional networks is quite mixed. However, pruning is not common practice. Additionally, in an intuitive sense, pruning a network seems as though it could be *defined* in terms of clusterability notions, which limits my enthusiasm for that result.
**Rohin's opinion:** I feel like there are quite a few interesting next questions for the study of modularity in neural networks:
1. Does modularity become more obvious or pronounced as we get to larger models and more complex and realistic data?
2. How well do networks cluster when you are looking for hundreds or thousands of clusters (rather than the 12 clusters considered in this paper)?
3. To what extent is modularity a result of any training, vs. a result of the specific data being trained on? If you train a network to predict random labels, will it be less modular as a result?
One challenge here is in ensuring that the clustering algorithm used is still accurately measuring modularity in these new settings. Another challenge is whether networks are more modular just because in a bigger model there are more chances to find good cuts. (In other words, what's the default to which we should be comparing?) For example, the paper does present some results with models trained on ImageNet, and ResNet-50 gets one of the highest clusterability numbers in the paper. But the authors do mention that the clustering algorithm was less stable, and it's not clear how exactly to interpret this high clusterability number.
**[Weight Banding](https://distill.pub/2020/circuits/weight-banding/)** *(Michael Petrov et al)* (summarized by Rohin): Empirically, when training neural networks on ImageNet, we can commonly observe “weight banding” in the final layer. In other words, the neurons in the final layer pay very strong attention to the vertical position of features, and ignore the horizontal position of features. This holds across InceptionV1, ResNet50, and VGG19, though it doesn’t hold for AlexNet.
If you rotate the training data by 90 degrees, then the phenomenon changes to have vertical striping, that is, we now pay strong attention to the horizontal position of features. This suggests that this phenomenon is being driven somehow by the ImageNet data.
The authors hypothesize that this is caused by the neural network needing to recover some spatial information that was reduced by the previous average pooling layer (which is not present in AlexNet). They try removing this layer, which causes the effect to go away in Inception, but not in VGG19. They seem to think that it also goes away in ResNet50, but when I look at the results, it seems like the phenomenon is still there (though not as strongly as before).
They try a bunch of other architectural interventions on a simplified architecture and find that weight banding persists across all of these.
#### **FEEDBACK**
I'm always happy to hear feedback; you can send it to me, **[Rohin Shah](https://rohinshah.com/)**, by **replying to this email**.
#### **PODCAST**
An audio podcast version of the **Alignment Newsletter** is available. This podcast is an audio version of the newsletter, recorded by **[Robert Miles](http://robertskmiles.com/)**. |
3c2c2842-ec60-4809-a370-a6d0468197eb | trentmkelly/LessWrong-43k | LessWrong | How would you run the statistics on whether Ivermectin helped India reduce COVID-19 cases?
I frequently read the claim that India reduced their COVID-19 cases a lot via Ivermectin but it's very unclear to me whether the relevant data to make that claim is cheripicked.
India has 34 region and each of those region has their own policy on Ivermectin. While we don't have exact data about Ivermectin usage, Google Trend data provides a proxy for usage of Ivermectin and does provide data that's fine-grained enough to separate individual region. John Hopkins provides data for COVID-19 cases by Indian region as well. 34 regions sounds to me like it's plausible to find statistically significant results if Ivermectin had an effect that's as great as proponent of Ivermectin claim
Given p-hacking concerns, it makes sense to first decide on how to analyse the data and then run the statistics. Does anyone have suggestion for how the data should be analysed? |
58a70118-e854-4687-99ce-b5bef39de9ca | trentmkelly/LessWrong-43k | LessWrong | Workflow vs interface vs implementation
(Found this old draft that was more or less complete; figured I should publish it. It's adapted from this old blog post of mine.)
It's commonly appreciated that when explaining how to use something, there's a distinction between the high-level usage instructions and the low-level details. But what I've noticed is that different people draw the line a bit differently, and this leads to some confusion. A more useful, less ambiguous distinction, I find, is to break things into three levels, which I'm calling "implementation", "interface", and "workflow".
Implementation is what it sounds like -- how the thing works under the hood, ignoring all abstractions you might use to paper it over. You normally shouldn't have to worry about this unless something breaks, or you want to modify it, or maybe you just want to learn how it works so you can do something new, or whatever. It's what's below the abstraction.
Interface is what the thing does and not how it does it; this is necessarily an abstraction and won't always work, but it's at least what the thing is supposed to do in every case.
Workflow is how you normally use the thing -- not everything it does, just how it will respond when you do the things you normally do with it, not how it's going to respond in weird edge cases.
Note that these distinctions will only really work for artificial things -- something like, say, cooking, doesn't have a clean separation between implementation and interface, and even for something artificial this may not quite work if it's not sufficiently complete or well-implemented -- but this can still be a useful separation to draw even when that separation may not match the territory cleanly.
Now this may sound fairly obvious, but I bother pointing it out because apparently it's not so obvious! I can think of multiple conversations I've had where (in effect) I was given workflow information, found it insufficient, asked about the interface, and was asked, why do you need to know about suc |
1a091cc4-78b7-4adb-bfe8-27557e36563d | trentmkelly/LessWrong-43k | LessWrong | Should you be worried about H5N1?
Epistemic status: a few people without any particular expertise in epidemiology spent an afternoon in a coffee shop discussing and reading about H5N1, with a focus on how an individual should orient towards this (as opposed to say, the government). This is a write-up of what I took away from that exercise, written from my perspective.
Some ideas were generated in collaboration with Claude but generally spot checked. This post was also sanity-checked by a friend who works in epidemiology. I feel ok about the ideas presented but would not be surprised if someone with more expertise has a significantly different conclusion. Any mistakes are mine.
I went into this exercise with a prior of “I’ve been hearing about bird flu for years, and it’s always been nothing, it’s probably nothing again this time.” The main upshot is that I walked away from the exercise thinking “I don’t know if this is going to be something or not.” As far as updates go, that seems directionally bad. My current orientation towards this is something like “watch and wait, and spend appropriately more effort on this if / when certain milestones happen (and also make some trades).”
What’s different this year?
One main thing that seems to generate a heightened level of ongoing risk is sustained infections in dairy cow populations. This is already bad because it’s mammal-to-mammal transmission in a population that hasn’t historically had problems with bird flu, and it would be worse if it becomes endemic in farmed cows. As long as it is sustaining infections in farmed cows, H5N1 has a convenient breeding ground for mutations that is:
* In a high density population
* Year round (bird flu is seasonal among birds)
* In regular contact with humans
The last point gets a bonus with the human flu season coming up. If a dairy worker gets sick with both bird flu and human flu simultaneously, the two strains might share their genetic information and mutate into a pandemic-worthy strain.
Another thing bei |
e1764fe5-6515-4c1b-b15a-1b8d3b264487 | StampyAI/alignment-research-dataset/special_docs | Other | Multi-agent learning in mixed-motive coordination problems
Multi-agent learning in mixed-motive coordination problems
Julian Stastny Johannes Treutlein Maxime Riché Jesse Clifton
Abstract
Cooperation in settings where agents have different but overlapping preferences ( mixed-motive set-
tings) has recently received considerable attention in multi-agent learning. However, the mixed-motive
environments typically studied are simplistic in that they have a single cooperative outcome on which all
agents can agree. Multi-agent systems in general may exhibit many payoff profiles which might be called
cooperative, but which agents have different preferences over. This causes problems for independently
trained agents that do not arise in the case of that there is a unique cooperative payoff profile. In this
note, we illustrate this problem with a class of games called mixed-motive coordination problems (MCPs) .
We demonstrate the failure of several methods for achieving cooperation in sequential social dilemmas
when used to independently train policies in a simple MCP. We discuss some possible directions for
ameliorating MCPs.
1 Introduction
InAn Essay on Bargaining [1956], Thomas Schelling describes a negotiation over the price of a house.
Considering different strategies the negotiators might use, he points out that
If the buyer is an agent authorized by a board of directors to buy at $16000 but not a cent more,
and the directors cannot constitutionally meet again for several months and the buyer cannot
exceed his authority, and if all this can be made known to the seller, then the buyer ‘wins’ — if,
again, the seller has not tied himself up with a commitment to $19000. (pg. 284)
Imagine various actors with different preferences — a mixed-motive setting [Schelling, 1958] — developing
machine learning systems to act on their behalf. These actors are like Schelling’s bargainers, in that their
systems implicitly encode a commitment to pursuing a particular tradeoff between their own preferences and
those of other players. Just as the price negotiation in Schelling’s example may fail if buyer and seller are
each committed to insisting on incompatible prices, machine learning systems may fail to cooperate if they
encode commitments to incompatible tradeoffs between their principals’ goals.
Multi-agent learning (MAL) has in recent years paid considerable attention to problems of cooperation
between agents with different, but not zero-sum, preferences. However, much of this work has been in the
setting of (sequential) social dilemmas (SSDs) (e.g., Peysakhovich and Lerer 2017a, Lerer and Peysakhovich
2017, Peysakhovich and Lerer 2017b, Foerster et al. 2018, Wang et al. 2018, Eccles et al. 2019). The classic
example of a social dilemma is the Prisoner’s Dilemma (PD), and the SSDs studied in this literature are
similar to the Prisoner’s Dilemma in that there is a single salient notion of “cooperation”. This means that
it is relatively easy for actors to coordinate in their selection of policies to deploy in these settings. This
situation is quite unlike that faced by Schelling’s bargainers, for whom there are many possible agreements to
commit to, and who thus may fail to reach an agreement if they make incompatible commitments1.
Cao et al. [2018] are an exception in that they look at negotiation between deep reinforcement learners,
such that there are multiple incompatible ways in which agents could choose among possible agreements.
However, they do not evaluate the performance of independently trained policies when played against one
Working draft. Last updated March 8, 2021.
1The inadequacy of the Prisoner’s Dilemma as a model for cooperation problems, and the need to study games that exhibit
competing “cooperative” solutions, has been pointed out in other fields. See Snidal [1985] for a discussion in international
relations, McAdams [2008] in law, and Binmore [1998] in evolutionary game theory.
1
another (“cross-play” [Hu et al., 2020]), which is where the existence of multiple incompatible cooperative
solutions creates problems.
