And I find it ironic that in this discussion of choosing words carefully, we still use the basically meaningless (because of all of the different ways to use them) words "conservative" and "liberal". The words have nonpolitical meanings, and political meanings. And even the political meanings are inconsistent.
Love Rust, but there's room for garbage collection over borrowchecking if we're talking about replacing scripting languages. It would be hard for a decent statically-typed FP language to be slower than Ruby, for example.
While I found your comment funny, I could stand to be a bit more charitable. I think it's an interesting side point they're trying to make about analog computation, but to be accurate should maybe be written more like "A theoretical perfectly-machined set of gears could perform calculations to infinite precision."
I don't know how to be charitable here, the side note is completely wrong, and I'm not sure what interesting point could be made here.
A theoretical perfectly-machined set of gears is a mathematical construction, not a physical construction. It seems reasonable to me that a mathematical construction could have infinite precision, but it is obvious that a physical device has limited precision. An analog computer is a physical device. We might as well pretend that a digital computer has infinite RAM and an infinitely fast CPU, if we are in the business of assuming that an analogue computer is infinitely precise.
In general you can make many of the same engineering tradeoffs with simple analogue computers. A cam wheel can encode a function, and by adjusting the size, manufacturing tolerances, and materials you can control the amount of error that the cam wheel gives you when calculating the function. Similarly, with a digital computer you can adjust the number of digits used to represent numbers, the size of lookup tables, the number of iteration for iterative methods, etc. and control the amount of precision a digital computer provides.
Everywhere I've lived with this "service", there's been no respect for potted plants either. They get the full violence of the wind.
There was even a group of these guys who came straight up to my door, open except for the screen, and blew right into the house. They arrived early and started blowing in that very spot, which is why I hadn't had time to close the door. Just jerks.
I agree it's better to let them decay in lots of cases.
Also, I try to give the benefit of the doubt to the people outside doing this every week, but I'm extremely annoyed. I don't really have a choice; I rent and everywhere one can rent, one has to be a part of this madness. I wouldn't by choice pay someone to do this.
Some trees (walnut) have poisonous leaves that can't be left to decay in place and must be treated and removed, but this is the best option in the majority of cases, yes.
I'm just wondering.. poisonous to who/what? Kids? Pets? How long are they poisonous for while decaying?
How long have people been 'treating and removing' them? Presumably the leaves were tolerable while on the tree, yet poisonous. It doesn't seem to stop people planting them. (A lot of questions, but there are a lot of experts on here!)
Normally to other plants, rarely for animals (see: poison ivy, poison oak and poison sumac). In really extreme cases can be very dangerous even to burn it (but nobody has a manchineel in their garden). Walnut can poison the soil for a couple of years so is better put its fallen leaves on a isolated compost pile. It depends on the species and the chemistry of the soil.
"Juglone is occasionally used as a herbicide. Traditionally, ...has been used as a natural dye for clothing and fabrics...and as ink. ...has also found use as a coloring agent for foods and...hair dyes. ...is currently being studied for its anticancer properties"
Except when the leaves form a coating over grass that grows mold, and you are allergic to mold, which will also kill the grass. You don't have to remove every last leaf, just enough to give the grass breathing room.
In fact I just spent six hours today doing that... with a rake. We had a window where the snow cover lifted just long enough. It'll rain tomorrow, then freeze and snow again later next week.
Hey, non-mathematician computer science type here.
If I follow correctly, the issue with randomly picking any real number in that interval is that irrational numbers would require infinite computational steps to resolve. So the probability is really 0 that you'll get an irrational. If you have a finite number of computations, you're guaranteed to resolve to a rational, while if you have an infinite number of computations, you never resolve to anything.
Uniformly at random picking a number from the interval [0, 1] isn't possible with a turing machine (even giving it access to random coins). I.e. it's not a computable function.
It doesn't even really make sense, you can't represent uncountably many numbers on a turing machine, so it isn't even possible to return all but a tiny subset of the space.
You're imagining some turing machine that attempts to compute it anyways and thinking about the output. You seem to think that you can make a turing machine that
- In the probability 0 case that we should output a rational, will output that number
- Will otherwise infinite loop
This is randomized, so we are getting our randomness from some kind of "coin flip" like process. To know that we are in the that probability 0 case of outputting a rational, we will need to have seen infinitely many coin flips. If we've seen only n coin flips, there is still 1/2^n > 0 of the probability space that we haven't explored. So in fact any such turing machine has to loop infinitely in the rational case as well.
I'd say it's perfectly possible for a Turing machine + a source of randomness to generate a random real number in [0, 1). We just lazily generate more and more digits to the required precision. It's no different from, say, pi being computable.
