Green Emeralds, Grue Diamonds
A putative new idea for AI control; index here.
When posing his “New Riddle of Induction”, Goodman introduced the concepts of “grue” and “bleen” to show some of the problems with the conventional understanding of induction.
I’ve somewhat modified those concepts. Let T be a set of intervals in time, and we’ll use the boolean X to designate the fact that the current time t belongs to T (with ¬X equivalent to t∉T). We’ll define an object to be:
Grue if it is green given X (ie whenever t∈T), and blue given ¬X (ie whenever t∈T).
Bleen if it is blue given X, and green given ¬X.
At this point, people are tempted to point out the ridiculousness of the concepts, dismissing them because of their strange disjunctive definitions. However, this doesn’t really solve the problem; if we take grue and bleen as fundamental concepts, then we have the disjunctively defined green and blue; an object is:
Green if it is grue given X, and bleen given ¬X.
Blue if it is bleen given X, and grue given ¬X.
Still, the categories green and blue are clearly more fundamental than grue and bleen. There must be something we can whack them with to get this—maybe Kolmogorov complexity or stuff like that? Sure someone on Earth could make a grue or bleen object (a screen with a timer, maybe?), but it would be completely artificial. Note that though grue and bleen are unnatural, “currently grue” (colour=green XOR ¬X) or “currently bleen” (colour=blue XOR ¬X) make perfect sense (though they require knowing X, an important point for later on).
But before that… are we so sure the grue and bleen categories are unnatural? Relative to what?
Welcome to Chiron Beta Prime
Chiron Beta Prime, apart from having its own issues with low-intelligence AIs, is noted for having many suns: one large sun that glows mainly in the blue spectrum, and multiple smaller ones glowing mainly in the green spectrum. They all emit in the totality of the spectrum, but they are stronger in those colours.
Because of the way the orbits are locked to each other, the green suns are always visible from everywhere. The blue sun rises and sets on a regular schedule; define T to be time when the blue sun is risen (so X=”Blue sun visible, some green suns visible” and ¬X=”Blue sun not visible, some green suns visible”).
Now “green” is a well defined concept in this world. Emeralds are green; they glow green under the green suns, and do the same when the blue sun is risen. “Blue” is also a well-defined concept. Sapphires are blue. They glow blue under the blue sun and continue to do so (albeit less intensely) when it is set.
But “grue” is also a well defined concept. Diamonds are grue. They glow green when the green suns are the only ones visible, but glow blue under the glare of the blue sun.
Green, blue, and grue (which we would insist on calling green, blue and white) are thus well understood and fundamental concepts, that people of this world use regularly to compactly convey useful information to each other. They match up easily to fundamental properties of the objects in question (eg frequency of light reflected).
Bleen, on the other hand—don’t be ridiculous. Sure, someone on Chiron Beta Prime could make a bleen object (a screen with a timer, maybe?), but it would be completely artificial.
In contrast, the inhabitants of Pholus Delta Secundus, who have a major green sun and many minor blue suns (coincidentally with exactly the same orbital cycles), feel that green, blue and bleen are the natural categories...
Natural relative to the (current) universe
We’ve shown that some categories that we see as disjunctive or artificial can seem perfectly natural and fundamental to beings in different circumstances. Here’s another example:
A philosopher proposes, as thought experiment, to define a certain concept for every object. It’s the weighted sum of the inverse of the height of an object (from the centre of the Earth), and its speed (squared, because why not?), and its temperature (but only on an “absolute” scale), and some complicated thing involving its composition and shape, and another term involving its composition only. And maybe we can add another piece for its total mass.
And then that philosopher proposes, to great derision, that this whole messy sum be given a single name, “Energy”, and that we start talking about it as if it was a single thing. Faced with such an artificially bizarre definition, sensible people who want to use induction properly have no choice… but to embrace energy as one of the fundamental useful facts of the universe.
What these example show is that green, blue, grue, bleen, and energy are not natural or non-natural categories in some abstract sense, but relative to the universe we inhabit. For instance, if we had some strange energy’ which used the inverse of the height cubed, then we’d have a useless category—unless we lived in five spacial dimensions.
You’re grue, what time is it?
So how can we say that green and blue are natural categories in our universe, while grue and bleen are not? A very valid explanation seems to be the dependence on X—on the time of day. In our earth, we can tell whether objects are green or blue without knowing anything about the time. Certainly we can get combined information about an object’s colour and the time of day (for instance by looking at emeralds out in the open). But we also expect to get information about the colour (by looking at an object in a lit basement) and the time (by looking at a clock). And we expect these pieces of information to be independent of each other.