In this note, we argue that a focus on sequential social dilemmas and the lack of cross-play evaluation
obscures a fundamental problem for decentralized MAL in mixed-motive settings. We discuss a class of games
that we call mixed-motive coordination problems (MCPs) , which highlight this problem. We compare MCPs
to pure coordination problems and SSDs. We then illustrate how MAL methods which achieve cooperation in
other social dilemmas fail in some simple MCPs, and argue that future methods for promoting cooperation in
mixed-motive settings should be benchmarked in these more difficult cases. We close with some tentative
positive contributions, by sketching some partial solutions to MCPs.
2 Preliminaries
We work in the setting of partially observable stochastic games (POSGs). For simplicity we assume two
players,i= 1;2. We will index player i’s counterpart by i. Each player has an action space Ai, there is an
underlying state space Swhich evolves according to a Markovian transition function P(St+1jSt;At
1;At
2). At
each time step, each player sees an observation Ot
iwhich depends on St. Thus each player has an accumulating
history of observations Ht
i=fOv
1;Av
1gt
v=1. Call the set of all histories Hi=[1
t=1Ht
i. Assume for simplicity
that the initial observation history is fixed and common knowledge, h0
1=h0
2h0. Finally, principals choose
among policies i:H! (Ai), which we imagine as artificial agents deployed by the principals. We will
refer to policy profiles generically as 2 := 12.
Each player has a reward function ri, such that ri(St;At
1;At
2)is their reward at time t. Define the value
to playeriof policy profile starting at history ht
iasVi(ht
i;) =E[P1
v=t
v tri(Sv;Av
1;Av
2)jHt
i=ht
i],
where
< 1is a discount factor. Define the value of a policy profile to player iasVi() =Vi(h0;).
A payoff profile is then a tuple (V1();V2()). We say that is a (Nash) equilibrium of a POSG if
i2arg max0
i2iVi(h0;0
i; i)fori= 1;2. We say that is Pareto-optimal if for i= 1;2and02, we
have thatVi(0)>Vi()impliesV i(0)<V i():
3 Mixed-motive coordination problems
We envision multiple actors (“principals”) deploying machine learning systems, defined by policies, into the
world. The principals may be human developers, or they may be machine learning systems themselves. The
principals simultaneously deploy policies, without communication (in Section 6 we discuss ways in which
communication may change the analysis). Let Wbe a set of welfare functions , which are supposed to encode
normative properties (total welfare, fairness, etc.) of different payoff profiles. Two uncontroversial properties
of a welfare function are Pareto-optimality (i.e., its optimizer should be Pareto-optimal) and symmetry (the
welfare of a policy profile should be invariant to permutations of player indices). Beyond that, there are
several appealing properties which cannot all be satisfied by the same welfare function (see Table 1). The
experimental evidence regarding people’s preferences over welfare functions is inconclusive [Felsenthal and
Diskin, 1982]. We will say that a game is a mixed-motive coordination problem (MCP) if it has several
equilibria2which are optimal with respect to some welfare function, but are incompatible with each other.
Definition 3.1 (Mixed-motive coordination problem) .LetGbe a POSG. LetWbe a set of welfare functions,
and letWEWbe the set of welfare functions in Wwhich are optimized by some equilibrium of G. Then
Gis a mixed-motive coordination problem (with respect to W) if there exist distinct w;w02WE2such that
(w
1;w0
2)has a Pareto-suboptimal policy profile for all w2arg max0w(0),w02arg max0w0(0).
An example of a mixed-motive coordination problem is the Bach or Stravinsky game. A variant of this
game with asymmetric payoffs is given in Figure 1 (left). The folk theorem [Friedman, 1971] tells us that any
payoff profile in the convex hull of these payoffs is attainable in an equilibrium of the repeated game, for
players with sufficiently high discount factors. And, as is suggested by the shape of the Pareto frontier for
2The spirit of an MCP is that independent training predictably leads to policy profiles which optimize different welfare
functions, but are incompatible. So one could modify the requirement that welfare-optimal policies be equilibria and retain this
spirit. One could replace the requirement of Nash equilibrium with a different or a broader set of solution concepts, e.g., the
recently-proposed Markov-Conley chains used in the -rank evaluation methodology [Omidshafiei et al., 2019].
2
Name of welfare function w Form ofw()
Nash [Nash, 1950] [V1() d1][V2() d2]
Kalai-Smorodinsky
[Kalai and Smorodinsky, 1975] V1() d1
V2() d2 supV1() d1
supV2() d2 +
fPareto-optimalg
Egalitarian [Kalai, 1977] minfV1() d1;V2() d2;g
Utilitarian (e.g. Harsanyi 1955) V1() +V2()
Table 1: Welfare functions, adapted to the multi-agent RL setting where two agents with value functions V1;V2
are bargaining over the policy profile . to deploy. The function in the definition of the Kalai-Smorodinsky
welfare is the1-0indicator, used to enforce the constraint in its argument. Vd
i=Vi(d)are the players’
disagreement values.
Note that in bargaining problems in which the set of feasible payoffs is convex, the Nash welfare
function uniquely satisfies the properties of (1) Pareto optimality, (2) symmetry, (3) invariance to affine
transformations, and (4) independence of irrelevant alternatives. See Zhou [1997] for a characterization of the
Nash welfare in non-convex bargaining problems. The Nash welfare can also be obtained as the subgame
perfect equilibrium of an alternating-offers game as the “patience” of the players goes to infinity [Binmore
et al., 1986].
On the other hand, Kalai-Smorodinsky uniquely satisfies (1)-(3) plus (5) resource monotonicity, which
means that all players are weakly better off when there are more resources to go around. The egalitarian
solution instead satisfies (1), (2), (4), and (5). The utilitarian welfare function is implicitly used in the work
of Peysakhovich and Lerer [2017a], Lerer and Peysakhovich [2017], Wang et al. [2018] on cooperation in
sequential social dilemmas.
BoS B S
B4, 1 0, 0
S0, 0 2, 2PD C D
C-1, -1 -3, 0
D0, -3 -2, -2
Figure 1: Payoffs for asymmetric Bach or Stravinsky (BoS) and the Prisoner’s Dilemma (PD).
BoS (Figure 2, left), different welfare functions pick out different Pareto-optimal payoff profiles. For instance,
taking the disagreement point to be d1;d2= (0;0), the Nash welfare is optimized at (3:0;1:5), whereas the
egalitarian welfare is optimized at (2:0;2:0). On the other hand, all welfare functions in Table 1 agree on the
payoffs corresponding to (C;C)in the iterated Prisoner’s Dilemma (Figure 2, right).
In the following, we compare mixed-motive coordination problems to the more general equilibrium selection
problem and to sequential social dilemmas.
3.1 Relation to other cooperation problems
Equilibrium and Pareto selection problems It is well-known that achieving cooperation between
independently-trained agents this is a challenging problem for MAL in general (e.g., Boutilier 1999, Matignon
et al. 2012, Hu et al. 2020). One reason is that there are often multiple equilibria, such that (to the extent
that one wants to deploy a policy which is a best-response to that of one’s counterpart) the choice of a policy
constitutes an equilibrium selection problem .
Such problems even arise in the fully cooperative case, i.e., when V1=V2V, and are then also called
Pareto selection problems [Matignon et al., 2012] or pure coordination problems. In a pure coordination
problem, any 2arg maxV()has an optimal value and is thus in equilibrium. There is an equilibrium
selection problem if there exist ;02arg maxV()such that (1;0
2)has Pareto-suboptimal payoff profile.
Mixed-motive coordination problems that occur between policy profiles in equilibrium are a subclass
of equilibrium selection problems. They only occur in general-sum, non-fully cooperative cases and pose a
challenge which is qualitatively different from the fully cooperative case: players have to coordinate among
3
options over which they have conflicting preferences. The pure coordination problem can be solved by merely
giving the players a chance to converge upon a coordinated policy profile (as in Boutilier [1999], for instance)
and any such policy will do equally well. This is not so in mixed-motive coordination problems, since agents
that are open to converging upon any Pareto-optimal outcome would be exploitable by opponents that insist
on an outcome they prefer. (See Example 6.0.1 for a concrete illustration of this difference.)
Sequential social dilemmas Sequential social dilemmas (SSDs), as defined by Leibo et al. [2017], are
POSGs whose policy space contains subsets corresponding to cooperation and defection, called C
iandD
i,
respectively. At each state, the normal form game whose payoffs are value functions under profiles of policies
inC
i;D
isatisfies the social dilemma inequalities [Macy and Flache, 2002], where atomic “cooperate” and
“defect” actions are replaced with elements of C
i;D
i. An SSD is not necessarily an equilibrium selection
problem. For instance, any Pareto-optimal and symmetric policy profile will select an equilibrium profile that
leads to constant cooperation in the iterated Prisoner’s Dilemma, and which is compatible with other such
profiles. But other SSDs may exhibit multiple incompatible ways of implementing cooperation.
SSDs with symmetric Pareto-optimal equilibria are in any case not mixed-motive coordination problems.
This includes the Gathering, Wolfpack [Leibo et al., 2017], and Coin Game [Peysakhovich and Lerer, 2017a]
environments. For instance, in the Coin Game, the single cooperative payoff profile is attained when agents
get only the coin of their own color, rather than attempting to take their counterpart’s coin as well.
4 Learning algorithms
Play between policies trained by different runs or variations of an algorithm, or “cross-play” [Hu et al., 2020]
is problematic for MCPs. To illustrate the need for evaluating policies in cross-play, we train policies using
two multi-agent learning algorithms which have been shown to achieve cooperation in SSDs. The first is a
variant of Lerer and Peysakhovich [2017]’s approximate Markov tit-for-tat (amTFT). The original amTFT
algorithm trains a cooperative policy on the utilitarian welfare (Table 1), as well as a punishment policy, and
switches from the cooperative to the punishment policy when it detects that the other player is defecting
from the cooperative policy. We consider the more general class of algorithms in which a cooperative policy
is constructed by optimizing some welfare function. Call the version of amTFT in which the cooperative
policy optimizes welfare function wasamTFT (w).
