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I am trying to study for my exams by using looking at my midterm. One thing I do not understand fully is the Master Theorem. I understand that there are three cases, and can apply them when they are in this form
T(n) = 25T(n/5) + n^(2)
but my professor likes to give some in this form
T(n) = {n+2 if n=0,1,2,3
T(n) = {4T(n-1) - 6T(n-2) + 4T(n-3) - T(n-4) otherwise
So I am confused if there is a different way to do Master Theorem, or if I am meant to somehow change this into into the format I understand.
n^lg25=n^2. and at the non recursive part we have this. So we should mult n^2 *log n. so the solution is o(n^2 log n)
I'm reading a lot of things about "correctness proof"* in algorithms.
Some say that proofs apply to algorithms and not implementations, but the demonstrations are done with code source most of the time, not math. And code source may have side effects. So, i would like to know if any basic principle prevent someone to prove an impure function correct.
I feel it's true, but cannot say why. If such principle exists, could you explain it ?
Thanks
* Sorry if wording is incorrect, not sure what the good one would be.
The answer is that, although there are no side effects in math, it is possible to mathematically model code that has side effects.
In fact, we can even pull this trick to turn impure code into pure code (without having to go to math in the first place. So, instead of the (psuedocode) function:
f(x) = {
y := y + x
return y
}
...we could write:
f(x, state_before) = {
let old_y = lookup_y(state_before)
let state_after = update_y(state_before, old_y + x)
let new_y = lookup_y(state_after)
return (new_y, state_after)
}
...which can accomplish the same thing with no side effects. Of course, the entire program would have to be rewritten to explicitly pass these state values around, and you'd need to write appropriate lookup_ and update_ functions for all mutable variables, but it's a theoretically straightforward process.
Of course, no one wants to program this way. (Haskell does something similar to simulate side effects without having them be part of the language, but a lot of work went into making it more ergonomic than this.) But because it can be done, we know that side-effects are a well-defined concept.
This means that we can prove things about languages with side-effects, provided that their specifications provide us with enough information to know how to rewrite programs in them into state-passing style.
How can I insert a difference equation in Xcos diagram, like:
y(k+1) = y(k)[a sqrt(y(k-1))] + b *y(k-1) ;
?
Thanks, Best Regards
EDIT
Finding on Internet I think that the best way to address my problem is use:
scifunck block.
I would actually do something like that:
where the delay blocks are set to the sample time of your model (I assume we are dealing with discrete signals here) and appropriate initial conditions. Make sure to choose a discrete solver and use the same (appropriate for your problem) sample time for the model and the delay blocks.
So I'm trying to teach myself Haskell. I am currently on the 11th chapter of Learn You a Haskell for Great Good and am doing the 99 Haskell Problems as well as the Project Euler Problems.
Things are going alright, but I find myself constantly doing something whenever I need to keep track of "variables". I just create another function that accepts those "variables" as parameters and recursively feed it different values depending on the situation. To illustrate with an example, here's my solution to Problem 7 of Project Euler, Find the 10001st prime:
answer :: Integer
answer = nthPrime 10001
nthPrime :: Integer -> Integer
nthPrime n
| n < 1 = -1
| otherwise = nthPrime' n 1 2 []
nthPrime' :: Integer -> Integer -> Integer -> [Integer] -> Integer
nthPrime' n currentIndex possiblePrime previousPrimes
| isFactorOfAnyInThisList possiblePrime previousPrimes = nthPrime' n currentIndex theNextPossiblePrime previousPrimes
| otherwise =
if currentIndex == n
then possiblePrime
else nthPrime' n currentIndexPlusOne theNextPossiblePrime previousPrimesPlusCurrentPrime
where currentIndexPlusOne = currentIndex + 1
theNextPossiblePrime = nextPossiblePrime possiblePrime
previousPrimesPlusCurrentPrime = possiblePrime : previousPrimes
I think you get the idea. Let's also just ignore the fact that this solution can be made to be more efficient, I'm aware of this.
So my question is kind of a two-part question. First, am I going about Haskell all wrong? Am I stuck in the imperative programming mindset and not embracing Haskell as I should? And if so, as I feel I am, how do avoid this? Is there a book or source you can point me to that might help me think more Haskell-like?
Your help is much appreciated,
-Asaf
Am I stuck in the imperative programming mindset and not embracing
Haskell as I should?
You are not stuck, at least I don't hope so. What you experience is absolutely normal. While you were working with imperative languages you learned (maybe without knowing) to see programming problems from a very specific perspective - namely in terms of the van Neumann machine.