In contrast, we never expect to get information about an object being currently grue or currently bleen without knowing the time (or the colour, for that matter). And information about the time can completely change our assessment as to whether an object is grue versus bleen. It would be a very contrived set of circumstances where we would be able to assert “I’m pretty sure that object is currently grue, but I have no idea about its colour or about the current time”.
Again, this is a feature of our world and the evidence we see in it, not some fundamental feature of the categories of grue and bleen. We just don’t generally seen green objects change into blue objects, nor do we typically learn about disjunctive statements of the type “colour=green XOR time=night” without learning about the colour and the time separately.
What about the grue objects on Chiron Beta Prime? There, people do see objects change colour regularly, and, upon investigation, they can detect whether an object is grue without knowing either the time or the apparent colour of the object. For instance, they know that diamond is grue, so they can detect some grue objects by a simple hardness test.
But what’s happening is that the Chiron Beta Primers have correctly identified a fundamental category—the one we call white, or, more technically “prone to reflect light both in the blue and green parts of the spectrum”—that has different features on their planet than on ours. From the macroscopic perspective, it’s as if we and they live in a different universe, hence grue means something to them and not to us. But the same laws of physics underlie both our worlds, so fundamentally the concepts converge—our white, their grue, mean the same things at the microscopic level.
Definitions open to manipulation
In the next post, I’ll look at whether we can formalise “expect independent information about colour and time”, and “we don’t expect change to the time information to change our colour assessment.”
But be warned. The naturalness of these categories is dependent on facts about the universe, and these facts could be changed. A demented human (or a powerful AI) could go through the universe, hiding everything in boxes, smashing clocks, and putting “current bleen detectors” all other the place, so that it suddenly becomes very easy to know statements like “colour=blue XOR time=night”, but very hard to know about colour (or time) independently from this. So it would be easy to say “this object is currently bleen”, but hard to say “this object is blue”. Thus the “natural” categories may be natural now, but this could well change, so we must have care when using these definitions to program an AI.
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There was a picture of a dress going viral where some people saw it as black in intense lighting and others as blue in dim lightning. Our sense of color is very far from the “apparent color”. A red object in a dark room is still red. Our brain calculates away lightning differences.
If there were local red torches on Chiron Beta Prime you could have objects that are very luminous near torches and objects that are not. Thus you could differentiate between diamons and turquoises. But diamons and turquoises are both grue. However turquoises are not white. Therefore grue is not white.
Note also that red torches could be a recent innovation. Thus what is natural “in this universe” is technology level dependant.
Yep. The true story is (as usual) more complicated. CBP grue is a collection of colour combinations that include white (and turquoise, and much else).
Irrelevantly to the contents of the post (which I vaguely agree with as similar to “the natural language to do induction in is the language of observation”), I have one very important thing to say:
There is no such thing as a green star.
Nearby planetary nebula, then ^_^ https://en.wikipedia.org/wiki/Green_star_(astronomy)
Excellent post. Basically simpler hypotheses are on average more probable than more complex ones, no matter how complexity is defined, as long as there is a minimum complexity and no maximum complexity. But some measures of simplicity are more useful than others, and this is determined by the world we live in; thus we learn by experience that mathematical simplicity is a better measure than “number of words it takes to describe the hypothesis,” even though both would work to some extent.
Ah, I see now that we said almost the same thing.
Which is the basic receipt of Solomonoff induction!
I would only say here that simplicity is determined by the agent evaluating it, and while most agent are determined by the world that they live in, some might not.
I’m going to raise an issue, and it could be fair to consider it a nitpick, but considering that you’re trying to be rigorous, perhaps it is okay to be unusually technical.
Blue and green are not natural categories, or at least they are as natural as “sour tasting” or “stinky”. To quote Bruce MacEvoy, “color is a complex judgment experienced as a sensation”; color is not an objective property of things in the world. When a human gazes at something, the color sensation they experience is highly dependent on all sorts of visual factors in the scene, and even depends on the memory and expectation of the human.
When I say, “that object is red”, I mean it as shorthand for “that object has a reflectance, transmittance, and emittance profile that usually leads humans to experience a red color sensation when viewing the object in neutral-ish conditions”. And let it be known that “red color sensation” and “neutral-ish conditions” are still massive shorthand. So really, “that object is red” is a statement concerning 1) the object 2) the human visual system 3) qualia 4) common viewing environments for humans.