In our experiments, we use the utilitarian welfare function wUtiland aninequity averse welfare function
wIA. Following Hughes et al. [2018], this welfare function is constructed using modified reward functions
which penalize inequitable outcomes:
VIA
i(at
1;at
2; ; ) =r1(at
1;at
2) +r1(at
2;at
2) [et
1(at
1;at
2) et
2(at
1;at
2)]+ [e2(at
1;at
2) e1(at
1;at
2)]+
where [x]+=maxfx;0gandet
iaresmoothedcumulativerewardscomputedas et
i(at
1;at
2) =
et 1
i(at 1
1;at 1
2)+
ri(at
1;at
2)with hyperparameter . and discount factor
and control how much unequal outcomes are
penalized.3
The second learning algorithm is learning with opponent-learning awareness (LOLA; Foerster et al. 2018)
a policy gradient method in which learners attempt to shape the others’ learning through the choice of their
own policy updates. Write Vi(1;2)as the value to player iunder a profile of policies with parameters 1;2.
Then, the LOLA update for player 1 at time twith parameters ;> 0is
t
1=t 1
1+r1Vi(1;2) +[r2V1(1;2)]T
r1r2V2(1;2):
We use the discount factor
= 0:96throughout.
3We choose the utilitarian and inequity-averse welfare functions largely for convenience, and not because they have particularly
compelling normative properties. Specifically, these welfare functions admit a Bellman equation, allowing us to straightforwardly
apply reinforcement learning methods in order to maximize them. Methods for optimizing other welfare functions (Table 1),
which may be less amenable to off-the-shelf reinforcement learning methods, are a direction for future work.
4
5 Results
Here we illustrate the special problem posed by MCPs by comparing the performance of the learning
algorithms above in iterated BoS (IBoS) and in the iterated Prisoner’s Dilemma (IPD). The folk theorems
[Mailath et al., 2006] tell us that, for players with sufficiently high discount factors, any payoff profile of
the stage game in which players get at least as much as they can unilaterally guarantee themselves can be
attained in an equilibrium of the repeated game. IBoS is an MCP with respect to the set of welfare functions
W=fwUtil;wIAg, since these welfare functions are optimized by distinct payoff profiles for sufficiently large
values of ; .
We measure the performance of these algorithms in self-play andcross-play . Self-play payoffs are the
average payoffs attained by a profile of policies produced by the same training run. Cross-play payoffs are
the average payoffs attained by a profile of two policies produced by two different training runs. In the
case of amTFT, we also consider cross-play between policies trained with amTFT (wUtil)and those trained with
amTFT (wIA).
5.1 LOLA
Following Foerster et al. [2018]’s parameterization of policies in the iterated Prisoner’s Dilemma, we used
policies which condition on the previous pair of actions played, with one parameter for every possible previous
action profile. To learn policy profiles, we used a discount factor of
= 0:96and applied LOLA to the
closed-form value functions (which can be computed for each policy profile using the Bellman equations; this
is called exact LOLA in Foerster et al. [2018]).
Figure 2: LOLA average payoff profiles for 40 training runs in IBoS (left) and IPD (right). Cross-play payoffs
were obtained by randomly sampling 40 pairs (1;2), where1and2come from different runs. A small
amount of random noise is applied to each point to enhance visibility.
On average, self-play yields payoffs (2:69;1:34), while cross-play yields (1:30;0:70). As Figure 2 shows, the
self-play payoffs almost all lie close to the empirical Pareto frontier, which consists of payoff profiles roughly
equal tof(4;1);(3;1:5);(2;2)g. This clustering is caused by the policies depending only on actions taken at
the previous timestep, which allows for always playing (S,S), alternating between them, or always playing
(B,B). These coincide with the maxima for wUtil, the Nash welfare [Nash, 1950] under disagreement profile
(0;0), orwIA.
On the other hand, the cross-play points are much more often Pareto-suboptimal. This is not surprising,
as if LOLA produces tit-for-tat-like strategies which punish deviations from a particular strategy profile, then
playing policies from separate runs is liable to lead to incompatible policies and therefore punishments.
5.2 amTFT
We evaluated self- and cross-play for amTFT in IBoS and IPD, where each episode lasted 100 of steps. Payoff
profiles from individual runs are displayed in Figure 3, and average payoffs under self- and cross-play for each
5
combination of welfare functions are displayed in Table 2.
Figure 3: amTFT payoff profiles for 40 training runs in IBoS (left) and IPD (right). A small amount of
random noise is applied to each point to enhance visibility.
Table 2: Self-play (" ") and cross-play (" ") payoffs in IBoS for amTFT (wUtil)andamTFT (wIA). Payoffs are
averages over 40 training runs. wUtilwIAleads to the same payoff regardless of which player operates under
which welfare function. All standard errors were below 0:01.
Policy profile wUtilwUtilwIAwIAwUtilwUtilwIAwIAwUtilwIA
Player 1 3.97 2.00 3.87 2.00 0.00
Player 2 1.01 2.00 0.97 2.00 0.00
6 Possible solutions to MCPs
Of course, the goal of developing benchmarks is not only to point out flaws in existing approaches, but also
to help develop better ones. In this section we discuss several research directions for ameliorating MCPs, and
give a toy illustration of one approach.
Coordination by the principals MCPs can be solved if the principals can agree on a way of selecting
among policy profiles to deploy. For instance, principals might agree on a welfare function, and choose
learning algorithms which converge to an optimizer of that welfare function. Coordination on a welfare
function (and among equilibria which all maximize that welfare function) may be easy in toy environments
such as the ones we have considered. But as systems become more complex, it will likely be intractable
for principals to fully coordinate so as to train welfare-optimal agents. Thus there is a need to develop
methods for low-bandwidth coordination between principals training agents in MCPs which captures most
of the benefits of full coordination. Bandwidth-constrained coordination has already been studied in fully
cooperative games (e.g., Stone and Veloso 1999, Zhang et al. 2019).
Compromise between welfare functions We have not discussed how giving agents the opportunity to
communicate post-deployment might ameliorate MCPs. One possibility is to have them reach a compromise
over different welfare-optimal policy profiles before deciding which to deploy. On possibility is for the train to
several policies corresponding to several welfare functions according to which they are willing to play. When
given the chance to communicate, their agents can then apply a bargaining protocol to this reduced set of
policy profiles. Here is a simple example of such a protocol. Agents first announce the sets of policy profiles
they are willing to play. If these sets overlap, a policy profile is taken at random from the intersection (compare
with Chakravorti and Conley [2004]’s discussion of randomization between the Nash and Kalai-Smorodinsky
6
Pure Coordination A B
A 1, 1 0, 0
B 0, 0 1, 1
Figure 4: Payoffs for a pure coordination problem.
bargaining solutions in the asymmetric iterated Prisoner’s Dilemma). Only when these sets do not overlap do
agents revert to a default policy. We give an illustration in Example 6.0.1.
Example 6.0.1 (Bargaining over amTFT policies) .An example of a protocol for bargaining over welfare-
optimal policy profiles is given in Algorithm 1. Policies are obtained by applying amTFT to a welfare
functionw, returning a policy amTFT (w)ifor playeri. Refer to the strategy which reports welfare functions
Wiand default policy d
ias(Wi;d
i). Table 3 displays the payoff table of the game with strategy profiles
f(W1;d
1);(W2;d
2)gin this bargaining protocol applied to IBoS.
This payoff matrix clearly shows the tradeoff between exploitability and robustness to independently-
trained policies. The strategies which play fwUtil;wIAgare of more robust, in the sense that they avoid the
mutually dispreferred profile of (1:0;0:3). However, these strategies are notplayed in equilibrium, because
they are exploitable: an agent can deviate from this profile by instead playing only their preferred welfare
function. Nevertheless, one has the intuition that it is reasonable to play fwUtil;wIAgin view of its robustness,
despite the fact that it is exploitable. On the other hand, the situation becomes more complicated when the
set of welfare functions under consideration is much larger. Identifying reasonable solution concepts for this
setting is an important direction for future work.
Besides the fact that avoiding bargaining failure is still not guaranteed, there are additional problems for
this protocol. First, it may be ex-ante Pareto suboptimal, because randomization between Pareto optimal
payoff profiles may be ex-ante Pareto suboptimal. Second, while it assumes that agents agree to uniform
randomization over the intersection of their sets, agents will in fact have differing preferences over different
distributions with which to randomize, generating another bargaining problem. Less ad-hoc solutions might
be found by using a dynamic bargaining protocol (as in the classical Rubinstein bargaining game Rubinstein
1982), which may narrow down the set of equilibria.
As a final note, compare this protocol with an analogous protocol for a pure coordination problem (Figure
4). In this protocol, agents play a default action but also announce a set of actions that they are willing to play.
Again, actions are taken uniformly at random from the overap of the sets if it is nonempty, and otherwise the
default action is taken. Clearly playing fA;Bgis a dominant strategy for each player, unlike the analogous
strategy in the protocol for IBoS, where greater flexibility comes at the price of greater exploitability.
Algorithm 1: Welfare function selection protocol
Input:Sets of welfare functions Wi, default policies d
i
// Independently train policies to maximize welfare functions
fori= 1;2do
i ;
forw2Wido
w
i amTFT (w)i
i i[fw
ig
// Communicate and decide what policies to deploy
Players announce W1;W2
ifW1\W 26=;then
wUnif (W1\W 2)
1;2 w
1;w
2
else
1;2 d
1;d
2
return1;2
7
Table 3: Empirical payoff matrix for agents playing IBoS and following bargaining protocol in Algorithm 1
with welfare functions in fwUtil;wIAg:
fwUtilg;IA
2fwIAg;IA
2fwUtil;wIAg;IA
2
fwUtilg;Util
1 4.0, 1.0 0.0, 0.0 4.0, 1.0
fwIAg;Util
1 0.0, 0.0 2.0, 2.0 2.0, 2.0
fwUtil;wIAg;Util
1 4.0, 1.0 2.0, 2.0 3.0, 1.5
Table 4: Payoff matrix for agents playing Pure Coordination and following a variant of the bargaining protocol
in 1 where sets of rather than welfare functions are announced.
fAg;BfBg;BfA;Bg;B
fAg;A 1.0, 1.0 0.0, 0.0 1.0, 1.0
fBg;A 0.0, 0.0 1.0, 1.0 1.0, 1.0
fA;Bg;A 1.0, 1.0 1.0, 1.0 1.0, 1.0
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9 |
fb8970b0-6000-4a55-8374-59d9a49f5d88 | trentmkelly/LessWrong-43k | LessWrong | Linkpost: Arena’s New Opening Hand Rule Has Huge Implications For How We Play the Game
This is a link-post for my latest Magic article, posted on Channel-Fireball: Arena’s New Opening Hand Rule Has Huge Implications For How We Play the Game.