We can even prove that there's zero probability that a rational number is generated: the decimal representation of a rational always has a repeating group of digits, but since each digit is generated randomly with 100% probability there will be no periodic pattern.
Computers are not type 2 turing machines, nor are any other physically existing thing that we know of. They aren't really turing machines either because they have a finite tape, but since we are only interested in running the turing machine for a finite amount of time and thus accessing a finite amount of tape that distinction is unimportant.
The standard definition of computable is on a turing machine, not a type 2 turing machine. Of course we can define an alternate model where more things are computable. Edit: And the standard definition of computable is relevant because it happens to be the exact set of functions we can compute on real computers.
While Weihrauch [0] does introduce a different definition of the word computable, that would in a randomized setting allow for sampling from the interval [0, 1] (and not just for rationals as I understand it either). Any algorithm on his "oracle turing machines" will still have to take an infinite amount of time, even to return the rationals. He just allows that in his definition of computable.
Type 2 Turing Machines are a conservative model of computation. They are as realistic as Type 1 Machines. Both models involve an infinite tape. Your previous comment used the Turing Machine abstraction, so my response was entirely valid, suggesting that replacing your abstraction with another equally valid, and I believe more appropriate for this purpose, abstraction eliminates the problem.
Also, by the way, the distinction between "complete" and "potential" infinity is useful here. The Type 2 Turing Machines features only a "potential" infinity, the same type that's common throughout Theoretical Computer Science. A real number is a only a "potential" infinity -- a process of sorts. You seem to be demanding that a real number be represented as a "complete" infinity, but this isn't needed for anything in physics or engineering or anything else. The demand you're making, which would imply that an infinite amount of time is needed, is unreasonable.
And by the way, the downvoter is somebody who can't argue with facts.
I suppose the argument you are trying to make here is basically "having a machine that if we run it for long enough, will tell us any particular digit of a number, is as good as having that number".
In the concrete example, you would argue that if a machine which after n time steps specifies which 1/2^n sized interval the randomly generated number lies in, then having that machine is equivalent to having the randomly generated number.
I disagree. If I come up with any property of the number that requires seeing arbitrarily many digits to specify (e.g. "is rational", "contains more 1s than 0s", etc) you can never tell me whether or not the number your machine specifies has that property. That said, I can see where you are coming from. There is at least some argument that under this model it's not the number which can't be computed, but the "is rational" function.
Personally, I wouldn't worry about the downvotes. Internet points aren't important in life anyways.
A program isn't a black box. The number specified (in binary) by this program
emit "0."
loop forever:
emit "110"
is both rational and contains more 1s than 0s. You no more need "run the program forever" to determine these of the number it presents than you need to perform an "infinite amount of long division" to determine it of the number presented by 6/7.
So? That's no different than a number specified in a "conventional" way, ie. by some mathematical formula in propositional logic. Let x be 0 is P if true and pi otherwise.
The conventional way of knowing a number is specifying it in a way that we can quickly determine what it is and operate on it.
If I say "the next prime after 9^9^9^9^9^9^9^9^9", or indeed "the next prime after busy beaver(1000)" I have specified a precise number. But you don't think I have it in any useful sense, because I can't compute it quickly (or in my second example at all).
Edit: And it should be noted that the above is more akin to the busy beaver example, no matter how long you operate that turing machine, if M' happens to be of the sort that doesn't halt but doesn't provably not halt, then you will never be able to tell me whether the number I "have" is 0 or pi.
But you haven't cleared anything up at all! What do you mean "determine what it is"? Do you mean compute its digits? Can you have an irrational number? a rational number with a non-finite decimal expansion? And what do you mean "operate on it"? By which operation? And what do you mean "quickly"?
In any case, the relevant program (assuming a fast random oracle)
emit "0."
loop forever:
x := query random oracle for one bit
emit x
seems to fit all your criterion. You can compute as many digits as you like very quickly. If you can have pi I don't see why you can't have this number (if you can have any random number at all).
We have branched into two different discussions it seems.
Does having a turing machine that will eventually output a number x suffice for having the number x.
And does that specific turing machine suffice for having a random number in range [0, 1].
As for the second, I have misgivvings about it but there's certainly an argument that that machine does work. The argument against that I currently find most convincing (that you don't have a classical representation that you can make two isolated copies of) is outlined here: https://news.ycombinator.com/item?id=18424725
The first discussion is more nuanced. The meaning of knowing/having a number is not something I claim to have an exact answer to (and indeed since it's an english term it probably doesn't have an exact meaning). It's clear however that having a turing machine that eventually outputs the value is not the same as knowing the value.
What operations is really the most fundamental question, I'd argue that it's clear that if you know x you can at least tell me what the n^th digit of x is. An arbitrary turing machine fails this, an arbitrary turing machine fails being able to tell me what the first digit is, let alone the n^th one.
