I had previously asked where the Winery types are indexed. I noticed that in the serialization for the schema for Bool, which is [4,6], the 4 is the version number, and 6 is the index of SBool in SchemaP. I verified the hypothesis using other "primitive" types like Integer (serialization: 16), Double (18), Text (20). Also, [Bool] will be SVector SBool, serialized to [4,2,6], which makes sense: the 2 is for SVector, the 6 is for SBool.
But SchemaP also contains constructors that I don't intuitively see how are used: SFix, SVar, STag and SLet. What are they, and which type would I need to construct the schema for, to see them used? Why is SLet at the end, but SFix at the beginning?
SFix looks like a µ quantifier for a recursive type. The type µx. T is the type T where x refers to the whole type µx. T. For example, a list data List a = Nil | Cons a (List a) can be represented as L(a) = µr. 1 + a × r, where the recursive occurrence of the type is replaced with the variable r. You could probably see this with a recursive user-defined type like data BinTree a = Leaf | Branch a (BinTree a) (BinTree a).
This encoding doesn’t explicitly include a variable name, because the next constructor SVar specifies that “SVar n refers to the nth innermost fixpoint”, where n is an Int in the synonym type Schema = SchemaP Int. This is a De Bruijn index. If you had some nested recursive types like µx. µy. … = SFix (SFix …), then the inner variable would be referenced as SVar 0 and the outer one as SVar 1 within the body …. This “relative” notation means you can freely reorganise terms without worrying about having to rename variables for capture-avoiding substitution.
SLet is a let binding, and since it’s specified as SLet !(SchemaP a) !(SchemaP a), I presume that SLet e1 e2 is akin to let x = e1 in e2, where the variable is again implicit. So I suspect this may be a deficiency of the docs, and SVar can also refer to Let-bound variables. I don’t know how they use this constructor, but it could be used to make sharing explicit in the schema.
Finally, STag appears to be a way to attach extra “tag” metadata within the schema, in some way that’s specific to the library.
The ordering of these constructors might be maintained for compatibility with earlier versions, so adding new constructors at the end would avoid disturbing the encoding, and I figure the STag and SLet constructors at the end were simply added later.
As I'm new to alloy, this is most likely a simple question. I've been through the on-line tutorials and am now reading the Software Abstractions, revised edition. On page 34 there is an example at the bottom of the page:
r' = {b:B, a:A, c:C | a->b->c in r}
where the text says that this defines a new relation of B->A->C. I don't see how an explicit order for r' is achieved by this statement.
It's the property of set comprehension
{a: A | somePredicate1[a]} is of type A and returns a set containing all atoms for which somePredicate1 holds;
{a: A, b: B | somePredicate2[a, b]} is of type A->B and returns a relation containing all a->b tuples for which somePredicate2 holds;
and so on
The syntax of set comprehension basically consists of two parts (1) type declaration (before the | character), and (2) predicate which must hold for every element in the returned set.
I am trying to design an API for some database system in Haskell, and I would like to model the columns of this database in a way such that interactions between columns of different tables cannot get mixed up.
More precisely, imagine that you have a type to represent a table in a database, associated to some type:
type Table a = ...
and that you can extract the columns of the table, along with the type of the column:
type Column col = ...
Finally, there are various extractors. For example, if your table contains descriptions of frogs, a function would let you extract the column containing the weight of the frog:
extractCol :: Table Frog -> Column Weight
Here is the question: I would like to distinguish the origin of the columns so that users cannot do operations between tables. For example:
bullfrogTable = undefined :: Table Frog
toadTable = undefined :: Table Frog
bullfrogWeights = extractCol bullfrogTable
toadWeights = extractCol toadTable
-- Or some other columns from the toad table
toadWeights' = extractCol toadTable
-- This should compile
addWeights toadWeights' toadWeights
-- This should trigger a type error
addWeights bullfrogWeights toadWeights
I know how to achieve this in Scala (using path-dependent types, see [1]), and I have been thinking of 3 options in Haskell:
not using types, and just doing a check at runtime (the current solution)
the TypeInType extension to add a phantom type on the Table type itself, and pass this extra type to the columns. I am not keen on it, because the construction of such a type would be very complicated (tables are generated through complex DAG operations) and probably slow to compile in this context.
wrapping the operations using a forall construct similar to the ST monad, but in my case, I would like the extra tagging type to actually escape the construction.
I am happy to have a very limited valid scoping for the construction of the same columns (i.e. columns from table and (id table) not being mixable), and I mostly care about the DSL feel of the API rather than the safety.
[1] What is meant by Scala's path-dependent types?
My current solution
Here is what I ended up doing, using RankNTypes.