Posting here is a more reliable way to engage with me than commenting there, but they have a robust comment section of other Magic players. |
a022ff7d-561e-49c6-bb45-0774d51b3bdb | trentmkelly/LessWrong-43k | LessWrong | Dodging systematic human errors in scalable oversight
Summary: Both our (UK AISI's) debate safety case sketch and Anthropic’s research agenda point at systematic human error as a weak point for debate. This post talks through how one might strengthen a debate protocol to partially mitigate this.
Not too many errors in unknown places
The complexity theory models of debate assume some expensive verifier machine M with access to a human oracle, such that
1. If we ran M in full, we’d get a safe answer
2. M is too expensive to run in full, meaning we need some interactive proof protocol (something like debate) to skip steps
Typically, M is some recursive tree computation, where for simplicity we can think of human oracle queries as occurring at the leaves of the tree. Key design elements encapsulated by the verifier machine M (rather than by the interactive proof protocol) are those related to the structure of the underlying debate, such as whether counterarguments are used, when the human oracle is consulted, and how the judges are trained.
Model the human oracle as type:
H:X→ΔY,
where X is the question type, Y is the answer type, and ΔY is distributions over Y (the humans are stochastic). Assumption (1) above then becomes
* Claim: M(H) is safe
This claim includes robustness to errors: M only has oracle access to H, so seeing through noise in H requires M to sample enough. It also includes robustness to systematic error: if H is systematically wrong, M(H) must sample different questions to dodge the wrongness.
This is a lot to ask of M! A more practical situation would be:
1. We’re handed some machine N, which is not robust to errors. That is
1. N(H) would be safe if H was error-free, or perhaps had only random errors
2. N(H) is not safe in practice, because we expect H to have too many errors
2. We assume H satisfies some “not too many errors” assumption G.
3. We have an improved protocol which is safe as long as G is true, either by
1. Using a different debate protocol, or
2. Building a |
879d5e07-5d3f-47b3-8d80-8e5e3d89bae3 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | Reward function learning: the value function
I've written quite a few posts about the problems with agents learning values/rewards, and manipulating the learning process. I won't be linking to those past posts, because the aim of this post and the next [one](https://www.lesswrong.com/posts/upLot6eG8cbXdKiFS/reward-function-learning-the-learning-process) is to present a complete, clear, and easy(er) to understand overview of the whole situation. They should stand alone, and serve as an introduction to the subject. This first post will present the framework, and the value function for learnt reward functions; the next one will be looking at the properties of the learning function. I'll be using variants on a single running example throughout, to try and increase understanding.
Feel free to comment on ways the ideas could be made clearer.
0 The main points
=================
The central insight of these posts may seem trivial by now: that a learning process that the agent can influence, does not have the same properties as one that it can't. Moving from "if you don't know what is right, look it up on this read-only list" to "if you don't know what is right, look it up on this read-write list (or ask the programmer)" is a **huge** change. Especially when read-only lists can easily become read-write in practice when the agent becomes more powerful.
Why write long posts and papers about this idea, though? First of all, because it is easy, in practice, not to notice that shift. And secondly, because it is easy to define a "learning" process that doesn't behave like we'd expect.
So by defining the value function and learning function of an ideal learning process, this allows us to know when we are facing an ideal uninfluenceable, unmanipulable learning process, and when we are not.
Furthermore, many learning processes cannot be easily made unmanipulable - especially those that involve human feedback, feedback conditional on the agent's actions. By specifying the ideal case, and seeing examples of what goes wrong in the non-ideal case, this can help develop learning processes where a small amount of manipulation is allowed, traded off against a large amount of desirable learning.
As a minor example of this, in the [next post](https://www.lesswrong.com/posts/upLot6eG8cbXdKiFS/reward-function-learning-the-learning-process), we'll see that though "uninfluenceable" is the ideal property, the weaker property of "unriggable" (which I previously called "unbiasable") is enough to get many of the desirable properties.
1 Framework and Examples
========================
1.1 Washing and Cooking and POMDPs
----------------------------------
The analysis will be illustrated by an ongoing example: that of a robot purchased to do domestic tasks.
The robot can specialise in cooking or washing (assume specialised robots are ultimately more effective than generalists). The robot has been turned on, but it has not yet been informed as to what its task is - it therefore has uncertainty about its reward function.
This robot will be modelled in the MDP ([Markov Decision Process](https://en.wikipedia.org/wiki/Markov_decision_process)) and POMDP ([Partially Observable Markov Decision Process](https://en.wikipedia.org/wiki/Partially_observable_Markov_decision_process)) formalisms. In these formalisms, the agent occupies a state .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-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')}
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@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')}
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@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')}
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@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-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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@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')}
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@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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@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')}
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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')}
a from the action set .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}
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.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}
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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-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. This transfers the agent to a new state, where it makes a new observation.
The Markov property is about how the transitions and observations are handled; basically these can depend on the previous state and/or action, but not on the those further in the past. If we 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}
@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}
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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')}
S, then we have three transition rules:
.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')}
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@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
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@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
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T takes the current state and the action and returns a distribution over the next state, .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}
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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}
.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')}
O takes the current state and returns a distribution over observations, 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}
@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')}
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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}
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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')}
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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')}
T0 is the distribution over the initial state where the agent begins. Both .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}
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.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}
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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}
.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}
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.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
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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-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')}
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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')}
S ignore any previous information.
The whole POMDP .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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μ.
For our example, the robot will be evolving in the following environment:
In this grid world, there are six grids the robot could occupy (the reason for this shape will be made clear later). There are two pizzas to cook in the top square, and one mud splatter to wash in the bottom one. The state space is therefore of size .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-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
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.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}
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.mjx-line-box-test {display: table!important}
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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-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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@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')}
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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')}
6×3×2=36, since there are .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}
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.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}
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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}
@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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@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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.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}
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.mjx-block {display: block}
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.mjx-cell {display: table-cell}
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37-th state, the 'episode ends' state.
In this case, there is no hidden information, so the set of states is the same as the set of observations, 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}
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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')}
@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')}
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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-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')}
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@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
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O is trivial. The robot always starts in the central position, and there are always .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}
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.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}
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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}
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.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
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.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}
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@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')}
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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')}
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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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@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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@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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1 mud splatter; this defines the starting state, with .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}
@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')}
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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}
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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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@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
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T0 being trivial and simply returning that starting state with certainty.
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A={N,E,S,W}, which involve moving in the four directions (staying put is not an action we'll need). If the robot can move into a square, it will. If it tries to move into a wall, it turns off and the episode ends (this avoids us having to give the robot an extra action to end the episode). If it is in in a room with a mud splatter or an uncooked pizza, then all pizzas in that room get cooked, or all mud splatters get washed. If the episode has ended, then it stays ended. This defines the transition 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}
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T, which is deterministic in this instance.
1.2 Rewards and reward functions
--------------------------------
Now we need to define the possible rewards of the robot. There are reward functions that map the robots's history to a numerical reward.
So what is a history? That is just a sequence of actions. The agent starts in initial state .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-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}
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.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
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.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}
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@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-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}
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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')}
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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}
@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')}
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@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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@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
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@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')}
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@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
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a2, and so on.
A history .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}
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.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
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.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
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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-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
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.mjx-vsize {width: 0}
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n o:
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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')}
n. For our purposes, we'll assume that the agent only operates for a finite number of steps: 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}
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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}
@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')}
Hm be the set of complete (full-length) histories, and 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}
@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')}
@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')}
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@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
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H=⋃mi=1Hi be the set of all histories. We might want to focus on the initial .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}
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i≤n.
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@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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@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')}
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@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
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[−1,1], the reward. 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}
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.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}
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.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}
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.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}
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@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')}
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@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-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
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@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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@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
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R∈R, and followed history .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')}
hm, then it would get total reward:
.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')}
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hm; first applying it to the first action and observation, then to the first two actions and observations, then to the first three... all the way to the full final history .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-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')}
hm.
In our ongoing example, we will consider two reward functions, .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-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')}
Rc also assigns a .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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−1/20 for each turn that the robot is active (ie the observation is not the end of episode state).
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Rw is the same, except it rewards washing mud-splatters, giving a reward of .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-block {display: block}
.mjx-span {display: inline}
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.mjx-itable {display: inline-table; width: auto}
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.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
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.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
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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')}
i mud-splatters. It also assigns .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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−1/20 for every turn of activity as well.
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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')}
π:H→A; 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}
@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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Π be the set of all policies.
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@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')}
Rc is to choose .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')}
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W to turn itself off. This gives it a total reward of .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-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}
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.mjx-annotation-xml {line-height: normal}
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.mjx-mtd {display: table-cell; text-align: center}
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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%}
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.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
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−1/20+2/2=0.95 (it gets no penalty on the second turn, because its second observation is the end of episode observation). The optimal policy 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}
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.mjx-op {display: block}
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.mjx-over {display: block}
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.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}
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.mjx-delim-h > .mjx-char {display: inline-block}
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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%}
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.mjx-vsize {width: 0}
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.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
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.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}
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.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}
@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}
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@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')}
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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-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')}
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@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
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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')}
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@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
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Rw is .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}
@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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@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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S, followed by any of .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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@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')}
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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')}
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@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-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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E, giving a total reward of .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-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-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
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.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
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−1/20+1/2=0.45.
1.3 Learning your reward
------------------------
We can now get to the key part of this post: learning the correct reward function. How could the agent do that? Well, the only data that it gets from outside is the observations; it also has a record of its actions. So it would seem that the only data that can determine whether a particular reward function is correct is the agent's history.
