I've bought and read the Software Abstractions book (great book actually) a couple of months if not 1.5 years ago. I've read online tutorials and slides on Alloy, etc. Of course, I've also done exercises and a few models of my own. I've even preached for Alloy in some confs. Congrats for Alloy btw!
Now, I am wondering if one can model and solve maximizing problems over integers in Alloy. I don't see how it could be done but I thought asking real experts could give me a more definitive answer.
For instance, say you have a model similar to this:
open util/ordering[State] as states
sig State {
i, j, k: Int
}{
i >= 0
j >= 0
k >= 0
}
pred subi (s, s': State) {
s'.i = minus[s.i, 2]
s'.j = s.j
s'.k = s.k
}
pred subj (s, s': State) {
s'.i = s.i
s'.j = minus[s.j, 1]
s'.k = s.k
}
pred subk (s, s': State) {
s'.i = s.i
s'.j = s.j
s'.k = minus[s.k, 3]
}
pred init (s: State) {
// one example
s.i = 10
s.j = 8
s.k = 17
}
fact traces {
init[states/first]
all s: State - states/last | let s' = states/next[s] |
subi[s, s'] or subj[s, s'] or subk[s, s']
let s = states/last | (s.i > 0 => (s.j = 0 and s.k = 0)) and
(s.j > 0 => (s.i = 0 and s.k = 0)) and
(s.k > 0 => (s.i = 0 and s.j = 0))
}
run {} for 14 State, 6 Int
I could have used Naturals but let's forget it. What if I want the trace which leads to the maximal i, j or k in the last state? Can I constrain it?
Some intuition is telling me I could do it by trial and error, i.e., find one solution and then manually add a constraint in the model for the variable to be stricly greater than the one value I just found, until it is unsatisfiable. But can it be done more elegantly and efficiently?
Thanks!
Fred
EDIT: I realize that for this particular problem, the maximum is easy to find, of course. Keep the maximal value in the initial state as-is and only decrease the other two and you're good. But my point was to illustrate one simple problem to optimize so that it can be applied to harder problems.
Your intuition is right: trial and error is certainly a possible approach, and I use it regularly in similar situations (e.g. to find minimal sets of axioms that entail the properties I want).
Whether it can be done more directly and elegantly depends, I think, on whether a solution to the problem can be represented by an atom or must be a set or other non-atomic object. Given a problem whose solutions will all be atoms of type T, a predicate Solution which is true of atomic solutions to a problem, and a comparison relation gt which holds over atoms of the appropriate type(s), then you can certainly write
pred Maximum[ a : T ] {
Solution[a]
and
all s : T | Solution[s] implies
(gt[a,s] or a = s)
}
run Maximum for 5
Since Alloy is resolutely first-order, you cannot write the equivalent predicate for solutions which involve sets, relations, functions, paths through a graph, etc. (Or rather, you can write them, but the Analyzer cannot analyze them.)
But of course one can also introduce signatures called MySet, MyRelation, etc., so that one has one atom for each set, relation, etc., that one needs in a problem. This sometimes works, but it does run into the difficulty that such problems sometimes need all possible sets, relations, functions, etc., to exist (as in set theory they do), while Alloy will not, in general, create an atom of type MySet for every possible set of objects in univ. Jackson discusses this technique in sections 3.2.3 (see "Is there a loss of expressive power in the restriction to flat relations?"), 5.2.2 "Skolemization", and 5.3 "Unbounded universal quantifiers" of his book, and the discussion has thus far repaid repeated rereadings. (I have penciled in an additional index entry in my copy of the book pointing to these sections, under the heading 'Second-order logic, faking it', so I can find them again when I need them.)
All of that said, however: in section 4.8 of his book, Jackson writes "Integers are not actually very useful. If you think you need them, think again; ... Of course, if you have a heavily numerical problem, you're likely to need integers (and more), but then Alloy is probably not suitable anyway."
To declare constant variables I can do following in Ruby
class COLOR
RED = 10
BLUE = 20
GREEM = 30
end
COLOR::RED returns 10, COLOR::BLUE returns 20, and so on. How do I accomplish that in Haskell?