If you have the problem of, say, making a list that contains some sequence of numbers (lets say we want the first 1000 even numbers), you immediately think of: a linked list implementation (perhaps from the standard library of your programming language), a loop and a variable that you'd set to a starting value and then you would loop for a while, updating the variable by adding 2 and putting it to the end of the list.
See how you mostly think to serve the machine? Memory locations, loops, etc.!
In imperative programming, one thinks about how to manipulate certain memory cells in a certain order to arrive at the solution all the time. (This is, btw, one reason why beginners find learning (imperative) programming hard. Non programmers are simply not used to solve problems by reducing it to a sequence of memory operations. Why should they? But once you've learned that, you have the power - in the imperative world. For functional programming you need to unlearn that.)
In functional programming, and especially in Haskell, you merely state the construction law of the list. Because a list is a recursive data structure, this law is of course also recursive. In our case, we could, for example say the following:
constructStartingWith n = n : constructStartingWith (n+2)
And almost done! To arrive at our final list we only have to say where to start and how many we want:
result = take 1000 (constructStartingWith 0)
Note that a more general version of constructStartingWith is available in the library, it is called iterate and it takes not only the starting value but also the function that makes the next list element from the current one:
iterate f n = n : iterate f (f n)
constructStartingWith = iterate (2+) -- defined in terms of iterate
Another approach is to assume that we had another list our list could be made from easily. For example, if we had the list of the first n integers we could make it easily into the list of even integers by multiplying each element with 2. Now, the list of the first 1000 (non-negative) integers in Haskell is simply
[0..999]
And there is a function map that transforms lists by applying a given function to each argument. The function we want is to double the elements:
double n = 2*n
Hence:
result = map double [0..999]
Later you'll learn more shortcuts. For example, we don't need to define double, but can use a section: (2*) or we could write our list directly as a sequence [0,2..1998]
But not knowing these tricks yet should not make you feel bad! The main challenge you are facing now is to develop a mentality where you see that the problem of constructing the list of the first 1000 even numbers is a two staged one: a) define how the list of all even numbers looks like and b) take a certain portion of that list. Once you start thinking that way you're done even if you still use hand written versions of iterate and take.
Back to the Euler problem: Here we can use the top down method (and a few basic list manipulation functions one should indeed know about: head, drop, filter, any). First, if we had the list of primes already, we can just drop the first 1000 and take the head of the rest to get the 1001th one:
result = head (drop 1000 primes)
We know that after dropping any number of elements form an infinite list, there will still remain a nonempty list to pick the head from, hence, the use of head is justified here. When you're unsure if there are more than 1000 primes, you should write something like:
result = case drop 1000 primes of
[] -> error "The ancient greeks were wrong! There are less than 1001 primes!"
(r:_) -> r
Now for the hard part. Not knowing how to proceed, we could write some pseudo code:
primes = 2 : {-an infinite list of numbers that are prime-}
We know for sure that 2 is the first prime, the base case, so to speak, thus we can write it down. The unfilled part gives us something to think about. For example, the list should start at some value that is greater 2 for obvious reason. Hence, refined:
primes = 2 : {- something like [3..] but only the ones that are prime -}
Now, this is the point where there emerges a pattern that one needs to learn to recognize. This is surely a list filtered by a predicate, namely prime-ness (it does not matter that we don't know yet how to check prime-ness, the logical structure is the important point. (And, we can be sure that a test for prime-ness is possible!)). This allows us to write more code:
primes = 2 : filter isPrime [3..]
See? We are almost done. In 3 steps, we have reduced a fairly complex problem in such a way that all that is left to write is a quite simple predicate.
Again, we can write in pseudocode:
isPrime n = {- false if any number in 2..n-1 divides n, otherwise true -}
and can refine that. Since this is almost haskell already, it is too easy:
isPrime n = not (any (divides n) [2..n-1])
divides n p = n `rem` p == 0
Note that we did not do optimization yet. For example we can construct the list to be filtered right away to contain only odd numbers, since we know that even ones are not prime. More important, we want to reduce the number of candidates we have to try in isPrime. And here, some mathematical knowledge is needed (the same would be true if you programmed this in C++ or Java, of course), that tells us that it suffices to check if the n we are testing is divisible by any prime number, and that we do not need to check divisibility by prime numbers whose square is greater than n. Fortunately, we have already defined the list of prime numbers and can pick the set of candidates from there! I leave this as exercise.