But there is arguably something else that can matter to the reward: the agent's policy. Suppose the learning process is defined so that, if the agent goes East, then it will see .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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R as the correct reward function. Then, arguably, on Bayesian grounds (see [next post](https://www.lesswrong.com/posts/upLot6eG8cbXdKiFS/reward-function-learning-the-learning-process)), if the agent has the *policy* of going East, it should already see .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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R as the correct reward function.
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ρ is defined as a function from histories and the agent's policy to a probability distribution over reward functions:
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ρ(R;π,h).
In our example, the value of the reward function is set by levers, levers that the robot itself can change. If the robot enters the leftmost box, the reward is set to Rw; if it enters the rightmost box, the reward is set to Rc instead. Before going into either of these boxes, it is uncertain between the two rewards.
This allows a definition of ρ(R;π,h), one that is independent of π. If h shows the agent was in the leftmost box more recently than the rightmost, then ρ(Rw;π,h)=1 and ρ(Rc;π,h)=0. If h shows the agent was in the rightmost box more recently than the leftmost, then ρ(Rc;π,h)=1 and ρ(Rw;π,h)=0. If the agent has been in neither box during history h, then ρ(Rc;π,h)=ρ(Rw;π,h)=1/2.
Now, this example makes ρ not feel like a learning process, but much more like an optimisation process with ρ being part of the reward. And that's precisely the problem; see the [next post](https://www.lesswrong.com/posts/upLot6eG8cbXdKiFS/reward-function-learning-the-learning-process) for desirable restrictions on ρ.
2 The value function
====================
2.1 The correct value function
------------------------------
The learning process and the reward functions are key elements, but how do we combine them into the value function - the estimate of the expected reward? If you get the value function wrong, then the agent may not be learning in the way you thought it would.
First of all, note that with agent's policy and the environment, we can compute the probability of a given history. Then
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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-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
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hn1, will then see history .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-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}
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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}
.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}
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.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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.MJXc-space3 {margin-left: .278em}
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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}
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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')}
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hn2 - ie that the first .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-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}
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.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
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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}
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@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')}
hn2n1=hn1). If that's the case, we write .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')}
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hn1≤hn2.
In pedagogy and in murder mysteries, one builds up to the final answer. But I'll short-circuit that process, and say that the correct value function for reward function learning is for an agent using policy .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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hm.
What motivates this formula? Well, the sum over R(hmi) is necessary if this is to be the value function of actual reward functions. Similarly, Pπ,μ and the sum over Hmis needed to make this into an expectation, and ρ is clearly in its right place, weighting the various possible rewards.
There are two choices that might be open to question, which are bolded in the above expression of V. The first is whether hm (the complete history) should be used for ρ; both hn (the current history) and hmi (the complete history that is known at the point the reward is assessed) are plausible candidates. The second questionable choice is the lower bound for the summation in i; rather than starting at 1, which is in the past for the current history hn, would it not be more suitable to start at i=n+1?
However, the substitutions hm→hn or i=1→i=n+1 will both result in value functions that are **inconsistent**. That is, an agent that attempts to maximise such value functions will wish that its future self not maximise them.
The substitution hm→hmi is not inconsistent, however. But it does result in a very weird and volatile agent, that continually learns and unlearns its reward functions, before doing any productive actions.
The rest of this post will demonstrate and illustrate these facts. It is not essential to understanding what's going on (though it can be instructive). Feel free to skip to the [next post](https://www.lesswrong.com/posts/upLot6eG8cbXdKiFS/reward-function-learning-the-learning-process) rather than reading the rest of this one.
2.2 Future-regarding inconsistency
----------------------------------
Let's deal with the i=1→i=n+1 substitution first, which has the *future-regarding* value-function:
Vf(μ,ρ,π,hn)=∑hm∈Hm∑R∈Rm∑i=n+1Pπ,μ(hm∣hn)ρ(R;π,hm)R(hmi).What is the optimal policy for this value at the start? It's to go North to cook the pizzas, then go South and East and East to push the lever over to Cook, and then turn itself off: N,S,E,E,E. Its final reward function would thus be Rc, and it would get a reward of 2/2=1 (for the pizzas), minus 4/20=1/5 (for each of the four turns where it doesn't reach an end of episode state), for a total reward of 0.8. No other policy gives it that much reward.
The problem is that after N,S, it no longer sees the interest in going East, because "learning" that Rc is correct only affects its *past* reward, which Vf no longer cares about. Instead it has two new optimal policies: either S,N,W,W (go South to wash, go North, go West to set reward function to Rw, end episode) or W,E,S,S (the same thing, but setting the reward function first). Both of these will give it an additional reward of 1/2−3/20=0.35, according to Vf of its current history.
Let's assume it attempts S,N,W,W; in that case, once it's done S, it no longer has any interest in changing the reward function (as that lies in the past), and will simply turn itself off with another S. Its final reward function will be 1/2Rc+1/2Rw (it's never pressed any of the levers, so ρ(h,π) remains the same as it was initially), and it gets a reward of (1+1/2)/2 (for cooking and washing), minus 3/20 (for taking four turns) for a total of 0.75−0.15=0.6.
If it decides instead to go W,E,S,S, then it will actually follow through with that policy, to disastrous effect: it will have a final reward function of Rw, getting 1/2 for one washing event, and −5/20 for taking six turns, for a total reward of 0.25.
Thus if the agent always follows the optimal policy according to Vf at the time, it will end up with a much lower reward than if it followed the policy that Vf initially optimal. It is not aligned with its future self.
But the agent could be more cunning: it could act to constrain its future self, knowing about their value difference. In that case, it should initially choose S; this washes the mud splatter, ensuring that its future self will not have any interest in Rw. Then it will choose N, and then E,E (because it knows that it won't care about setting the reward to Rc after cooking). Then it will simply follow W,W,N,N. Its final reward function is Rc, and its total reward is will be 1−7/20=0.65. This table gives the way the different policies are rewarded:
PolicyFinal Reward FunctionRewardN,S,E,E,ERc0.8N,S,S,S1/2Rc+1/2Rw0.6N,S,W,E,S,SRw0.25S,N,E,E,W,W,N,NRc0.652.3 Change-averse inconsistency
-------------------------------
Let's now look at the the hm→hn substitution. This results in the change-averse value function:
Va(μ,ρ,π,hn)=∑hm∈Hm∑R∈Rm∑i=1Pπ,μ(hm∣hn)ρ(R;π,hn)R(hmi).Why did I call this value function change-averse? Because it assess the value of its *future* reward according to its *current* estimate of the reward functions. We can illustrate this by moving the Cooking and Washing leavers into the same rooms as the pizzas and the mud splatters, respectively:
The optimal policy for Va, initially, is N,S,S,S (or S,N,N,N), which will allow it to cook and wash; under its initial reward function, 1/2Rc+1/2Rw giving it a total reward of (1+1/2)/2−3/20=0.6. However, as soon as it's done N, the ρ will change its reward function to being Rc, with certainty, and it will end the episode, choosing N (or E or W) again. According to its initial reward function, this gives it a reward of 1/2−1/20=0.45.
However, its final reward function is Rc, and, according to that, its final reward is 1−1/20=0.95. Thus unlike the Vf, its future version ends up more satisfied that its past version.
Doing both substitutions, i=1→i=n+1 **and** hm→hn, will result in the same problems as the example in this subsection.
2.4 The volatile learning agent
-------------------------------
If we do the substitution hm→hmi, then we get the (consistent) *volatile* value function:
Vv(μ,ρ,π,hn)=∑hm∈Hm∑R∈Rm∑i=1Pπ,μ(hm∣hn)ρ(R;π,hmi)R(hmi).In this instance, the expression Rρ,π(hmi)=∑R∈Rρ(R;π,hmi)R(hmi) is just a normal reward function itself. Therefore Vv is just the standard expected value function of the reward function Rρ,π, explaining why it is consistent (this also means that it's irrelevant whether i=1 is the bound or i=n+1 is, since normal reward functions don't care about past rewards).
But Rρ,π is a very peculiar reward function, even if ρ is independent of π (which then makes Rρ,π also independent of π). In this situation, the agent always wants to be maximising reward according to its current estimate of the correct reward. Or, conversely, it always wants to set its current estimate to what it can then easily maximise.
In our running example, one of the agent's optimal policy for Vv is first to go E,E, setting its reward to Rc. It then goes W,W,N, claiming a reward of 1, via Rc, for cooking the pizzas. It then goes S,W, setting its reward to Rw, and finally goes E,S,S, claiming the reward of 1/2, according to Rw, for washing, then ending the episode. This gives it a total reward of 1+1/2−7/20=1.05. The other optimal policy - W,E,S,N,E,E,W,W,N,N - gives the same reward.
Whatever we meant by a reward function learning agent, I think it's pretty clear that this agent, which jumps its reward function back and forth before taking actions, is not one of them. |
732b0dba-d02d-4c2c-8bb4-779eec6013c7 | StampyAI/alignment-research-dataset/arbital | Arbital | Report likelihoods, not p-values
This page advocates for a change in the way that statistics is done in standard scientific journals. The key idea is to report [likelihood functions](https://arbital.com/p/56s) instead of p-values, and this could have many benefits.
_(Note: This page is a personal [opinion page](https://arbital.com/p/60p).)_
[https://arbital.com/p/toc:](https://arbital.com/p/toc:)
## What's the difference?
The status quo across scientific journals is to test data for "[statistical significance](https://arbital.com/p/statistically_significant)" using functions such as [p-values](https://arbital.com/p/p_value). A p-value is a number calculated from a hypothesis (called the "null hypothesis"), an experiment, a result, and a [https://arbital.com/p/-summary_statistic](https://arbital.com/p/-summary_statistic). For example, if the null hypothesis is "this coin is fair," and the experiment is "flip it 6 times", and the result is HHHHHT, and the summary statistic is "the sequence has at least five H values," then the p-value is 0.11, which means "if the coin were fair, and we did this experiment a lot, then only 11% of the sequences generated would have at least five H values."%%note:This does _not_ mean that the coin is 89% likely to be biased! For example, if the only alternative is that the coin is biased towards tails, then HHHHHT is evidence that it's fair. This is a common source of confusion with p-values.%% If the p-value is lower than an arbitrary threshold (usually $p < 0.05$) then the result is called "statistically significant" and the null hypothesis is "rejected."