I want to have a namespace name in front of my variable name. Maybe the example above is not a good example. For the case below, you can see including a namespace name will make a variable much easier to understand.
class BASEBALL_TEAM
GIANTS = 15
METS = 30
REDS = 45
...
end
BASEBALL_TEAM::GIANTS is much clear than GIANTS.
based on the comments below, it seems the only way I can accomplish it is by doing something like below:
module Color (Color) where
data Color = Red | Blue | Green deriving (Eq, Show, Ord, Bounded, Enum)
fromEnum' x = (fromEnum x) + 10
to get integer value of 10 for Color.Red, I have to write fromEnum Color.Red, the syntax is not very clean.
Untagged constants are bad. If you go with bunch of Int constants then you lose type-checking (think about possible values that Int -> whatever function takes as opposed to SomeConstType -> whatever) and possibly introduce bugs. You want a strong type instead:
data Colour = Red | Blue | Green deriving (Show, Eq, Enum)
Also, representing those values as integers is not really necessary in most cases. If you actually do need it, Enum typeclass provides toEnum and fromEnum.
As for namespaces: modules are namespaces in Haskell. Put your type in a module and then you can import it qualified and you'll have your prefix:
-- Colours.hs
module Colours (Colour) where
data Colour = ...
-- SomeOtherModule.hs
module SomeOtherModule where
import qualified Colours
foo = Colours.Red
That said, creating modules just for this one type with constants (or importing them qualified) is not really necessary, because you can easily trace things by their types.
Things are constant in Haskell by default, so
red = 10
blue = 20
green = 30
would be equivalent.
A more interesting question would be why you want to do this? There are likely better ways to accomplish what you want in Haskell. The answer by #CatPlusPlus shows a good way of doing this.
This seems like a very basic Haskell question, so I'll politely point you to Learn you a Haskell, which, in my opinion, is the best resource to get started with Haskell.
Another promising resource for learning Haskell is FP complete's School of Haskell, which is currently in beta, but I haven't tried it myself. This is a more interactive setting, where you can directly try things out in the browser.
One of the arguments I've heard against functional languages is that single assignment coding is too hard, or at least significantly harder than "normal" programming.
But looking through my code, I realized that I really don't have many (any?) use patterns that can't be written just as well using single assignment form if you're writing in a reasonably modern language.
So what are the use cases for variables that vary within a single invocation of their scope? Bearing in mind that loop indexes, parameters, and other scope bound values that vary between invocations aren't multiple assignments in this case (unless you have to change them in the body for some reason), and assuming that you are writing in something a far enough above the assembly language level, where you can write things like
values.sum
or (in case sum isn't provided)
function collection.sum --> inject(zero, function (v,t) --> t+v )
and
x = if a > b then a else b
or
n = case s
/^\d*$/ : s.to_int
'' : 0
'*' : a.length
'?' : a.length.random
else fail "I don't know how many you want"
when you need to, and have list comprehensions, map/collect, and so forth available.
Do you find that you still want/need mutable variables in such an environment, and if so, what for?
To clarify, I'm not asking for a recitation of the objections to SSA form, but rather concrete examples where those objections would apply. I'm looking for bits of code that are clear and concise with mutable variables and couldn't be written so without them.
My favorite examples so far (and the best objection I expect to them):
Paul Johnson's Fisher-Yates algorithm answer, which is pretty strong when you include the big-O constraints. But then, as catulahoops points out, the big-O issue isn't tied to the SSA question, but rather to having mutable data types, and with that set aside the algorithm can be written rather clearly in SSA:
shuffle(Lst) ->
array:to_list(shuffle(array:from_list(Lst), erlang:length(Lst) - 1)).
shuffle(Array, 0) -> Array;
shuffle(Array, N) ->
K = random:uniform(N) - 1,
Ek = array:get(K, Array),
En = array:get(N, Array),
shuffle(array:set(K, En, array:set(N, Ek, Array)), N-1).