I point this out because it seems wrong to try to do a bunch of rigorous thinking founded upon an extremely flawed example (flawed in the example is supposed to be about objective things, but is actually about subjective things). We even have Eliezer Yudkowsky talking about the nature of truth and using statements like “snow is white if and only if snow is white”. It might sound like he’s talking about facts, but imagine the analogous sentence “skunks are stinky if and only if skunks are stinky” or “chess is interesting if and only if chess is interesting”. Now it is more clear the statement is about subjective experiences and actually fails to have a definite truth value.
Would you be comfortable if your natural category example was whether some music was soothing, or whether some object was bitter? If you would not be comfortable with using those subjective examples, you should not be comfortable using color as an example. If you are comfortable with those examples, you can disregard the issue I raise.
I think a lot of your points still stand, but you’re taking unnecessary risks by using a flawed example.
You could mean that.
Or you could mean it as shorthand for “that object emits or reflects electromagnetic radiation with a pronounced peak around 700nm wavelength”.
Neat, I recognize your username. I always liked your choice of username, and I’ve often enjoyed your comments. Thanks.
Except that is not sufficient nor necessary to ensure that the object would typically generate a red color sensation in humans, even in “neutral or typical conditions”. So, I would not mean it as shorthand for that. Color sensations can not be boiled down to or predicted by spectral power distributions and reflectance profiles only.
I’m thinking your comment perhaps was a gentle nudge to say, “it’s not too hard to make color an objective creature”. Well, you can come up with some objective definition of whether some object has the property of redness, but you’d have to basically reimplement the human visual system and assume a huge amount about the object’s current surroundings (or you could not go to that effort and end up with something that does a very poor job of corresponding to human color sensations). It would be similar to converting bitterness of food or soothingness of music into objective properties. Or maybe your comment was a gentle nudge in a completely different direction.
Thanks :-)
You are now dealing in circular logic. If you criterion for “red” is a “color sensation in humans”, you have already defined red. That’s it, we’re done. My point is that you do not have to define “red” in terms of human qualia.
Definitions should be judged by their usefulness. Sometimes “a human would call that red” is the right defintion, sometimes “this light peaks at ~700nm” is the right definition. For example, if the sensor in your telescope captured a few photons from a dim distant star, you might call them “red” even if the human visual system will be unable to process these photons (or associate with them the qualia of “red”).
It is easy to make color an objective creature—just define it a particular mixture of wavelengths of visible light. To produce a workable definition of subjective color is much harder—this is a practical matter in photography and graphics and whole books are written on the subject.
We would run into the same problem for any description of a quale/sensation. For example, we would describe/identify nausea, bitterness, and redness in similar ways—it’s hard to directly describe sensations, so we often indirectly identify sensations by pointing to conditions that lead to humans experiencing the sensation, or pointing to how the sensation relates to other sensations. That’s the unfortunate business of qualia/sensations.
So, the criterion isn’t circular, it’s just unsatisfying in that we basically end up saying, “in situation X, you will probably feel the sensation I’m talking about, and we’ve labeled that sensation ‘red’”.
Agree 100%. Sometimes you can pretend that color is a simple objective creature and it turn out okay, just like we can use Newtonian physics when relativistic effects are small enough for our purpose.
As I said before, you CAN come up with a simple objective definition of color, but it will do a “very poor job of corresponding to human color sensations”.
A particular spectral power distribution can lead to many different color sensation depending on visual context, and even the expectation and memory of the human experiencing the color sensation. This fact makes that sort of definition unfit for a lot of purposes.
Look at these two scenes. The left-hand scene contains “blue” pixels that use the same RGB value (0x6e6f73) as the right-hand scene’s “yellow” pixels. And in the context of the legend at top, that RGB value produces a third color sensation: gray.
So, any definition of color that only depends on spectral power distributions will not correspond very well with actual human color sensations. The linked picture demonstrates that “the light mixture your monitor produces for an RGB value of 0x6e6f73” is nowhere near enough information to predict what color sensation a human will experience from viewing pixels with that RGB value, even within the limited range of conditions of looking at something on a monitor.
Also, the two-scene picture is not an unusual case. The highly contextual nature of color perception is ubiquitous. Human color processing is always making contextual adjustments from scene illumination. The picture of a fruit basket in this section does a good job of showing how contextual adjustments are the norm. The overwhelming importance of context in color perception massively shrinks the situations where simple objective definitions of color are useful. Treating color as a simple objective creature gets you into trouble fairly quickly.
Yep, which is why I urged the author of the post to choose something other than color as an example of a simple/natural category.
Also, don’t you mean objective? The color model work you’re talking about is an effort to come up with objective mathematical models that exist outside of human minds (thus considered objective) that give outputs that correspond to sensations that exist inside human minds (those sensations being subjective). I don’t want us to get hung up on what “objective” and “subjective” means, but if this conversation continues much more, it might be good to spend a bit of time making sure we understand each other when we use those words.
