This is a question popped into my mind while reading the halting problem, collatz conjecture and Kolmogorov complexity. I have tried to search for something similar but I was unable to find a particular topic maybe because it is not of great value or it could just be a trivial question.
For the sake of simplicity I will give three examples of programs/functions.
function one(s):
return s
function two(s):
while (True):
print s
function three(s):
for i from 0 to 10^10:
print(s)
So my questions is, if there is a way to formalize the length of a program (like the bits used to describe it) and also the internal memory used by the program, to determine the minimum/maximum number of time/steps needed to decide whether the program will terminate or run forever.
For example, in the first function the program doesn't alter its internal memory and halts after some time steps.
In the second example, the program runs forever but the program also doesn't alter its internal memory. For example, if we considered all the programs with the same length as with the program two that do not alter their state, couldn't we determine an upper bound of steps, which if surpassed we could conclude that this program will never terminate ? (If not why ?)
On the last example, the program alters its state (variable i). So, at each step the upper bound may change.
[In short]
Kolmogorov complexity suggests a way of finding the (descriptive) complexity of an object such as a piece of text. I would like to know, given a formal way of describing the memory-space used by a program (computed in runtime), if we could compute a maximum number of steps, which if surpassed would allow us to know whether this program will terminate or run forever.
Finally, I would like to suggest me any source that I might find useful and help me figure out what I am exactly looking for.
Thank you. (sorry for my English, not my native language. I hope I was clear)
If a deterministic Turing machine enters precisely the same configuration twice (which we can detect b keeping a trace of configurations seen so far), then we immediately know the TM will loop forever.
If it known in advance that a deterministic Turing machine cannot possibly use more than some fixed constant amount of its input tape, then the TM must explicitly halt or eventually enter some configuration it has already visited. Suppose the TM can use at most k tape cells, the tape alphabet is T and the set of states is Q. Then there are (|T|+1)^k * |Q| unique configurations (the number of strings over (T union blank) of length k times the number of states) and by the pigeonhole principle we know that a TM that takes that many steps must enter some configuration it has already been to before.
one: because we are given that this function does not use internal memory, we know that it either halts or loops forever.
two: because we are given that this function does not use internal memory, we know that it either halts or loops forever.
three: because we are given that this function only uses a fixed amount of internal memory (like 34 bits) we can tell in fewer than 2^34 iterations of the loop whether the TM will halt or not for any given input s, guaranteed.
Now, knowing how much tape a TM is going to use, or how much memory a program is going to use, is not a problem a TM can solve. But if you have an oracle (like a person who was able to do a proof) that tells you a correct fixed upper bound on memory, then the halting problem is solvable.
Related
I've got a system that takes 15 points out of a 17 by 17 grid as input (order doesn't matter), and generates a single scalar as output. The system is not representable by a formal function.
The goal is to find the optimal 15 points so that the output scalar is minimum. Solving this problem exhaustively simply takes too much time to be practical as each run takes 14 seconds.
I've started taking a machine learning course online. But this problem does seem to be rather unsophisticated and I wonder if anyone can point me to the right direction. Any help is greatly appreciated!
Use simulated annealing. I guess this will be close to optimal here.
Therefore, start with a random distribution of the 15 points. Then, in each iteration change one and accept the new state if the resulting scalar value is lower. If it is larger, accept with a certain probability (a Boltzmann factor). Eventually you have to try this for a small number of randomly chosen initial states and afterwards accept the lowest value.
I've got an algorithm using a single (positive integer) number as an input to produce an output. And I've got the reverse function which should do the exact opposite, going back from the output to the same integer number. This should be a unique one-to-one reversible mapping.
I've tested this for some integers, but I want to be 100% sure that it works for all of them, up to a known limit.
The problem is that if I just test every integer, it takes an unreasonably long time to run. If I use 64-bit integers, that's a lot of numbers to check if I want to check them all. On the other hand, if I only test every 10th or 100th number, I'm not going to be 100% sure at the end. There might be some awkward weird constellation in one of the 90% or 99% which I didn't test.
Are there any general ways to identify edge cases so that just those "interesting" or "risky" numbers are checked? Or should I just pick numbers at random? Or test in increasing increments?
Or to put the question another way, how can I approach this so that I gain 100% confidence that every case will be properly handled?
The approach for this is generally checking every step of the computation for potential flaws. Concerning integer math, that is overflows, underflows and rounding errors from division, basically that the mathematical result can't be represented accurately. In addition, all operations derived from this suffer similar problems.