jpalecek's area of a polygon example:
def area(figure : List[Point]) : Float = {
if(figure.empty) return 0
val last = figure(0)
var first= figure(0)
val ret = 0
for (pt <- figure) {
ret+=crossprod(last - first, pt - first)
last = pt
}
ret
}
which might still be written something like:
def area(figure : List[Point]) : Float = {
if figure.length < 3
0
else
var a = figure(0)
var b = figure(1)
var c = figure(2)
if figure.length == 3
magnitude(crossproduct(b-a,c-a))
else
foldLeft((0,a,b))(figure.rest)) {
((t,a,b),c) => (t+area([a,b,c]),a,c)
}
Or, since some people object to the density of this formulation, it could be recast:
def area([]) = 0.0 # An empty figure has no area
def area([_]) = 0.0 # ...nor does a point
def area([_,_]) = 0.0 # ...or a line segment
def area([a,b,c]) = # The area of a triangle can be found directly
magnitude(crossproduct(b-a,c-a))
def area(figure) = # For larger figures, reduce to triangles and sum
as_triangles(figure).collect(area).sum
def as_triangles([]) = [] # No triangles without at least three points
def as_triangles([_]) = []
def as_triangles([_,_]) = []
def as_triangles([a,b,c | rest) = [[a,b,c] | as_triangles([a,c | rest])]
Princess's point about the difficulty of implementing O(1) queues with immutable structures is interesting (and may well provide the basis for a compelling example) but as stated it's fundamentally about the mutability of the data structure, and not directly about the multiple assignment issue.
I'm intrigued by the Sieve of Eratosthenes answer, but unconvinced. The proper big-O, pull as many primes as you'd like generator given in the paper he cited does not look easy to implement correctly with or without SSA.
Well, thanks everyone for trying. As most of the answers turned out to be either 1) based on mutable data structures, not on single-assignment, and 2) to the extent they were about single assignment form easily countered by practitioners skilled in the art, I'm going to strike the line from my talk and / or restructure (maybe have it in backup as a discussion topic in the unlikely event I run out of words before I run out of time).
Thanks again.
The hardest problem I've come across is shuffling a list. The Fisher-Yates algorithm (also sometimes known as the Knuth algorithm) involves iterating through the list swapping each item with a random other item. The algorithm is O(n), well known and long-since proven correct (an important property in some applications). But it requires mutable arrays.
That isn't to say you can't do shuffling in a functional program. Oleg Kiselyov has written about this. But if I understand him correctly, functional shuffling is O(n . log n) because it works by building a binary tree.
Of course, if I needed to write the Fisher-Yates algorithm in Haskell I'd just put it in the ST monad, which lets you wrap up an algorithm involving mutable arrays inside a nice pure function, like this:
-- | Implementation of the random swap algorithm for shuffling. Reads a list
-- into a mutable ST array, shuffles it in place, and reads out the result
-- as a list.
module Data.Shuffle (shuffle) where
import Control.Monad
import Control.Monad.ST
import Data.Array.ST
import Data.STRef
import System.Random
-- | Shuffle a value based on a random seed.
shuffle :: (RandomGen g) => g -> [a] -> [a]
shuffle _ [] = []
shuffle g xs =
runST $ do
sg <- newSTRef g
let n = length xs
v <- newListArray (1, n) xs
mapM_ (shuffle1 sg v) [1..n]
getElems v
-- Internal function to swap element i with a random element at or above it.
shuffle1 :: (RandomGen g) => STRef s g -> STArray s Int a -> Int -> ST s ()
shuffle1 sg v i = do
(_, n) <- getBounds v
r <- getRnd sg $ randomR (i, n)
when (r /= i) $ do
vi <- readArray v i
vr <- readArray v r
writeArray v i vr
writeArray v r vi
-- Internal function for using random numbers
getRnd :: (RandomGen g) => STRef s g -> (g -> (a, g)) -> ST s a
getRnd sg f = do
g1 <- readSTRef sg
let (v, g2) = f g1
writeSTRef sg g2
return v
If you want to make the academic argument, then of course it's not technically necessary to assign a variable more than once. The proof is that all code can be represented in SSA (Single Static Assignment) form. Indeed, that's the most useful form for many kinds of static and dynamic analysis.
At the same time, there are reasons we don't all write code in SSA form to begin with:
It usually takes more statements (or more lines of code) to write code this way. Brevity has value.