I still want to give users the ability to use columns how they see fit without having some strong type checks, and opt in if they want some stronger type guarantees: this is a DSL for data scientist who will not know the power of the Haskell side
Tables are still tagged by their content:
type Table a = ...
and columns are now tagged with some extra reference types, on top of the type of the data they contain:
type Column ref col = ...
Projections from tables to columns are either tagged or untagged. In practice, this is hidden behind a lens-like DSL.
extractCol :: Table Frog -> Column Frog Weight
data TaggedTable ref a = TaggedTable { _ttTable :: Table a }
extractColTagged :: Table ref Frog -> Column ref Weight
withTag :: Table a -> (forall ref. TaggedTable ref a -> b) -> b
withTag tb f = f (TaggedTable tb)
Now I can write some code as following:
let doubleToadWeights = withTag toadTable $ \ttoadTable ->
let toadWeights = extractColTagged ttoadTable in
addWeights toadWeights toadWeights
and this will not compile, as desired:
let doubleToadWeights =
toadTable `withTag` \ttoads ->
bullfrogTable `withTag` \tbullfrogs ->
let toadWeights = extractColTagged ttoads
bullfrogWeights = extractColTagged tbullfrogs
in addWeights toadWeights bullfrogWeights -- Type error
From a DSL perspective, I believe it is not as straightforward as what one could achieve with Scala, but the type error message is understandable, which is paramount for me.
Haskell does not (as far as I know) have path dependent types, but you can get some of the way by using rank 2 types. For instance the ST monad has a dummy type parameter s that is used to prevent leakage between invocations of runST:
runST :: (forall s . ST s a) -> a
Within an ST action you can have an STRef:
newSTRef :: a -> ST s (STRef s a)
But the STRef you get carries the s type parameter, so it isn't allowed to escape from the runST.
Are these two equivalent:
r: A -> B
r: A set -> set B
That is, is set the default multiplicity?
If yes, then I will quibble with the definition of the arrow operator in the Software Abstractions book. The book says on page 55:
The arrow product (or just product) p->q of two relations p and q is
the relation you get by taking every combination of a tuple from p and
a tuple from q and concatenating them.
I interpret that definition to mean the only valid instance for p->q is one that has every possible combination of tuples from p with tuples from q. But that's not right (I think). Any instance containing mappings between p and q is valid. For example, on page 56 is this example,
Name = {(N0), (N1)}
Addr = {(D0), (D1)}
The book says this is a valid relation for Name->Addr
{(N0, D0), (N0, D1), (N1, D0), (N1, D1)}
But that's not the only valid relation, right? For example, this is a valid relation:
{(N0, D0), (N1, D1)}
Is that right?
The declaration r : A->B means r is a subset of A->B. The expression A->B has just one value, which is the cross product of A and B. The declaration results in a set of possible values for r, which would include both the example given in the book that you cite, and the example that you ask about.
I am learning Haskell from learnyouahaskell.com. I am having trouble understanding type constructors and data constructors. For example, I don't really understand the difference between this:
data Car = Car { company :: String
, model :: String
, year :: Int
} deriving (Show)
and this:
data Car a b c = Car { company :: a
, model :: b
, year :: c
} deriving (Show)
I understand that the first is simply using one constructor (Car) to built data of type Car. I don't really understand the second one.
Also, how do data types defined like this:
data Color = Blue | Green | Red
fit into all of this?
From what I understand, the third example (Color) is a type which can be in three states: Blue, Green or Red. But that conflicts with how I understand the first two examples: is it that the type Car can only be in one state, Car, which can take various parameters to build? If so, how does the second example fit in?
Essentially, I am looking for an explanation that unifies the above three code examples/constructs.
In a data declaration, a type constructor is the thing on the left hand side of the equals sign. The data constructor(s) are the things on the right hand side of the equals sign. You use type constructors where a type is expected, and you use data constructors where a value is expected.
Data constructors
To make things simple, we can start with an example of a type that represents a colour.
data Colour = Red | Green | Blue
Here, we have three data constructors. Colour is a type, and Green is a constructor that contains a value of type Colour. Similarly, Red and Blue are both constructors that construct values of type Colour. We could imagine spicing it up though!
data Colour = RGB Int Int Int
We still have just the type Colour, but RGB is not a value – it's a function taking three Ints and returning a value! RGB has the type
RGB :: Int -> Int -> Int -> Colour
RGB is a data constructor that is a function taking some values as its arguments, and then uses those to construct a new value. If you have done any object-oriented programming, you should recognise this. In OOP, constructors also take some values as arguments and return a new value!
In this case, if we apply RGB to three values, we get a colour value!