The process of auditing then looks at single steps in turn. For example, if you want to allocate memory for N integers, you need N times the size of an integer in bytes and this multiplication can overflow. You now determine those values where the multiplication overflows and create according tests that exercise these. Note that for the example of allocating memory, proper handling typically means that the function does not allocate memory but fail.
The principle behind this is that you determine the ranges for every operation where the outcome is somehow different (like e.g. where it overflows) and then make sure via tests that both variants work. This reduces the number of tests from all possible input values to just those where you expect a significant difference.
I'm thinking more about how much system memory my programs will use nowadays. I'm currently doing A level Computing at college and I know that in most programs the difference will be negligible but I'm wondering if the following actually makes any difference, in any language.
Say I wanted to output "True" or "False" depending on whether a condition is true. Personally, I prefer to do something like this:
Dim result As String
If condition Then
Result = "True"
Else
Result = "False"
EndIf
Console.WriteLine(result)
However, I'm wondering if the following would consume less memory, etc.:
If condition Then
Console.WriteLine("True")
Else
Console.WriteLine("False")
EndIf
Obviously this is a very much simplified example and in most of my cases there is much more to be outputted, and I realise that in most commercial programs these kind of statements are rare, but hopefully you get the principle.
I'm focusing on VB.NET here because that is the language used for the course, but really I would be interested to know how this differs in different programming languages.
The main issue making if's fast or slow is predictability.
Modern CPU's (anything after 2000) use a mechanism called branch prediction.
Read the above link first, then read on below...
Which is faster?
The if statement constitutes a branch, because the CPU needs to decide whether to follow or skip the if part.
If it guesses the branch correctly the jump will execute in 0 or 1 cycle (1 nanosecond on a 1Ghz computer).
If it does not guess the branch correctly the jump will take 50 cycles (give or take) (1/200th of a microsecord).
Therefore to even feel these differences as a human, you'd need to execute the if statement many millions of times.
The two statements above are likely to execute in exactly the same amount of time, because:
assigning a value to a variable takes negligible time; on average less than a single cpu cycle on a multiscalar CPU*.
calling a function with a constant parameter requires the use of an invisible temporary variable; so in all likelihood code A compiles to almost the exact same object code as code B.
*) All current CPU's are multiscalar.
Which consumes less memory
As stated above, both versions need to put the boolean into a variable.
Version A uses an explicit one, declared by you; version B uses an implicit one declared by the compiler.
However version A is guaranteed to only have one call to the function WriteLine.
Whilst version B may (or may not) have two calls to the function WriteLine.
If the optimizer in the compiler is good, code B will be transformed into code A, if it's not it will remain with the redundant calls.
How bad is the waste
The call takes about 10 bytes for the assignment of the string (Unicode 2 bytes per char).
But so does the other version, so that's the same.
That leaves 5 bytes for a call. Plus maybe a few extra bytes to set up a stackframe.
So lets say due to your totally horrible coding you have now wasted 10 bytes.
Not much to worry about.
From a maintainability point of view
Computer code is written for humans, not machines.
So from that point of view code A is clearly superior.
Imagine not choosing between 2 options -true or false- but 20.
You only call the function once.
If you decide to change the WriteLine for another function you only have to change it in one place, not two or 20.
How to speed this up?
With 2 values it's pretty much impossible, but if you had 20 values you could use a lookup table.
Obviously that optimization is not worth it unless code gets executed many times.
If you need to know the precise amount of memory the instructions are going to take, you can use ildasm on your code, and see for yourself. However, the amount of memory consumed by your code is much less relevant today, when the memory is so cheap and abundant, and compilers are smart enough to see common patterns and reduce the amount of code that they generate.
A much greater concern is readability of your code: if a complex chain of conditions always leads to printing a conditionally set result, your first code block expresses this idea in a cleaner way than the second one does. Everything else being equal, you should prefer whatever form of code that you find the most readable, and let the compiler worry about optimization.
P.S. It goes without saying that Console.WriteLine(condition) would produce the same result, but that is of course not the point of your question.
Y
I have 6 parameters for which I know maxi and mini values. I have a complex function that includes the 6 parameters and return a 7th value (say Y). I say complex because Y is not directly related to the 6 parameters; there are many embeded functions in between.
I would like to find the combination of the 6 parameters which returns the highest Y value. I first tried to calculate Y for every combination by constructing an hypercube but I have not enough memory in my computer. So I am looking for kinds of markov chains which progress in the delimited parameter space, and are able to overpass local peaks.
when I give one combination of the 6 parameters, I would like to know the highest local Y value. I tried to write a code with an iterative chain like a markov's one, but I am not sure how to process when the chain reach an edge of the parameter space. Obviously, some algorythms should already exist for this.