It's almost always less efficient. Yes I know you're talking about higher languages -- a fair scoping -- but even in the world of Java and C#, far away from assembly, speed matters. There are few applications where speed is irrelevant.
It's not as easy to understand. Although SSA is "simpler" in a mathematical sense, it's more abstract from common sense, which is what matters in real-world programming. If you have to be really smart to grok it, then it has no place in programming at large.
Even in your examples above, it's easy to poke holes. Take your case statement. What if there's an administrative option that determines whether '*' is allowed, and a separate one for whether '?' is allowed? Also, zero is not allowed for the integer case, unless the user has a system permission that allows it.
This is a more real-world example with branches and conditions. Could you write this as a single "statement?" If so, is your "statement" really different from many separate statements? If not, how many temporary write-only variables do you need? And is that situation significantly better than just having a single variable?
I've never identified such a case. And while you can always just invent new names, as in conversion to SSA form, I actually find it's easy and natural for each value to have its own name. A language like Haskell gives me a lot of choices about which values to name, and two different places to put name bindings (let and where). I find the single-assignment form quite natural and not at all difficult.
I do occasionally miss being able to have pointers to mutable objects on the heap. But these things have no names, so it's not the same objection. (And I also find that when I use mutable objects on the heap, I tend to write more bugs!)
I think you'll find the most productive languages allow you to mix functional and imperative styles, such as OCaml and F#.
In most cases, I can write code which is simply a long line of "map x to y, reduce y to z". In 95% of cases, functional programming simplifies my code, but there is one area where immutability shows its teeth:
The wide disparity between the ease of implementing and immutable stack and an immutable queue.
Stacks are easy and mesh well with persistence, queues are ridiculous.
The most common implementations of immutable queues use one or more internal stacks and stack rotations. The upside is that these queues run in O(1) most of the time, but some operations will run in O(n). If you're relying on persistence in your application, then its possible in principle that every operation runs in O(n). These queues are no good when you need realtime (or at least consistent) performance.
Chris Okasaki's provides an implementation of immutable queues in his book, they use laziness to achieve O(1) for all operations. Its a very clever, reasonably concise implementation of a realtime queue -- but it requires deep understanding of its underlying implementation details, and its still an order of magnitude more complex than an immutable stack.
In constrast, I can write a stack and queue using mutable linked lists which run in constant time for all operations, and the resulting code would be very straightforward.
Regarding the area of a polygon, its easy to convert it to functional form. Let's assume we have a Vector module like this:
module Vector =
type point =
{ x : float; y : float}
with
static member ( + ) ((p1 : point), (p2 : point)) =
{ x = p1.x + p2.x;
y = p1.y + p2.y;}
static member ( * ) ((p : point), (scalar : float)) =
{ x = p.x * scalar;
y = p.y * scalar;}
static member ( - ) ((p1 : point), (p2 : point)) =
{ x = p1.x - p2.x;
y = p1.y - p2.y;}
let empty = { x = 0.; y = 0.;}
let to_tuple2 (p : point) = (p.x, p.y)
let from_tuple2 (x, y) = { x = x; y = y;}
let crossproduct (p1 : point) (p2 : point) =
{ x = p1.x * p2.y; y = -p1.y * p2.x }
We can define our area function using a little bit of tuple magic:
let area (figure : point list) =
figure
|> Seq.map to_tuple2
|> Seq.fold
(fun (sum, (a, b)) (c, d) -> (sum + a*d - b*c, (c, d) ) )
(0., to_tuple2 (List.hd figure))
|> fun (sum, _) -> abs(sum) / 2.0
Or we can use the cross product instead
let area2 (figure : point list) =
figure
|> Seq.fold
(fun (acc, prev) cur -> (acc + (crossproduct prev cur), cur))
(empty, List.hd figure)
|> fun (acc, _) -> abs(acc.x + acc.y) / 2.0
I don't find either function unreadable.
That shuffle algorithm is trivial to implement using single assignment, in fact it's exactly the same as the imperative solution with the iteration rewritten to tail recursion. (Erlang because I can write it more quickly than Haskell.)
shuffle(Lst) ->
array:to_list(shuffle(array:from_list(Lst), erlang:length(Lst) - 1)).
shuffle(Array, 0) -> Array;
shuffle(Array, N) ->
K = random:uniform(N) - 1,
Ek = array:get(K, Array),
En = array:get(N, Array),
shuffle(array:set(K, En, array:set(N, Ek, Array)), N-1).