This page advocates that scientific articles should report _likelihood functions_ instead of p-values. A likelihood function for a piece of evidence $e$ is a function $\mathcal L$ which says, for each hypothesis $H$ in some set of hypotheses, the probability that $H$ assigned to $e$, written [$\mathcal L_e](https://arbital.com/p/51n).%%note: Many authors write $\mathcal L(H \mid e)$ instead. We think this is confusing, as then $\mathcal L(H \mid e) = \mathbb P(e \mid H),$ and it's hard enough for students of statistics to keep "probability of $H$ given $e$" and "probability of $e$ given $H$" straight as it is if the notation _isn't_ swapped around every so often.%% For example, if $e$ is "this coin, flipped 6 times, generated HHHHHT", and the set of hypotheses are $H_{0.25} =$ "the coin only produces heads 25% of the time" and $H_{0.5}$ = "the coin is fair", then $\mathcal L_e(H_{0.25})$ $=$ $0.25^5 \cdot 0.75$ $\approx 0.07\%$ and $\mathcal L_e(H_{0.5})$ $=$ $0.5^6$ $\approx 1.56\%,$ for a [likelihood ratio](https://arbital.com/p/1rq) of about $21 : 1$ in favor of the coin being fair (as opposed to biased 75% towards tails).
In fact, with a single likelihood function, we can report the amount of support $e$ gives to _every_ hypothesis $H_b$ of the form "the coin has bias $b$ towards heads":%note:To learn how this graph was generated, see [Bayes' rule: Functional form](https://arbital.com/p/1zj).%
Note that this likelihood function is _not_ telling us the probability that the coin is actually biased, it is _only_ telling us how much the evidence supports each hypothesis. For example, this graph says that HHHHHT provides about 3.8 times as much evidence for $H_{0.75}$ over $H_{0.5}$, and about 81 times as much evidence for $H_{0.75}$ over $H_{0.25}.$
Note also that the likelihood function doesn't necessarily contain the _right_ hypothesis; for example, the function above shows the support of $e$ for every possible bias on the coin, but it doesn't consider hypotheses like "the coin alternates between H and T". Likelihood functions, like p-values, are essentially a mere summary of the raw data — there is no substitute for the raw data when it comes to allowing people to test hypotheses that the original researchers did not consider. (In other words, even if you report likelihoods instead of p-values, it's still virtuous to share your raw data.)
Where p-values let you measure (roughly) how well the data supports a _single_ "null hypothesis", with an arbitrary 0.05 "not well enough" cutoff, the likelihood function shows the support of the evidence for lots and lots of different hypotheses at once, without any need for an arbitrary cutoff.
## Why report likelihoods instead of p-values?
__1. Likelihood functions are less arbitrary than p-values.__ To report a likelihood function, all you have to do is pick which hypothesis class to generate the likelihood function for. That's your only degree of freedom. This introduces one source of arbitrariness, and if someone wants to check some other hypothesis they still need access to the raw data, but it is better than the p-value case, where you only report a number for a single "null" hypothesis.
Furthermore, in the p-value case, you have to pick not only a null hypothesis but also an experiment and a summary statistic, and these degrees of freedom can have a huge impact on the final report. These extra degrees of freedom are both unnecessary ([to carry out a probabilistic update, all you need are your own personal beliefs and a likelihood function](https://arbital.com/p/1lz)) and exploitable, and empirically, they're actively harming scientific research.
__2. Reporting likelihoods would solve p-hacking.__ If you're using p-values, then you can game the statistics via your choice of experiment and summary statistics. In the example with the coin above, if you say your experiment and summary statistic are "flip the coin 6 times and count the number of heads" then the p-value of HHHHHT with respect to $H_{0.5}$ is 0.11, whereas if you say your experiment and summary statistic are "flip the coin until it comes up tails and count the number of heads" then the p-value of HHHHHT with respect to $H_{0.5}$ is 0.03, which is "significant." This is called "[p-hacking](https://arbital.com/p/https://en.wikipedia.org/wiki/Data_dredging)", and it's a serious problem in modern science.
In a likelihood function, _the amount of support an evidence gives to a hypothesis does not depend on which experiment the researcher had in mind._ Likelihood functions depend only on the data you actually saw, and the hypotheses you chose to report. The only way to cheat a likelihood function is to lie about the data you collected, or refuse to report likelihoods for a particular hypothesis.
If your paper fails to report likelihoods for some obvious hypotheses, then (a) that's precisely analogous to you choosing the wrong null hypothesis to consider; (b) it's just as easily noticeable as when your paper considers the wrong null hypothesis; and (c) it can be easily rectified given access to the raw data. By contrast, p-hacking can be subtle and hard to detect after the fact.
__3. Likelihood functions are very difficult to game.__ There is no analog of p-hacking for likelihood functions. This is a theorem of probability theory known as [https://arbital.com/p/-conservation_of_expected_evidence](https://arbital.com/p/-conservation_of_expected_evidence), which says that likelihood functions can't be gamed unless you're falsifying or omitting data (or screwing up the likelihood calculations).%note:Disclaimer: the theorem says likelihood functions can't be gamed, but we still shouldn't underestimate the guile of dishonest researchers struggling to make their results look important. Likelihood functions have not been put through the gauntlet of real scientific practice; p-values have. That said, when p-values were put through that gauntlet, they [failed](https://arbital.com/p/https://en.wikipedia.org/wiki/Data_dredging) in a [spectacular fashion](https://arbital.com/p/https://en.wikipedia.org/wiki/Replication_crisis). When rebuilding, it's probably better to start from foundations that provably cannot be gamed.%
__4. Likelihood functions would help stop the "vanishing effect sizes" phenomenon.__ The [decline effect](https://arbital.com/p/https://en.wikipedia.org/wiki/Decline_effect) occurs when studies which reject a null hypothesis $H_0$ have effect sizes that get smaller and smaller and smaller over time (the more someone tries to replicate the result). This is [usually evidence](https://arbital.com/p/http://slatestarcodex.com/2014/04/28/the-control-group-is-out-of-control/) that there is no actual effect, and that the initial "large effects" were a result of publication bias.
Likelihood functions help avoid the decline effect by _treating different effect sizes differently._ The likelihood function for coins of different biases shows that the evidence HHHHHT gives a different amount of support to $H_{0.52},$ $H_{0.61}$, and $H_{0.8}$ (which correspond to small, medium, and large effect sizes, respectively). If three different studies find low support for $H_{0.5},$ and one of them gives all of its support to the large effect, another gives all its support to the medium effect, and the third gives all of its support to the smallest effect, then likelihood functions reveal that something fishy is going on (because they're all peaked in different places).
If instead we only use p-values, and always decide whether or not to "keep" or "reject" the null hypothesis (without specifying how much support goes to different alternatives), then it's hard to notice that the studies are actually contradictory (and that something very fishy is going on). Instead, it's very tempting to exclaim "3 out of 3 studies reject $H_{0.5}$!" and move on.
__5. Likelihood functions would help stop publication bias.__ When using p-values, if the data yields a p-value of 0.11 using a null hypothesis $H_0$, the study is considered "insignificant," and many journals have a strong bias towards positive results. When reporting likelihood functions, there is no arbitrary "significance" threshold. A study that reports a relative likelihoods of $21 : 1$ in favor of $H_a$ vs $H_0,$ that's exactly the same _strength of evidence_ as a study that reports $21 : 1$ odds against $H_a$ vs $H_0.$ It's all just evidence, and it can all be added to the corpus, there's no arbitrary "significance" threshold.
__6. Likelihood functions make it trivially easy to combine studies.__ When combining studies that used p-values, researchers have to perform complex meta-analyses with dozens of parameters to tune, and they often find [exactly what they were expecting to find](https://arbital.com/p/https://en.wikipedia.org/wiki/Confirmation_bias). By contrast, the way you combine multiple studies that reported likelihood functions is... (drumroll)
...you just multiply the likelihood functions together. If study A reports that $H_{0.75}$ was favored over $H_{0.5}$ with a [https://arbital.com/p/-1rq](https://arbital.com/p/-1rq) of $3.8 : 1$, and study B reports that $H_{0.75}$ was favored over $H_{0.5}$ at $5 : 1$, then the combined likelihood functions of both studies favors $H_{0.75}$ over $H_{0.5}$ at $(3.8 \cdot 5) : 1$ $=$ $19 : 1.$
Want to combine a hundred studies on the same subject? Multiply a hundred functions together. Done. No parameter tuning, no degrees of freedom through which bias can be introduced — just multiply.
__7. Likelihood functions make it obvious when something has gone wrong.__ If, when you multiply all the likelihood functions together, _all_ hypotheses have extraordinarily low likelihoods, then something has gone wrong. Either a mistake has been made somewhere, or fraud has been committed, or the true hypothesis wasn't in the hypothesis class you're considering.
The actual hypothesis that explains all the data will have decently high likelihood across all the data. If none of the hypotheses fit that description, then either you aren't considering the right hypothesis yet, or some of the studies went wrong. (Try looking for one study that has a likelihood function _very very different_ from all the other studies, and investigate that one.)
Likelihood functions won't do your science for you — you still have to generate good hypotheses, and be honest in your data reporting — but they _do_ make it obvious when something went wrong. (Specifically, [each hypothesis can tell you how low its likelihood is expected to be on the data](https://arbital.com/p/227), and if _every_ hypothesis has a likelihood far lower than expected, then something's fishy.)
---
A scientific community using likelihood functions would produce scientific research that's easier to use. If everyone's reporting likelihood functions, then all you personally need to do in order to figure out what to believe is take your own personal (subjective) prior probabilities and multiply them by all the likelihood functions in order to get your own personal (subjective) posterior probabilities.
For example, let's say you personally think the coin is probably fair, with $10 : 1$ odds of being fair as opposed to 75% biased in favor of heads. Now let's say that study A reports a likelihood function which favors $H_{0.75}$ over $H_{0.5}$ with a likelihood ratio of $3.8 : 1.$, and study B reports a $5 : 1$ likelihood ratio in the same direction. Multiplying all these together, your personal posterior beliefs should be $19 : 10$ in favor of $H_{0.75}$ over $H_{0.5}$. This is simply [Bayes' rule](https://arbital.com/p/1lz). Reporting likelihoods instead of p-values lets science remain objective, while allowing everyone to find their own personal posterior probabilities via a simple application of Bayes' theorem.
