If I had random C program print me out 2,3,5,7,11 , I would still assume VERY low probability that it is going to print out primes correctly up to reasonably big number. Even more so for Turing machines or anything of this kind. Ditto for any natural processes. Ditto for seeing those numbers in idk child doodling when child didn’t learn the primes yet. Child might have invented primes but that list is not remotely enough evidence.
If a human writer of a test tells me this sequence, I would guess that he knows of primes, and is telling primes to see if I know of primes too. But if he didn’t get taught primes, I would not assume he invented primes from just this sequence.
The Kolmogorov’s complexity really is dependent to language. If we do it informally with human language, then ‘primes’ is an answer that is shorter than ‘two three five seven eleven’. But then the complexity depends to language and expectations of what’s more complex.
Consider the ‘petals around the roses’ joke as very extreme example. Most educated individuals just try to search some giant solution space and are blind to the dots themselves, seeing it as numbers. Unless they have encountered something of this kind before (e.g. the other joke about counting loops in numbers), in which case they solve it quite easily.
edit: There is other slightly less related example with regards to how distribution is different for humans. If you look at natural data, it most often begins with 1. If you look at human-forged data, the frequencies of first digits are much more equal unless the person committing the forgery is very clever.
All the cases in your first paragraph provide context. After the first few, the context essentially tells you whether it’s possible for the sequence to be an enumeration of primes.
In the first few cases, of unknown computer programs, do you really think that the prime number hypothesis should be struck with a 40 decibel probability penalty? I’d love to bet with you. Lots and lots of money, as often as possible.
Well, if the programming language had some primes(); function that prints primes, then no, it shouldn’t. If this is a random choice of a program written by a human being, ditto.
If we are talking of some programming language like C, or assembly, or especially Turing machine, and randomly generated programs, then i’m pretty sure if you see 2,3,5,7,11 it is still quite unlikely (on order of 10^-4 at least) that program would print primes correctly. (However the chance that program prints 13 next would be way higher than 10^-4)
In general, the generated programs have a tendency to do really weird stuff. There was an example posted right here:
And the likehood btw depends on programming language. With wolfram alpha, ‘5 primes’ will print you the primes. With x-86 assembly, division instruction may be used. With z-80 assembly, there’s no division or multiplication instruction. With Turing machine or anything of this sort even the addition needs to be ‘reinvented’.
With a language that has huge library of functions, with huffman-coded names (approximately the human language), the complexity will greatly depend to how often people who made that library expected to use primes.
Well, I dunno.
If I had random C program print me out 2,3,5,7,11 , I would still assume VERY low probability that it is going to print out primes correctly up to reasonably big number. Even more so for Turing machines or anything of this kind. Ditto for any natural processes. Ditto for seeing those numbers in idk child doodling when child didn’t learn the primes yet. Child might have invented primes but that list is not remotely enough evidence.
If a human writer of a test tells me this sequence, I would guess that he knows of primes, and is telling primes to see if I know of primes too. But if he didn’t get taught primes, I would not assume he invented primes from just this sequence.
The Kolmogorov’s complexity really is dependent to language. If we do it informally with human language, then ‘primes’ is an answer that is shorter than ‘two three five seven eleven’. But then the complexity depends to language and expectations of what’s more complex.
Consider the ‘petals around the roses’ joke as very extreme example. Most educated individuals just try to search some giant solution space and are blind to the dots themselves, seeing it as numbers. Unless they have encountered something of this kind before (e.g. the other joke about counting loops in numbers), in which case they solve it quite easily.
edit: There is other slightly less related example with regards to how distribution is different for humans. If you look at natural data, it most often begins with 1. If you look at human-forged data, the frequencies of first digits are much more equal unless the person committing the forgery is very clever.
Very low probability, but much, much larger than most other specific sequences of comparable length!
All the cases in your first paragraph provide context. After the first few, the context essentially tells you whether it’s possible for the sequence to be an enumeration of primes.
In the first few cases, of unknown computer programs, do you really think that the prime number hypothesis should be struck with a 40 decibel probability penalty? I’d love to bet with you. Lots and lots of money, as often as possible.
Well, if the programming language had some primes(); function that prints primes, then no, it shouldn’t. If this is a random choice of a program written by a human being, ditto.
If we are talking of some programming language like C, or assembly, or especially Turing machine, and randomly generated programs, then i’m pretty sure if you see 2,3,5,7,11 it is still quite unlikely (on order of 10^-4 at least) that program would print primes correctly. (However the chance that program prints 13 next would be way higher than 10^-4)
In general, the generated programs have a tendency to do really weird stuff. There was an example posted right here:
http://lesswrong.com/lw/9pl/automatic_programming_an_example/
And the likehood btw depends on programming language. With wolfram alpha, ‘5 primes’ will print you the primes. With x-86 assembly, division instruction may be used. With z-80 assembly, there’s no division or multiplication instruction. With Turing machine or anything of this sort even the addition needs to be ‘reinvented’.
With a language that has huge library of functions, with huffman-coded names (approximately the human language), the complexity will greatly depend to how often people who made that library expected to use primes.