If the efficiency of those array operations is a concern, then that's a question about mutable data structures and has nothing to do with single assignment.
You won't get an answer to this question because no examples exist. It is only a question of familiarity with this style.
In response to Jason --
function forbidden_input?(s)
(s = '?' and not administration.qmark_ok) ||
(s = '*' and not administration.stat_ok) ||
(s = '0' and not 'root node visible' in system.permissions_for(current_user))
n = if forbidden_input?(s)
fail "'" + s + "' is not allowed."
else
case s
/^\d*$/ : s.to_int
'' : 0
'*' : a.length
'?' : a.length.random
else fail "I don't know how many you want"
I would miss assignments in a non-purely functional language. Mostly because they hinder the usefulness of loops. Examples (Scala):
def quant[A](x : List[A], q : A) = {
var tmp : A=0
for (el <- x) { tmp+= el; if(tmp > q) return el; }
// throw exception here, there is no prefix of the list with sum > q
}
This should compute the quantile of a list, note the accumulator tmp which is assigned to multiple times.
A similar example would be:
def area(figure : List[Point]) : Float = {
if(figure.empty) return 0
val last = figure(0)
var first= figure(0)
val ret = 0
for (pt <- figure) {
ret+=crossprod(last - first, pt - first)
last = pt
}
ret
}
Note mostly the last variable.
These examples could be rewritten using fold on a tuple to avoid multiple assignments, but that would really not help the readability.
Local (method) variables certainly never have to be assigned to twice. But even in functional programming re-assigning a variable is allowed. It's changing (part of) the value that's not allowed. And as dsimcha already answered, for very large structures (perhaps at the root of an application) it doesn't seem feasible to me to replace the entire structure. Think about it. The state of an application is all contained ultimately by the entrypoint method of your application. If absolutely no state can change without being replaced, you would have to restart your application with every keystroke. :(
If you have a function that builds a lazy list/tree then reduces it again, a functional compiler may be able to optimize it using deforestation.
If it's tricky, it might not. Then you're sort of out of luck, performance & memory wise, unless you can iterate and use a mutable variable.
Thanks to the Church-Turing Thesis, we know that anything that can be written in a Turing-complete language can be written in any Turing-complete language. So, when you get right down to it, there's nothing you can't do in Lisp that you couldn't do in C#, if you tried hard enough, or vice versa. (More to the point, either one is going to get compiled down to x86 machine language in most cases anyway.)
So, the answer to your question is: there are no such cases. All there are are cases that are easier for humans to comprehend in one paradigm/language or another-- and the ease of comprehension here is tied to training and experience.
Perhaps the main issue here is the style of looping in a language. In langauges where we use recursion, any values changing over the course of a loop are re-bound when the function is called again. Languages using iterators in blocks (e.g., Smalltalk's and Ruby's inject method) tend to be similar, though many people in Ruby would still use each and a mutable variable over inject.
When you code loops using while and for, on the other hand, you don't have the easy re-binding of variables that comes automatically when you can pass in several parameters to your chunk of code that does one iteration of the loop, so immutable variables would be rather more inconvenient.
Working in Haskell is a really good way to investigate the necessity of mutable variables, since the default is immutable but mutable ones are available (as IORefs, MVars, and so on). I've been recently, er, "investigating" in this way myself, and have come to the following conclusions.
In the vast majority of cases, mutable variables are not necessary, and I'm happy living without them.
For inter-thread communication, mutable variables are essential, for fairly obvious reasons. (This is specific to Haskell; runtime systems that use message passing at the lowest level don't need them, of course.) However, this use is rare enough that having to use functions to read and write them (readIORef fooRef val etc.) is not much of a burden.
I have used mutable variables within a single thread, because it seemed to make certain things easier, but later regretted it as I realized that it became very hard to reason about what was happening to the value stored there. (Several different functions were manipulating that value.) This was a bit of an eye-opener; in typical frog-in-the-pot-of-warming-water style, I'd not realized how easy Haskell had made it for me to reason about the use of values until I ran into an example of how I used to use them.