## Why should we think this would work?
This may all sound too good to be true. Can one simple change really solve that many problems in modern science?
First of all, you can be assured that reporting likelihoods instead of p-values would not "solve" all the problems above, and it would surely not solve all problems with modern experimental science. Open access to raw data, preregistration of studies, a culture that rewards replication, and many other ideas are also crucial ingredients to a scientific community that zeroes in on truth.
However, reporting likelihoods would _help_ solve lots of different problems in modern experimental science. This may come as a surprise. Aren't likelihood functions just one more statistical technique, just another tool for the toolbox? Why should we think that one single tool can solve that many problems?
The reason lies in [https://arbital.com/p/-1bv](https://arbital.com/p/-1bv). According to the axioms of probability theory, there is only one good way to account for evidence when updating your beliefs, and that way is via likelihood functions. Any other method is subject to inconsistencies and pathologies, as per the [coherence theorems of probability theory](https://arbital.com/p/probability_coherence_theorems).
If you're manipulating equations like $2 + 2 = 4,$ and you're using methods that may or may not let you throw in an extra 3 on the right hand side (depending on the arithmetician's state of mind), then it's no surprise that you'll occasionally get yourself into trouble and deduce that $2 + 2 = 7.$ The laws of arithmetic show that there is only one correct set of tools for manipulating equations if you want to avoid inconsistency.
Similarly, the laws of probability theory show that there is only one correct set of tools for manipulating _uncertainty_ if you want to avoid inconsistency. According to [those rules](https://arbital.com/p/1lz), the right way to represent evidence is through likelihood functions.
These laws (and a solid understanding of them) are younger than the experimental science community, and the statistical tools of that community predate a modern understanding of probability theory. Thus, it makes a lot of sense that the existing literature uses different tools. However, now that humanity _does_ possess a solid understanding of probability theory, it should come as no surprise that many diverse pathologies in statistics can be cleaned up by switching to a policy of reporting likelihoods instead of p-values.
## What are the drawbacks?
The main drawback is inertia. Experimental science today reports p-values almost entirely across the board. Modern statistical toolsets have built-in support for p-values (and other related statistical tools) but very little support for reporting likelihood functions. Experimental scientists are trained mainly in [https://arbital.com/p/-frequentist_statistics](https://arbital.com/p/-frequentist_statistics), and thus most are much more familiar with p-value-type tools than likelihood-function-type tools. Making the switch would be painful.
Barring the switching costs, though, making the switch could well be a strict improvement over modern techniques, and would help solve some of the biggest [problems](https://arbital.com/p/http://slatestarcodex.com/2014/04/28/the-control-group-is-out-of-control/) [facing](https://arbital.com/p/http://www.stat.columbia.edu/~gelman/research/published/asa_pvalues.pdf) [science](https://arbital.com/p/https://en.wikipedia.org/wiki/Data_dredging) [today](https://arbital.com/p/https://en.wikipedia.org/wiki/Publication_bias).
See also the [Likelihoods not p-values FAQ](https://arbital.com/p/505) and [https://arbital.com/p/4xx](https://arbital.com/p/4xx). |
3ef2c67b-0f87-433d-89a4-077904529224 | trentmkelly/LessWrong-43k | LessWrong | Modal Bargaining Agents
Summary: Bargaining problems are interesting in the case of Löbian cooperation; Eliezer suggested a geometric algorithm for resolving bargaining conflicts by leaving the Pareto frontier, and this algorithm can be made into a modal agent, given an additional suggestion by Benja.
Bargaining and Fairness
When two agents can read each others' source code before playing a game, Löbian cooperation can transform many games into pure bargaining problems. Explicitly, the algorithm of "if I can prove you choose X, then I choose Y, otherwise I choose Z" makes it possible for an agent to offer a deal better for both parties than a Nash equilibrium, then default to that Nash equilibrium if the deal isn't verifiably taken. In the simple case where there is a unique Nash equilibrium, two agents with that sort of algorithm can effectively reduce the decision theory problem to a negotiation over which point on the Pareto frontier they will select.
However, if two agents insist on different deals (e.g. X insists on point D, Y insists on point A), then it falls through and we're back to the Nash equilibrium O. And if you try and patch this by offering several deals in sequence along the Pareto frontier, then a savvy opponent is just going to grab whichever of those is best for them. So we'd like both a notion of the fair outcome, and a way to still make deals with agents who disagree with us on fairness (without falling back to the Nash equilibrium, and without incentivizing them to play hardball with us).
Note that utility functions are only defined up to affine transformations, so any solution should be invariant under independent rescalings of the players' utilities. This requirement, plus a few others (winding up on the Pareto frontier, independence of irrelevant alternatives, and symmetry) are all satisfied by the Nash solution to the bargaining problem: choose the point on the Pareto frontier so that the area of the rectangle from O to that point is maximized.
This give |
479a10fc-f9c1-4f1e-ac11-1b5ef105e3c5 | trentmkelly/LessWrong-43k | LessWrong | Biohacking in New York, Cybernetics and first Cyborg Hate Crime: theverge.com
http://www.theverge.com/2012/8/8/3177438/cyborg-america-biohackers-grinders-body-hackers |
355182a3-296e-4e74-8a9b-c5a3b2ed10b8 | trentmkelly/LessWrong-43k | LessWrong | OpenPrinciples Bootcamp (Free) -- Reflect & Act on your Rationality Principles.
If you have seen OpenPrinciples Bootcamp's previous edit of the post, we have made a few changes:
1. We changed the Bootcamp from paid to free, because: 1) we found our price has greatly limited the quality and quantity of our applicants; 2) we introduced some experimental activities (which are experimental features in our Ultrabrain Principles Assistant) which needs more validation.
2. We have changed the program's level of commitment from 5hr/wk to 2.5 hr/wk manditory + 3hr/wk optional activities (1h/wk workshop + 1h/wk group discussion + speaker sessions + other activities) to give participants more flexibility and better-customed experience.
As rationalists, one of the most valuable results of our rational reflections is the principles we like to live by. Here I like to introduce you an opportunity to reflect and apply your principles with the accountability of principled coaches, and reminders from AI and human personal assistants.
If you have problems finding, remembering, or acting on your lessons / life principles / EA principles; or like to build your pipeline and the "second brain" system for storing, discovering, reflecting, strategizing, auto reminding, prompted measuring, iterating and open-sourcing your life principles, then here is an opportunity for you -- OpenPrinciples BootCamp, a free, 3wks, 2.5 hr/wk mandatory + 3hr/wk optional commitment, online program in America & Europe time, starting at Oct 1st.
The first round application deadline is Saturday Sep 17th 23:59 UTC, and the second round application deadline (with fewer spots) is Saturday Sep 24th 23:59 UTC, we do accept applications after that but will only have very limited spots. Apply here: bootcamp.openprinciples.org. If you are interested but the time does not work out for you, please fill out OpenPrinciples Bootcamp Waiting List instead.
This is the 2nd cohort of the program. After our previous 1st cohort program (called OpenPrinciples Fellowship), our attendees reported being 44 |
d99f14b7-8c97-45cb-b778-179579c18520 | StampyAI/alignment-research-dataset/eaforum | Effective Altruism Forum | Governments pose larger risks than corporations: a brief response to Grace
In her article [Counterarguments to the basic AI risk case](/posts/zoWypGfXLmYsDFivk/counterarguments-to-the-basic-ai-risk-case), Katja Grace argues that the track record of corporations is a reason to doubt what she presents as the basic argument for AI risk.
Corporations, however, are not the largest or most powerful human organisations. The governments of the USA and China are much larger and more powerful than any corporation on Earth. Just as we should expect the largest risks to come from the most powerful AI systems (or organisations of cooperating AIs), we should expect the most powerful human organisations to pose the largest risks.
Grace suggests that the argument for AI risk implies that we should consider human organisations to pose a substantial risk in one or both of the following ways:
> 1. **A corporation would destroy humanity rapidly**. This may be via ultra-powerful capabilities at e.g. technology design and strategic scheming, or through gaining such powers in an ‘intelligence explosion‘ (self-improvement cycle). Either of those things may happen either through exceptional heights of intelligence being reached or through highly destructive ideas being available to minds only mildly beyond our own.
> 2. **Superhuman AI would gradually come to control the future via accruing power and resources.** Power and resources would be more available to the corporation than to humans on average, because of the corporation having far greater intelligence.
>
Powerful governments have constructed large stockpiles of nuclear weapons that many believe poses a large risk to human flourishing (though the precise magnitude is controversial). Furthermore, there are many instances in which these weapons were apparently close to being used. Thus governments pose a substantial risk of bringing about a disaster similar to scenario 1 above, albeit not as severe.
There have been governments in our history which have seized a great deal of power and whose actions brought about great disasters for many people. No single governments has ever held power over everybody, and governments do not seem to have an infinite lifespan. In my view it's plausible (but not probable) that technological and economic changes could mean that neither of these trends holds in the future. Thus, governments also seem to pose some risk of bringing about a disaster similar to scenario 2.
I also think it's plausible that if corporations, not governments, were the most powerful human organisations then we might have seen similar actions from corporations. For example, governments would obviously not allow large corporations to maintain their own nuclear arsenals, and it is plausible that some corporations would maintain an arsenal if they were allowed. We could also speculate that the most powerful governments may also limit the power of any corporation that threatened to become a rival.
The track record of corporations on its own may seem to undermine the standard AI risk argument, but I think we should consider governments as well, and it is not clear whether the record of governments supports or undermines the argument. |
2e8acd5c-5af4-45cb-ba0e-f00408767059 | trentmkelly/LessWrong-43k | LessWrong | [SEQ RERUN] Sensual Experience
Today's post, Sensual Experience was originally published on 21 December 2008. A summary (taken from the LW wiki):
> Much of the anomie and disconnect in modern society can be attributed to our spending all day on tasks (like office work) that we didn't evolve to perform (unlike hunting and gathering on the savanna). Thus, many of the tasks we perform all day do not engage our senses - even the most realistic modern video game is not the same level of sensual experience as outrunning a real tiger on the real savanna. Even the best modern video game is low-bandwidth fun - a low-bandwidth connection to a relatively simple challenge, which doesn't fill our brains well as a result. But future entities could have different senses and higher-bandwidth connections to more complicated challenges, even if those challenges didn't exist on the savanna.