Prelude> RGB 12 92 27
#0c5c1b
We have constructed a value of type Colour by applying the data constructor. A data constructor either contains a value like a variable would, or takes other values as its argument and creates a new value. If you have done previous programming, this concept shouldn't be very strange to you.
Intermission
If you'd want to construct a binary tree to store Strings, you could imagine doing something like
data SBTree = Leaf String
| Branch String SBTree SBTree
What we see here is a type SBTree that contains two data constructors. In other words, there are two functions (namely Leaf and Branch) that will construct values of the SBTree type. If you're not familiar with how binary trees work, just hang in there. You don't actually need to know how binary trees work, only that this one stores Strings in some way.
We also see that both data constructors take a String argument – this is the String they are going to store in the tree.
But! What if we also wanted to be able to store Bool, we'd have to create a new binary tree. It could look something like this:
data BBTree = Leaf Bool
| Branch Bool BBTree BBTree
Type constructors
Both SBTree and BBTree are type constructors. But there's a glaring problem. Do you see how similar they are? That's a sign that you really want a parameter somewhere.
So we can do this:
data BTree a = Leaf a
| Branch a (BTree a) (BTree a)
Now we introduce a type variable a as a parameter to the type constructor. In this declaration, BTree has become a function. It takes a type as its argument and it returns a new type.
It is important here to consider the difference between a concrete type (examples include Int, [Char] and Maybe Bool) which is a type that can be assigned to a value in your program, and a type constructor function which you need to feed a type to be able to be assigned to a value. A value can never be of type "list", because it needs to be a "list of something". In the same spirit, a value can never be of type "binary tree", because it needs to be a "binary tree storing something".
If we pass in, say, Bool as an argument to BTree, it returns the type BTree Bool, which is a binary tree that stores Bools. Replace every occurrence of the type variable a with the type Bool, and you can see for yourself how it's true.
If you want to, you can view BTree as a function with the kind
BTree :: * -> *
Kinds are somewhat like types – the * indicates a concrete type, so we say BTree is from a concrete type to a concrete type.
Wrapping up
Step back here a moment and take note of the similarities.
A data constructor is a "function" that takes 0 or more values and gives you back a new value.
A type constructor is a "function" that takes 0 or more types and gives you back a new type.
Data constructors with parameters are cool if we want slight variations in our values – we put those variations in parameters and let the guy who creates the value decide what arguments they are going to put in. In the same sense, type constructors with parameters are cool if we want slight variations in our types! We put those variations as parameters and let the guy who creates the type decide what arguments they are going to put in.
A case study
As the home stretch here, we can consider the Maybe a type. Its definition is
data Maybe a = Nothing
| Just a
Here, Maybe is a type constructor that returns a concrete type. Just is a data constructor that returns a value. Nothing is a data constructor that contains a value. If we look at the type of Just, we see that
Just :: a -> Maybe a
In other words, Just takes a value of type a and returns a value of type Maybe a. If we look at the kind of Maybe, we see that
Maybe :: * -> *
In other words, Maybe takes a concrete type and returns a concrete type.
Once again! The difference between a concrete type and a type constructor function. You cannot create a list of Maybes - if you try to execute
[] :: [Maybe]
you'll get an error. You can however create a list of Maybe Int, or Maybe a. That's because Maybe is a type constructor function, but a list needs to contain values of a concrete type. Maybe Int and Maybe a are concrete types (or if you want, calls to type constructor functions that return concrete types.)
Haskell has algebraic data types, which very few other languages have. This is perhaps what's confusing you.
In other languages, you can usually make a "record", "struct" or similar, which has a bunch of named fields that hold various different types of data. You can also sometimes make an "enumeration", which has a (small) set of fixed possible values (e.g., your Red, Green and Blue).
In Haskell, you can combine both of these at the same time. Weird, but true!
Why is it called "algebraic"? Well, the nerds talk about "sum types" and "product types". For example:
data Eg1 = One Int | Two String
An Eg1 value is basically either an integer or a string. So the set of all possible Eg1 values is the "sum" of the set of all possible integer values and all possible string values. Thus, nerds refer to Eg1 as a "sum type". On the other hand:
data Eg2 = Pair Int String
Every Eg2 value consists of both an integer and a string. So the set of all possible Eg2 values is the Cartesian product of the set of all integers and the set of all strings. The two sets are "multiplied" together, so this is a "product type".
Haskell's algebraic types are sum types of product types. You give a constructor multiple fields to make a product type, and you have multiple constructors to make a sum (of products).