You'll learn later how to use the standard library and the syntactic sugar like sections, list comprehensions, etc. and you will gradually give up to write your own basic functions.
Even later, when you have to do something in an imperative programming language again, you'll find it very hard to live without infinte lists, higher order functions, immutable data etc.
This will be as hard as going back from C to Assembler.
Have fun!
It's ok to have an imperative mindset at first. With time you will get more used to things and start seeing the places where you can have more functional programs. Practice makes perfect.
As for working with mutable variables you can kind of keep them for now if you follow the rule of thumb of converting variables into function parameters and iteration into tail recursion.
Off the top of my head:
Typeclassopedia. The official v1 of the document is a pdf, but the author has moved his v2 efforts to the Haskell wiki.
What is a monad? This SO Q&A is the best reference I can find.
What is a Monad Transformer? Monad Transformers Step by Step.
Learn from masters: Good Haskell source to read and learn from.
More advanced topics such as GADTs. There's a video, which does a great job explaining it.
And last but not least, #haskell IRC channel. Nothing can even come close to talk to real people.
I think the big change from your code to more haskell like code is using higher order functions, pattern matching and laziness better. For example, you could write the nthPrime function like this (using a similar algorithm to what you did, again ignoring efficiency):
nthPrime n = primes !! (n - 1) where
primes = filter isPrime [2..]
isPrime p = isPrime' p [2..p - 1]
isPrime' p [] = True
isPrime' p (x:xs)
| (p `mod` x == 0) = False
| otherwise = isPrime' p xs
Eg nthPrime 4 returns 7. A few things to note:
The isPrime' function uses pattern matching to implement the function, rather than relying on if statements.
the primes value is an infinite list of all primes. Since haskell is lazy, this is perfectly acceptable.
filter is used rather than reimplemented that behaviour using recursion.
With more experience you will find you will write more idiomatic haskell code - it sortof happens automatically with experience. So don't worry about it, just keep practicing, and reading other people's code.
Another approach, just for variety! Strong use of laziness...
module Main where
nonmults :: Int -> Int -> [Int] -> [Int]
nonmults n next [] = []
nonmults n next l#(x:xs)
| x < next = x : nonmults n next xs
| x == next = nonmults n (next + n) xs
| otherwise = nonmults n (next + n) l
select_primes :: [Int] -> [Int]
select_primes [] = []
select_primes (x:xs) =
x : (select_primes $ nonmults x (x + x) xs)
main :: IO ()
main = do
let primes = select_primes [2 ..]
putStrLn $ show $ primes !! 10000 -- the first prime is index 0 ...
I want to try to answer your question without using ANY functional programming or math, not because I don't think you will understand it, but because your question is very common and maybe others will benefit from the mindset I will try to describe. I'll preface this by saying I an not a Haskell expert by any means, but I have gotten past the mental block you have described by realizing the following:
1. Haskell is simple
Haskell, and other functional languages that I'm not so familiar with, are certainly very different from your 'normal' languages, like C, Java, Python, etc. Unfortunately, the way our psyche works, humans prematurely conclude that if something is different, then A) they don't understand it, and B) it's more complicated than what they already know. If we look at Haskell very objectively, we will see that these two conjectures are totally false:
"But I don't understand it :("
Actually you do. Everything in Haskell and other functional languages is defined in terms of logic and patterns. If you can answer a question as simple as "If all Meeps are Moops, and all Moops are Moors, are all Meeps Moors?", then you could probably write the Haskell Prelude yourself. To further support this point, consider that Haskell lists are defined in Haskell terms, and are not special voodoo magic.
"But it's complicated"
It's actually the opposite. It's simplicity is so naked and bare that our brains have trouble figuring out what to do with it at first. Compared to other languages, Haskell actually has considerably fewer "features" and much less syntax. When you read through Haskell code, you'll notice that almost all the function definitions look the same stylistically. This is very different than say Java for example, which has constructs like Classes, Interfaces, for loops, try/catch blocks, anonymous functions, etc... each with their own syntax and idioms.
You mentioned $ and ., again, just remember they are defined just like any other Haskell function and don't necessarily ever need to be used. However, if you didn't have these available to you, over time, you would likely implement these functions yourself when you notice how convenient they can be.
2. There is no Haskell version of anything
This is actually a great thing, because in Haskell, we have the freedom to define things exactly how we want them. Most other languages provide building blocks that people string together into a program. Haskell leaves it up to you to first define what a building block is, before building with it.