Question: Does anybody know what are the best functions in R to do these two things? I read that optim() could be appropriate to find the global peak but I am not sure that it can deal with complex functions (I prefer asking before engaging in a long (for me) process of code writing). And fot he local peaks? optim() should not be able to do this
In advance, thank you for any lead
Julien from France
Take a look at the Optimization and Mathematical Programming Task View on CRAN. I've personally found the differential evolution algorithm to be very fast and robust. It's implemented in the DEoptim package. The rgenoud package is another good candidate.
I like to use the Metropolis-Hastings algorithm. Since you are limiting each parameter to a range, the simple thing to do is let your proposal distribution simply be uniform over the range. That way, you won't run off the edges. It won't be fast, but if you let it run long enough, it will do a good job of sampling your space. The samples will congregate at each peak, and will spread out around them in a way that reflects the local curvature.
I found this on an "interview questions" site and have been pondering it for a couple of days. I will keep churning, but am interested what you guys think
"10 Gbytes of 32-bit numbers on a magnetic tape, all there from 0 to 10G in random order. You have 64 32 bit words of memory available: design an algorithm to check that each number from 0 to 10G occurs once and only once on the tape, with minimum passes of the tape by a read head connected to your algorithm."
32-bit numbers can take 4G = 2^32 different values. There are 2.5*2^32 numbers on tape total. So after 2^32 count one of numbers will repeat 100%. If there were <= 2^32 numbers on tape then it was possible that there are two different cases – when all numbers are different or when at least one repeats.
It's a trick question, as Michael Anderson and I have figured out. You can't store 10G 32b numbers on a 10G tape. The interviewer (a) is messing with you and (b) is trying to find out how much you think about a problem before you start solving it.
The utterly naive algorithm, which takes as many passes as there are numbers to check, would be to walk through and verify that the lowest number is there. Then do it again checking that the next lowest is there. And so on.
This requires one word of storage to keep track of where you are - you could cut down the number of passes by a factor of 64 by using all 64 words to keep track of where you're up to in several different locations in the search space - checking all of your current ones on each pass. Still O(n) passes, of course.
You could probably cut it down even more by using portions of the words - given that your search space for each segment is smaller, you won't need to keep track of the full 32-bit range.
Perform an in-place mergesort or quicksort, using tape for storage? Then iterate through the numbers in sequence, tracking to see that each number = previous+1.
Requires cleverly implemented sort, and is fairly slow, but achieves the goal I believe.
Edit: oh bugger, it's never specified you can write.
Here's a second approach: scan through trying to build up to 30-ish ranges of contiginous numbers. IE 1,2,3,4,5 would be one range, 8,9,10,11,12 would be another, etc. If ranges overlap with existing, then they are merged. I think you only need to make a limited number of passes to either get the complete range or prove there are gaps... much less than just scanning through in blocks of a couple thousand to see if all digits are present.
It'll take me a bit to prove or disprove the limits for this though.
Do 2 reduces on the numbers, a sum and a bitwise XOR.
The sum should be (10G + 1) * 10G / 2
The XOR should be ... something
It looks like there is a catch in the question that no one has talked about so far; the interviewer has only asked the interviewee to write a program that CHECKS
(i) if each number that makes up the 10G is present once and only once--- what should the interviewee do if the numbers in the given list are present multple times? should he assume that he should stop execting the programme and throw exception or should he assume that he should correct the mistake by removing the repeating number and replace it with another (this may actually be a costly excercise as this involves complete reshuffle of the number set)? correcting this is required to perform the second step in the question, i.e. to verify that the data is stored in the best possible way that it requires least possible passes.
(ii) When the interviewee was asked to only check if the 10G weight data set of numbers are stored in such a way that they require least paases to access any of those numbers;
what should the interviewee do? should he stop and throw exception the moment he finds an issue in the algorithm they were stored in, or correct the mistake and continue till all the elements are sorted in the order of least possible passes?
If the intension of the interviewer is to ask the interviewee to write an algorithm that finds the best combinaton of numbers that can be stored in 10GB, given 64 32 Bit registers; and also to write an algorithm to save these chosen set of numbers in the best possible way that require least number of passes to access each; he should have asked this directly, woudn't he?
I suppose the intension of the interviewer may be to only see how the interviewee is approaching the problem rather than to actually extract a working solution from the interviewee; wold any buy this notion?
Regards,
Samba