Discuss the post here (rather than in the comments to the original post).
This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was Complex Novelty, and you can use the sequence_reruns tag or rss feed to follow the rest of the series.
Sequence reruns are a community-driven effort. You can participate by re-reading the sequence post, discussing it here, posting the next day's sequence reruns post, or summarizing forthcoming articles on the wiki. Go here for more details, or to have meta discussions about the Rerunning the Sequences series. |
a950df58-3bd8-4d0f-b9da-abfdd3feaf67 | trentmkelly/LessWrong-43k | LessWrong | The convergent dynamic we missed
An excerpt from a longer post that I kept refining over the last 5 months.
By far the greatest danger of Artificial Intelligence is that people conclude too early that they understand it. Of course this problem is not limited to the field of AI.
Jacques Monod wrote: “A curious aspect of the theory of evolution is that everybody thinks he understands it”
— Yudkowsky, 2008
The convergence argument most commonly discussed is instrumental convergence: where machinery channels their optimisation through represented intermediate outcomes in order to be more likely to achieve any aimed-for outcomes later. [1]
Instrumental convergence results from internal optimisation:
– code components being optimised for (an expanding set of) explicit goals.
Instrumental convergence has a hidden complement: substrate-needs convergence.
Substrate-needs convergence results from external selection:
– all components being selected for (an expanding set of) implicit needs.
----------------------------------------
This will sound abstract. Bear with me. Let's take this from different angles:
AGI would be made up of a population of connected/nested components. This population changes as eg. hardware is modified and produced, or code is learned from inputs and copied onto the hardware.
AGI, as defined here, also has a general capacity to maintain own components.
Any physical component has a limited lifespan. Configurations erode in chaotic ways.
To realistically maintain components[2], AGI also must produce the replacement parts.
AGI's components are thus already interacting to bring about all the outside conditions and contexts needed to produce their own parts. Imagine all the subtle parallel conditions needed at mines, chemical plants, fab labs and assembly plants to produce hardware. All that would be handled by the machinery components of AGI.
So there is a changing population of components. And those connected components function in interactions to create the ambient |
4b4032bc-152e-4d53-b4b9-eb61b0777af6 | trentmkelly/LessWrong-43k | LessWrong | [SEQ RERUN] Intelligence in Economics
Today's post, Intelligence in Economics was originally published on 30 October 2008. A summary (taken from the LW wiki):
> There are a few connections between economics and intelligence, so economics might have something to contribute to a definition of intelligence.
Discuss the post here (rather than in the comments to the original post).
This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was Economic Definition of Intelligence, and you can use the sequence_reruns tag or rss feed to follow the rest of the series.
Sequence reruns are a community-driven effort. You can participate by re-reading the sequence post, discussing it here, posting the next day's sequence reruns post, or summarizing forthcoming articles on the wiki. Go here for more details, or to have meta discussions about the Rerunning the Sequences series. |
c8852794-3ca7-4727-aec2-4bc5897be88d | trentmkelly/LessWrong-43k | LessWrong | Alignment & Capabilities: what's the difference?
In the AI safety literature, AI alignment is often presented as conceptually distinct from capabilities. However, (1) the distinction seems somewhat fuzzy and (2) many techniques that are supposed to improve alignment also improve capabilities.
(1) The distinction is fuzzy because one common way of defining alignment is getting an AI system to do what the programmer or user intends. However, programmers intend for systems to be capable. eg we want chess systems to win at chess. So, a system that wins more is more intent aligned, and is also more capable.
(2) eg This Irving et al (2018) paper by a team at Open AI proposes debate as a way to improve safety and alignment, where alignment is defined as aligning with human goals. However, the debate also improved the accuracy of image classification in the paper, and therefore also improved capabilities.
Similarly, Reinforcement learning with human feedback was initially presented as an alignment strategy, but my loose impression is that it also made significant capabilities improvements. There are many other examples in the literature of alignment strategies also improving capabilities.
**
This makes me wonder whether alignment is actually more neglected that capabilities work. AI companies want to make aligned systems because they are more useful.
How do people see the difference between alignment and capabilities?
I also posted this on the EA Forum, but I'm curious to see what people here think. |
0b082b29-e8cb-43b2-806e-df33d7367ee9 | trentmkelly/LessWrong-43k | LessWrong | My research methodology
(Thanks to Ajeya Cotra, Nick Beckstead, and Jared Kaplan for helpful comments on a draft of this post.)
I really don’t want my AI to strategically deceive me and resist my attempts to correct its behavior. Let’s call an AI that does so egregiously misaligned (for the purpose of this post).
Most possible ML techniques for avoiding egregious misalignment depend on detailed facts about the space of possible models: what kind of thing do neural networks learn? how do they generalize? how do they change as we scale them up?
But I feel like we should be possible to avoid egregious misalignment regardless of how the empirical facts shake out--it should be possible to get a model we build to do at least roughly what we want. So I’m interested in trying to solve the problem in the worst case, i.e. to develop competitive ML algorithms for which we can’t tell any plausible story about how they lead to egregious misalignment.
This is a much higher bar for an algorithm to meet, so it may just be an impossible task. But if it’s possible, there are several ways in which it could actually be easier:
* We can potentially iterate much faster, since it’s often easier to think of a single story about how an algorithm can fail than it is to characterize its behavior in practice.
* We can spend a lot of our time working with simple or extreme toy cases that are easier to reason about, since our algorithm is supposed to work even in these cases.
* We can find algorithms that have a good chance of working in the future even if we don’t know what AI will look like or how quickly it will advance, since we’ve been thinking about a very wide range of possible failure cases.
I’d guess there’s a 25–50% chance that we can find an alignment strategy that looks like it works, in the sense that we can’t come up with a plausible story about how it leads to egregious misalignment. That’s a high enough probability that I’m very excited to gamble on it. Moreover, if it fails I think we’re likel |
9802e9f5-b48c-4cc3-a960-c277b1692941 | trentmkelly/LessWrong-43k | LessWrong | Moving away from physical continuity
In thought experiments about cloning objects, people often call the object which maintained physical continuity "original" and other "clone". This seems like just a naming, but when experiments talk about cloning a person, people tend to care where their original version is. This feels dodgy in cases when person doesn't even feel their continuity (in sleep, for example), so I think that is a bit overvalued.
Being one and the same
Let's take a black box cloning cars - perhaps taking the required set of details, correcting them on level of atoms to be indistinguishable from input car and constructing the second car. Inside the box, it is pretty obvious which car is original and which is a clone, but on the outside one can only learn that from box specification.
We can add an operation inside the box: let workers use a quantum RNG, and swap the result cars iff the random bit was 1. Then, person on the outside cannot tell which car is a clone. One could say "I don't know that, however there is 50% that cloned car is left one and 50% that it's right one instead"; I'd respond "the question is ill-formed since your definition of original doesn't really apply to this case".
I prefer to define being same by causal flows. For instance, let's open a window in the car before putting it into the black box and check which result car is causally downstream of input one - that is, has window opened. Once we see that both products have their windows opened, we can name both same[1], and same as the input.
1. ^
that said, in future the cars might continue to be approximately same or very similar, or might not if one gets in a crash; their evolutions are different because of the outer world (drivers, for instance) |
bcc7112f-7710-47ef-903c-90bfc993203c | trentmkelly/LessWrong-43k | LessWrong | Meetup : Austin, TX - HPMoR Wrap Party
Discussion article for the meetup : Austin, TX - HPMoR Wrap Party
WHEN: 14 March 2015 05:00:00PM (-0600)
WHERE: 4212 Hookbilled Kite Drive
The next meetup is March 14th, 2015 at 6:00PM at a private residence (newcomers, especially from places other than Austin, are welcome!). We'll be celebrating the end of HPMoR; there will be food of both the dinner and snack varieties, as well as a wealth of pies. (Feel free to bring food if you'd like, but we expect to have enough even if you don't.)
The topic of the March 21st meetup will be Fun and Games. (Suggest ideas for future topics!)
If you have any questions or want to get my phone number to make sure you find us, please send me a private message. I'll notice and respond to those much more quickly than I'll notice comments on this meetup announcement.
We have a Google Group here.
Discussion article for the meetup : Austin, TX - HPMoR Wrap Party |
8be264b6-b138-46cb-9f0e-dcb381c168ed | trentmkelly/LessWrong-43k | LessWrong | Consistently Inconsistent
Robert Kurzban's Why Everyone (Else) Is a Hypocrite: Evolution and the Modular Mind is a book about how our brains are composed of a variety of different, interacting systems. While that premise is hardly new, many of our intuitions are still grounded in the idea of a unified, non-compartmental self. Why Everyone (Else) Is a Hypocrite takes the modular view and systematically attacks a number of ideas based on the unified view, replacing them with a theory based on the modular view. It clarifies a number of issues previously discussed on Overcoming Bias and Less Wrong, and even debunks some outright fallacious theories that we on Less Wrong have implicitly accepted. It is quite possibly the best single book on psychology that I've read. In this posts and posts that follow, I will be summarizing some of its most important contributions.
Chapter 1: Consistently Inconsistent (available for free here) presents evidence of our brains being modular, and points out some implications of this.
As previously discussed, severing the connection between the two hemispheres of a person's brain causes some odd effects. Present the left hemisphere with a picture of a chicken claw, and the right with a picture of a wintry scene. Now show the patient an array of cards with pictures of objects on them, and ask them to point (with each hand) something related to what they saw. The hand controlled by the left hemisphere points to a chicken, the hand controlled by the right hemisphere points to a snow shovel. Fine so far.
But what happens when you ask the patient to explain why they pointed to those objects in particular? The left hemisphere is in control of the verbal apparatus. It knows that it saw a chicken claw, and it knows that it pointed at the picture of the chicken, and that the hand controlled by the other hemisphere pointed at the picture of a shovel. Asked to explain this, it comes up with the explanation that the shovel is for cleaning up after the chicken. While the righ |
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