Many beginners ask questions like "How do I do a For loop in Haskell?" and innocent people who are just trying to help will give an unfortunate answer, probably involving a helper function, and extra Int parameter, and tail recursing until you get to 0. Sure, this construct can compute something like a for loop, but in no way is it a for loop, it's not a replacement for a for loop, and in no way is it really even similar to a for loop if you consider the flow of execution. Similar is the State monad for simulating state. It can be used to accomplish similar things as static variables do in other languages, but in no way is it the same thing. Most people leave off the last tidbit about it not being the same when they answer these kinds of questions and I think that only confuses people more until they realize it on their own.
3. Haskell is a logic engine, not a programming language
This is probably least true point I'm trying to make, but hear me out. In imperative programming languages, we are concerned with making our machines do stuff, perform actions, change state, and so on. In Haskell, we try to define what things are, and how are they supposed to behave. We are usually not concerned with what something is doing at any particular time. This certainly has benefits and drawbacks, but that's just how it is. This is very different than what most people think of when you say "programming language".
So that's my take how how to leave an imperative mindset and move to a more functional mindset. Realizing how sensible Haskell is will help you not look at your own code funny anymore. Hopefully thinking about Haskell in these ways will help you become a more productive Haskeller.
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As far as variable naming conventions go, should iterators be named i or something more semantic like count? If you don't use i, why not? If you feel that i is acceptable, are there cases of iteration where it shouldn't be used?
Depends on the context I suppose. If you where looping through a set of Objects in some
collection then it should be fairly obvious from the context what you are doing.
for(int i = 0; i < 10; i++)
{
// i is well known here to be the index
objectCollection[i].SomeProperty = someValue;
}
However if it is not immediately clear from the context what it is you are doing, or if you are making modifications to the index you should use a variable name that is more indicative of the usage.
for(int currentRow = 0; currentRow < numRows; currentRow++)
{
for(int currentCol = 0; currentCol < numCols; currentCol++)
{
someTable[currentRow][currentCol] = someValue;
}
}
"i" means "loop counter" to a programmer. There's nothing wrong with it.
Here's another example of something that's perfectly okay:
foreach (Product p in ProductList)
{
// Do something with p
}
I tend to use i, j, k for very localized loops (only exist for a short period in terms of number of source lines). For variables that exist over a larger source area, I tend to use more detailed names so I can see what they're for without searching back in the code.
By the way, I think that the naming convention for these came from the early Fortran language where I was the first integer variable (A - H were floats)?
i is acceptable, for certain. However, I learned a tremendous amount one semester from a C++ teacher I had who refused code that did not have a descriptive name for every single variable. The simple act of naming everything descriptively forced me to think harder about my code, and I wrote better programs after that course, not from learning C++, but from learning to name everything. Code Complete has some good words on this same topic.
i is fine, but something like this is not:
for (int i = 0; i < 10; i++)
{
for (int j = 0; j < 10; j++)
{
string s = datarow[i][j].ToString(); // or worse
}
}
Very common for programmers to inadvertently swap the i and the j in the code, especially if they have bad eyesight or their Windows theme is "hotdog". This is always a "code smell" for me - it's kind of rare when this doesn't get screwed up.
i is so common that it is acceptable, even for people that love descriptive variable names.
What is absolutely unacceptable (and a sin in my book) is using i,j, or k in any other context than as an integer index in a loop.... e.g.
foreach(Input i in inputs)
{
Process(i);
}
i is definitely acceptable. Not sure what kind of justification I need to make -- but I do use it all of the time, and other very respected programmers do as well.
Social validation, I guess :)
Yes, in fact it's preferred since any programmer reading your code will understand that it's simply an iterator.
What is the value of using i instead of a more specific variable name? To save 1 second or 10 seconds or maybe, maybe, even 30 seconds of thinking and typing?
What is the cost of using i? Maybe nothing. Maybe the code is so simple that using i is fine. But maybe, maybe, using i will force developers who come to this code in the future to have to think for a moment "what does i mean here?" They will have to think: "is it an index, a count, an offset, a flag?" They will have to think: "is this change safe, is it correct, will I be off by 1?"
Using i saves time and intellectual effort when writing code but may end up costing more intellectual effort in the future, or perhaps even result in the inadvertent introduction of defects due to misunderstanding the code.
Generally speaking, most software development is maintenance and extension, so the amount of time spent reading your code will vastly exceed the amount of time spent writing it.