I'm not going to take the stance that being able to compute its digits quickly is sufficient for having a real number, but I also won't argue against it. Personally I'd like to ask for equality testing too. I'm willing to yield that since I'm pretty sure that man-and-laptop will disagree and argue that being able to test for closeness is good enough, and he has a point even if I'm not convinced.
As for your the other questions you put to me, they aren't relevant for the turing machine part but:
Quickly, means in at most polynomial time. It might actually mean something stricter, but polynomial time seems to be a pretty clear upper bound on the amount of computation time you can need to find something while still being able to claim to know it. (See the paper I linked above)
We can certainly have a rational number with a non finite decimal expansion, you just need more creative forms of representing the numbers than listing digits. For instance the common method of putting a bar on top of the repeated ones, or alternatively quote notation is rather cool: https://en.wikipedia.org/wiki/Quote_notation
> I'd argue that it's clear that if you know x you can at least tell me what the n^th digit of x is. An arbitrary turing machine fails this, an arbitrary turing machine fails being able to tell me what the first digit is, let alone the n^th one.
Sound good (assuming we accept the 999... problem of non-uniqueness). So let's assume the machine makes progress in finite time ie. there is a sequence of natural numbers a(n) such that after time a(n) the machine has emitted at least n digits.
There's another possibility you mentioned above about a machine taking an input n and finding a value within 2^(-n) of the number. A machine that keeps emitting numbers on either side of an integer, eg. 2.1, 1.99, 2.00001, etc. fails to tell the first digit. But these numbers are arguably even more physical that programs-emitting-digits. They're roughly the real numbers you'd get from doing actual (classical) physical measurements.
> Personally I'd like to ask for equality testing too.
You didn't answer the question about irrational numbers but do you think we can have pi? It seems infeasible to mechanically determine that an arbitrary program you are given is a valid pi-digit-calculator though.
If you don't think you can ever have irrational numbers, then I think I see where you're coming from now. Having a number could be having a finite representation of the number that can be mechanically tested for equality (in time polynomial in the sizes of representations): a string of digits, a ratio of strings of digits, a string of digits with decimal point and a bar over the repeated ones, etc. IOW a normal form in some finitary data type.
> Quickly, means in at most polynomial time.
In what? You want the nth digit printed by time O(P(n))? If so, that is strictly stronger than finite time progress so we could dispense with that. But polynomial time doesn't make any sense for a machine that emits a finite string of digits and halts because it doesn't have an input.
> or alternatively quote notation is rather cool
Hah, I was going to ask about p-adics (do I have -1 if I have the machine that emits an infinite string of 1s?).
I'm really not satisfied with saying that a machine that gives smaller intervals like that is fully sufficient, on the other hand that's really all we're doing when we specify digits...
> They're roughly the real numbers you'd get from doing actual (classical) physical measurements.
I'd argue that a series of physical measurements don't give you a number so much as a probability distribution, even classically.
> do you think we can have pi
I'm not sure.
The problem I have with denying pi is it doesn't make much more sense than denying 1. Base pi is a perfectly rational numbering system. It's perfectly possible to introduce a special 'pi' symbol (rather like 'i') and define rules of arithmetic so that things work with both rationals and rational multiples of pi. And everything I said just applies to infinitely many other irrationals as well (e.g. 2^(1/n))
> It seems infeasible to mechanically determine that an arbitrary program you are given is a valid pi-digit-calculator though.
Indeed, this is true even if you replace "pi" with "1" though, that's another reason why having a program that calculates the digits isn't sufficient.
> Having a number could be having a finite representation of the number that can be mechanically tested for equality (in time polynomial in the sizes of representations)
Yes.
> In what? You want the nth digit printed by time O(P(n))?
It's a vague definition anyways, polynomial in the sum of everything that is relevant... probably O(P(n + the number represented)).
It seems to me that we should have a different measure than lebesgue measure when we speak of picking numbers from an interval.
I wonder if there is a measure under which computable numbers have measure 1 while the others have measure 0. Would it have all the correct properties?
All this becomes pretty intuitive if you think about numbers in terms of their decimal expansions. A "random number" is one with a random digit in each of its (infinitely long) list of decimal digits. A rational number is one where, at some finite point, all of the digits start to repeat in some finite pattern. The odds of that happening by chance are zero.
Likewise, if you take any two randomly generated list of decimal digits, at some point there will be a digit in the same place that is different between the two. At that point, you can construct a rational between the two by choosing the smaller digit and then adding "1000....".
Can you clarify that last point? I don't think it's strictly true. Aside from countable and uncountable infinities, you can have larger and smaller infinites as well. Unless every set S of all rational numbers between any irrational x and irrational y is isomorphic to the set P of all rational numbers, I don't see that this is correct. And I don't immediately see that you can put them into 1-1 correspondence.