So these days I've come down fairly firmly on the side of immutable variables.
Since previous answers to this question have confused these things, I feel compelled to point out here quite forcefully that this issue is orthogonal to both purity and functional programming. I feel that Ruby, for example, would benefit from having single-assignment local variables, though possibly a few other changes to the language, such as adding tail recursion, would be necessary to make this truly convenient.
What about when you need to make small changes to large data structures? You don't really want to copy a whole array or large class every time you would modify a few elements.
I haven't really thought about this much except now that you're pointing it out.
Actually I try not using multiple assignments subconsciously.
Here's an example of what Im talking about, in python
start = self.offset%n
if start:
start = n-start
Written this way to avoid an unneccesary extra Modulo or subtraction. This is used with bignum style long ints, so its a worthwhile optimization. Thing about it, though, is that it really is a single assignment.
I wouldn't miss multiple assignment at all.
I know you asked for code that did show the benefits of mutable variables. And I wish I could provide it. But as pointed out before - there is no problem that can't be expressed in both fashions. And especially since you pointed out that jpalecek's area of a polygon example could be written with a folding algo (which is IMHO way messier and takes the problem to different level of complexity) - well it made me wonder why you are coming down on mutability so hard. So I'll try to make the argument for a common ground and an coexistence of immutable and mutable data.
In my opinion this question misses the point a bit. I know that us programmers are prone to liking things clean and simple but we sometimes miss that a mixture is possible as well. And that's probably why in the discussion about immutability there is seldom somebody taking the middle ground. I just wonder why, because let's face it - immutability is a great tool of abstracting all kinds of problems. But sometimes it is a huge pain in the ass. Sometimes it simply is too constraining. And that alone makes me stop and thing - do we really want to loose mutability? Is it really either-or? Isn't there some common ground we can arrive at? When does immutability help me achieve my goals faster, when does mutability? Which solution is easier to read and maintain? (Which for me is the biggest question)
A lot of these questions are influenced by a programmer's taste and by what they are used to program in. So I'll focus on one of the aspects that is usually the center of most pro-immutability arguments - Parallelism:
Often parallelism is thrown into the argument surrounding immutability. I've worked on problem sets that required 100+ CPUs to solve in a reasonable time. And it has taught me one very important thing: Most of the time parallelizing the manipulation of graphs of data is really not the kind of thing that will be the most efficient way to parallelize. It sure can benefit greatly, but imbalance is a real problem in that problem-space. So usually working on multiple mutable graphs in parallel and exchanging information with immutable messages is way more efficient. Which means, when I know that the graph is isolated, that I have not revealed it to the outside world, I would like to perform my operations on it in the most concise manner I can think of. And that usually involves mutating the data. But after these operation on the data I want to open the data up to the whole world - and that's the point where I usually get a bit nervous, if the data is mutable. Because other parts of the program could corrupt the data, the state becomes invalid, ... because after opening up to the world the data often does get into the world of parallelism.
So real world parallel programs usually have areas where data graphs are used in definitive single thread operations - because they simply are not known to the outside - and areas where they could be involved in multi-threaded operations (hopefully just supplying data not being manipulated). During those multi-threaded parts we never want them to change - it simply is better to work on outdated data than on inconsistent data. Which can be guaranteed by the notion of immutability.
That made me come to a simple conclusion: The real problem for me is that non of the programming languages I am familiar with allow me to say: "After this point this whole data structure shal be immutable" and "give me a mutable copy of this immutable data structure here, please verify that only I can see the mutable copy". Right now I have to guarantee it myself by flipping a readonly bit or something similar. If we could have compiler support for it, not only would it guarantee for me that I did not do anything stupid after flipping said bit, but it could actually help the compiler do various optimizations that it couldn't do before. Plus - the language would still be attractive for programmers that are more familiar with the imperative programming model.
So to sum up. IMHO programs usually have a good reason to use both immutable and mutable representations of data graphs. I would argue that data should be immutable by default and the compiler should enforce that - but we should have the notion of private mutable representations, because there naturally are areas where multi-threading will never reach - and readability and maintainability could benefit from a more imperative structuring.