It's very easy to develop the habit of using meaningful names everywhere, and once you have that habit it takes only a few seconds more to write code with meaningful names, but then you have code which is easier to read, easier to understand, and more obviously correct.
I use i for short loops.
The reason it's OK is that I find it utterly implausible that someone could see a declaration of iterator type, with initializer, and then three lines later claim that it's not clear what the variable represents. They're just pretending, because they've decided that "meaningful variable names" must mean "long variable names".
The reason I actually do it, is that I find that using something unrelated to the specific task at hand, and that I would only ever use in a small scope, saves me worrying that I might use a name that's misleading, or ambiguous, or will some day be useful for something else in the larger scope. The reason it's "i" rather than "q" or "count" is just convention borrowed from mathematics.
I don't use i if:
The loop body is not small, or
the iterator does anything other than advance (or retreat) from the start of a range to the finish of the loop:
i doesn't necessarily have to go in increments of 1 so long as the increment is consistent and clear, and of course might stop before the end of the iterand, but if it ever changes direction, or is unmodified by an iteration of the loop (including the devilish use of iterator.insertAfter() in a forward loop), I try to remember to use something different. This signals "this is not just a trivial loop variable, hence this may not be a trivial loop".
If the "something more semantic" is "iterator" then there is no reason not to use i; it is a well understood idiom.
i think i is completely acceptable in for-loop situations. i have always found this to be pretty standard and never really run into interpretation issues when i is used in this instance. foreach-loops get a little trickier and i think really depends on your situation. i rarely if ever use i in foreach, only in for loops, as i find i to be too un-descriptive in these cases. for foreach i try to use an abbreviation of the object type being looped. e.g:
foreach(DataRow dr in datatable.Rows)
{
//do stuff to/with datarow dr here
}
anyways, just my $0.02.
It helps if you name it something that describes what it is looping through. But I usually just use i.
As long as you are either using i to count loops, or part of an index that goes from 0 (or 1 depending on PL) to n, then I would say i is fine.
Otherwise its probably easy to name i something meaningful it its more than just an index.
I should point out that i and j are also mathematical notation for matrix indices. And usually, you're looping over an array. So it makes sense.
As long as you're using it temporarily inside a simple loop and it's obvious what you're doing, sure. That said, is there no other short word you can use instead?
i is widely known as a loop iterator, so you're actually more likely to confuse maintenance programmers if you use it outside of a loop, but if you use something more descriptive (like filecounter), it makes code nicer.
It depends.
If you're iterating over some particular set of data then I think it makes more sense to use a descriptive name. (eg. filecounter as Dan suggested).
However, if you're performing an arbitrary loop then i is acceptable. As one work mate described it to me - i is a convention that means "this variable is only ever modified by the for loop construct. If that's not true, don't use i"
The use of i, j, k for INTEGER loop counters goes back to the early days of FORTRAN.
Personally I don't have a problem with them so long as they are INTEGER counts.
But then I grew up on FORTRAN!
my feeling is that the concept of using a single letter is fine for "simple" loops, however, i learned to use double-letters a long time ago and it has worked out great.
i asked a similar question last week and the following is part of my own answer:// recommended style ● // "typical" single-letter style
●
for (ii=0; ii<10; ++ii) { ● for (i=0; i<10; ++i) {
for (jj=0; jj<10; ++jj) { ● for (j=0; j<10; ++j) {
mm[ii][jj] = ii * jj; ● m[i][j] = i * j;
} ● }
} ● }
in case the benefit isn't immediately obvious: searching through code for any single letter will find many things that aren't what you're looking for. the letter i occurs quite often in code where it isn't the variable you're looking for.
i've been doing it this way for at least 10 years.
note that plenty of people commented that either/both of the above are "ugly"...
I am going to go against the grain and say no.
For the crowd that says "i is understood as an iterator", that may be true, but to me that is the equivalent of comments like 'Assign the value 5 to variable Y. Variable names like comment should explain the why/what not the how.
To use an example from a previous answer:
for(int i = 0; i < 10; i++)
{
// i is well known here to be the index
objectCollection[i].SomeProperty = someValue;
}
Is it that much harder to just use a meaningful name like so?
for(int objectCollectionIndex = 0; objectCollectionIndex < 10; objectCollectionIndex ++)
{
objectCollection[objectCollectionIndex].SomeProperty = someValue;
}
Granted the (borrowed) variable name objectCollection is pretty badly named too.