As an example of why that might be useful, suppose you have something that outputs data as either XML or JSON, and it takes a configuration record - but obviously, the configuration settings for XML and for JSON are totally different. So you might do something like this:
data Config = XML_Config {...} | JSON_Config {...}
(With some suitable fields in there, obviously.) You can't do stuff like this in normal programming languages, which is why most people aren't used to it.
Start with the simplest case:
data Color = Blue | Green | Red
This defines a "type constructor" Color which takes no arguments - and it has three "data constructors", Blue, Green and Red. None of the data constructors takes any arguments. This means that there are three of type Color: Blue, Green and Red.
A data constructor is used when you need to create a value of some sort. Like:
myFavoriteColor :: Color
myFavoriteColor = Green
creates a value myFavoriteColor using the Green data constructor - and myFavoriteColor will be of type Color since that's the type of values produced by the data constructor.
A type constructor is used when you need to create a type of some sort. This is usually the case when writing signatures:
isFavoriteColor :: Color -> Bool
In this case, you are calling the Color type constructor (which takes no arguments).
Still with me?
Now, imagine you not only wanted to create red/green/blue values but you also wanted to specify an "intensity". Like, a value between 0 and 256. You could do that by adding an argument to each of the data constructors, so you end up with:
data Color = Blue Int | Green Int | Red Int
Now, each of the three data constructors takes an argument of type Int. The type constructor (Color) still doesn't take any arguments. So, my favorite color being a darkish green, I could write
myFavoriteColor :: Color
myFavoriteColor = Green 50
And again, it calls the Green data constructor and I get a value of type Color.
Imagine if you don't want to dictate how people express the intensity of a color. Some might want a numeric value like we just did. Others may be fine with just a boolean indicating "bright" or "not so bright". The solution to this is to not hardcode Int in the data constructors but rather use a type variable:
data Color a = Blue a | Green a | Red a
Now, our type constructor takes one argument (another type which we just call a!) and all of the data constructors will take one argument (a value!) of that type a. So you could have
myFavoriteColor :: Color Bool
myFavoriteColor = Green False
or
myFavoriteColor :: Color Int
myFavoriteColor = Green 50
Notice how we call the Color type constructor with an argument (another type) to get the "effective" type which will be returned by the data constructors. This touches the concept of kinds which you may want to read about over a cup of coffee or two.
Now we figured out what data constructors and type constructors are, and how data constructors can take other values as arguments and type constructors can take other types as arguments. HTH.
As others pointed out, polymorphism isn't that terrible useful here. Let's look at another example you're probably already familiar with:
Maybe a = Just a | Nothing
This type has two data constructors. Nothing is somewhat boring, it doesn't contain any useful data. On the other hand Just contains a value of a - whatever type a may have. Let's write a function which uses this type, e.g. getting the head of an Int list, if there is any (I hope you agree this is more useful than throwing an error):
maybeHead :: [Int] -> Maybe Int
maybeHead [] = Nothing
maybeHead (x:_) = Just x
> maybeHead [1,2,3] -- Just 1
> maybeHead [] -- None
So in this case a is an Int, but it would work as well for any other type. In fact you can make our function work for every type of list (even without changing the implementation):
maybeHead :: [t] -> Maybe t
maybeHead [] = Nothing
maybeHead (x:_) = Just x
On the other hand you can write functions which accept only a certain type of Maybe, e.g.
doubleMaybe :: Maybe Int -> Maybe Int
doubleMaybe Just x = Just (2*x)
doubleMaybe Nothing= Nothing
So long story short, with polymorphism you give your own type the flexibility to work with values of different other types.
In your example, you may decide at some point that String isn't sufficient to identify the company, but it needs to have its own type Company (which holds additional data like country, address, back accounts etc). Your first implementation of Car would need to change to use Company instead of String for its first value. Your second implementation is just fine, you use it as Car Company String Int and it would work as before (of course functions accessing company data need to be changed).
The second one has the notion of "polymorphism" in it.
The a b c can be of any type. For example, a can be a [String], b can be [Int]
and c can be [Char].
While the first one's type is fixed: company is a String, model is a String and year is Int.
The Car example might not show the significance of using polymorphism. But imagine your data is of the list type. A list can contain String, Char, Int ... In those situations, you will need the second way of defining your data.
As to the third way I don't think it needs to fit into the previous type. It's just one other way of defining data in Haskell.
This is my humble opinion as a beginner myself.
Btw: Make sure that you train your brain well and feel comfortable to this. It is the key to understand Monad later.
It's about types: In the first case, your set the types String (for company and model) and Int for year. In the second case, your are more generic. a, b, and c may be the very same types as in the first example, or something completely different. E.g., it may be useful to give the year as string instead of integer. And if you want, you may even use your Color type.