> If I follow correctly, the issue with randomly picking any real number in that interval is that irrational numbers would require infinite computational steps to resolve.
In mathematics, doing things an infinite number of times is generally no problem, in fact it’s almost always done.
Of course, there are also many different sizes of infinity, and doing things a larger infinite size number of times requires some extra steps... but most branches of math only do things a countable infinite number of times (the smallest infinity).
"Rational" numbers are those that can be expressed as a fraction of two integers. For example, a square root of two is not rational.
"Computable" numbers are those that can be calculated by a computer (with unlimited memory, but finite speed) with arbitrary (not infinite) precision in finite time. As a rule of thumb, anything you can express using words like "plus", "minus", "square root", "logarithm" etc. is going to be computable.
> irrational numbers would require infinite computational steps to resolve
No, this is not the real reason. Both "1/3" and "square root of 2" have infinite number of decimal places, so neither can be fully written by a computer program. However, each of them can be approximated to e.g. one billion decimal places.
To show you how most numbers are not rational, consider only numbers of form "A + B * square root of two", where A and B are rational. Each different pair of A and B gives you a unique number. Among them, the numbers with B = zero are rational, and numbers with B <> zero are irrational. (Furthermore, all rational numbers can be expressed like this, but many irrational numbers, such as "square root of three" are outside of this set.) This should make it intuitively obvious why a randomly picked number is infinitely unlikely to be rational.
But the rabbit hole goes much deeper.
Let's ignore all technical details, and take a set of all numbers you can describe (unambiguously, and without any paradox) by a sentence of a finite length (including any finite number of equations of a finite length). If you need a whole book to define a number, so be it. Take all numbers humans can describe.
The point is that all these numbers are just an infinite minority in the vast ocean of real numbers.
How can that be? What else is missing? This seems tricky, because -- by definition -- I should be unable to give you an example of a number that cannot be described. But even if I can't give you a specific number, I can point you towards a concept: the numbers whose decimal digits are all randomly generated.
Any number that can be described by a finite description, contains a finite amount of information. A number with infinitely many randomly generated digits contains an infinite amount of information. To put it differently, when you generate numbers with infinite number of random decimal digits, you would have to be infinitely lucky to generate a number which only happens to contain a finite amount of information (for example, when all randomly generated digits happen to be zeroes). But the former are the real numbers, and the latter are the computable numbers. So if you take a random real number, you have to be infinitely lucky to get a computable one.
There is actually more, because "infinitely more" does not adequately describe the difference between comparing "rational" with "irrational, but still computable" numbers (both infinities have the same cardinality), and "computable" with "real" numbers (infinities of a different cardinality)... but this part, I am afraid, is beyond lay interpretations and requires paying attention to some technical definitions. A simple version is that the rational numbers and the computable numbers can both be numbered by integers, if we choose a sufficiently smart numbering scheme; but for real numbers, even this is impossible. But it takes some technical details to explain why it is so, and why it matters so much.
Exactly. I've used all 3 major OSs and when we get into these discussions, you have to look at how each has its own pain points to be fair. Overall, Linux is a clear win for me. But YMMV, indeed.
It's a big problem though that when pain points are brought up the response is often to just point at one of the other OSs's pain points and dismiss the criticism. Unfortunately we can do very little to make Windows or MacOS better other than complain to the deaf ears of the companies that own them, but theoretically Linux Desktop could be better if complaints weren't simply dismissed in this fashion.
Sure, maybe, if you have the time and expertise to fix those flaws. But if the community doesn't adopt your fixes then you're stuck maintaining them yourself forever.
Over the years I've seen several solutions to my personal gripes with Linux Desktop, and they are routinely ignored or rejected because the culture as a whole just doesn't seem to actually care about those problems, or doesn't even believe they are problems.
I haven't had the same experience. Which distros have you tried? Perhaps another would be a better fit technically or community-wise. I'd recommend Manjaro for a rolling release (no major reinstall every year), a focus on user-friendliness, a friendly community, as well as being based on Arch, which means the platinum-standard Arch Wiki is available as documentation for you.
I've tried many. The problems I have with the Linux Desktop are not solved by any of them, though Nitrux at least tries to deal with one, albeit poorly. This is because the problems I have are pretty much endemic to the thought process of people who use and develop for UNIX systems. Things like hardcoded paths, the weird file hierarchy (please don't try and "explain" it, I know how it works and the reasoning behind it, I just disagree with it), no separation between system and application, overly complex and convoluted abstractions, etc.
>a rolling release (no major reinstall every year)
In my experience, rolling release just break things more often.