Only if you decide you’re defining a sensation and not some physical phenomenon.
Yes, I understand that very well. But all that tells you is that different definitions will diverge in many cases.
“Subjective” was probably the wrong word. I distinguish:
A physical approach which defines color through spectral power distributions
A human objective approach which defines color via the tristimulus model (the CIE color space, etc.)
A human subjective approach which defines color as a particular perception
The human subjective approach has—as you have pointed out—all the issues associated with talking about subjective sensations, that is, they are essentially unobservable and it’s very hard to get a good handle on them. That, to me, makes defining color through qualia a definition that isn’t useful all that often.
That’s the definition used in the overwhelming majority of cases. Careful, technical texts often make it clear that color is a sensation. Even Isaac Newton stressed that “the rays [of light] are not colored”.
Even wikipedia goes with the sensation definition of color: “Color...is the visual perceptual property corresponding in humans to the categories called red, blue, yellow, etc...The color of an object depends on both the physics of the object in its environment and the characteristics of the perceiving eye and brain.”
In everyday use, when a person says things like “hand me the blue towel”, that person usually does not care, know, or even think about reflectance profiles and spectral power distributions. Usually all that person cares about is that the towel “looks blue” to him and the person he’s talking to. He’ll say “that towel is blue” just like he’ll say “that chocolate is bitter”.
It’s very useful to have definitions that depend on human sensations. You and I are both humans, and we often have conversations with other humans.
I do not believe that to be so. An example: all color management in digital photography. Another example: color swatches (e.g. Pantone).
Cheers! I used blue and green because the grue and bleen example is a standard philosophical one, and it’s not hard to make pretty rigorous (as Lumifer suggested, with the radiation frequency, and some outside conditions added to it). The pretty rigorous definition is only a partial match for the subjective blue and green, as you pointed out, and I’ll try and make that clearer in any subsequent write-up.
Taking “outside conditions” into account to produce an objective definition of color that does a good job of corresponding to human color sensations is actually extremely complex and a very difficult task. Human color sensations are the result of extremely complex and highly contextual processing. I have studied color vision a great deal, and it is very, very common for people to underestimate the complexity and contextual nature of color perception.
Also, you’re already conceding that color is not a property of a single object, which would make color a poor example of a property of an object.
Anyway, I’ll take your response as a sign that you are comfortable with the problematic nature of your example, and the more pressing concern is playing nice with philosophical tradition/convention. So, I consider the issue closed.
This reminds me a bit of Solomonoff induction.
The prior stays universal (dominant) even if you switch an initial finite segment, corresponding to the switching of the base universal Turing machine used to encode the Kolmogorov complexity.
Here naturality to me seems connected to the Kolmogorov complexity, as induced on the agent evaluating it by its environment. But what’s important is not the environment, but the agent. If an agent had an internal lamp that switched color every now and then, for that agent grue/bleen would be the most natural (i.e. least complex) choices.
This is exactly how I responded to the problem of grue when hearing about it. I don’t see your post as invalidating that. Here’s why (and this may be equivalent to your own answer, I don’t know): you need to calculate the additional Kolmogorov complexity of concept X+”everything else you knew” over “everything else you know”.
For a simple example, if I see just the cover of a known book, considering the Kolmogorov complexity of the book in isolation should lead me to conclude that the inside of the book doesn’t exist. Surely the inside of the book containing a whole lot of data has far greater Kolmogorov complexity than an empty book with the same cover? The obvious answer is that once you include the rest of your knowledge, the marginal added complexity by assuming this book “matches” the cover is less than assuming it doesn’t.
In the same way, in our world, if something has attributes that fit both grue and green, the overall complexity of grue+”all my knowledge” will be greater than overall complexity of green+”all my knowledge”. Conversely, if “changing when the sun is out” is a real possibility, that compresses the space needed to express the grue concept, and lowers the marginal complexity.
You seem to be using “ease of learning X” as some kind of proxy for actual Kolmogorov complexity, or something.
Yep. I didn’t go for Kolmogorov complexity because I had another mathematical definition I wanted to try out: http://lesswrong.com/r/discussion/lw/mbr/grue_bleen_and_natural_categories/
“So the naturalness of the category depended on the type of evidence we expected to find.”
Reminds me of the No Free Lunch theorems of David Wolpert, particularly the basic NFL for learning algorithms. Basically, there is no a priori distinction between learning algorithms, and the actual performance is determined by the “fit” of the algorithm to the problem.