Essentially, the idea is the following: consider a function f: 'a * 'b -> 'c. This function takes a pair of arguments, the first of type 'a, and the second of type 'b, and gives back a result in 'c. There is only one way f can be applied: by passing the two arguments together. Given x:'a, and y:'b, we write f(x,y) for the application of f to the pair(x,y).
We could imagine now a variant of f that, instead of taking the arguments together, takes them "one at the time", and gives the same result as f when it is supplied with both arguments. This variant is called curried version of f. Let us denote it by f_{c}. The type of this function is f_{c}: 'a -> 'b -> 'c, and, by definition, we have that for every x:'a, and y:'b, f(x,y) = f_{c} x y holds. The main difference between f and f_{c} is that the latter can be applied also to the first argument only: The expression f_{c} x (for x:'a) is perfectly legal and denotes a function of type 'b -> 'c (the function which, when provided with an input y:'b, will give as result f(x,y)).
In order to provide support for curried functions, a language needs to be Higher Order. The currying adds flexibility and expressivity to the language, and it is part of the general principle of abstraction. In a sense, currying is intrinsic to the philosophy and design of an higher-order language. Below we will see some examples of how this feature adds expressivity.
Let us see how the currying possibility can be used in ML.
- fun append([],k) = k | append(x::l,k) = x::append(l,k); val append = fn : 'a list * 'a list -> 'a listThe curried version of append can be defined in the following way:
- fun append_c [] k = k | append_c (x::l) k = x::(append_c l k); val append_c = fn : 'a list -> 'a list -> 'a listor, equivalently:
- fun append_c [] = (fn k => k) | append_c (x::l) = fn k => x::(append_c l k); val append_c = fn : 'a list -> 'a list -> 'a listWe can now write expressions like append_c [1], for instance in a declaration:
- val append_one = append_c [1]; val append_one = fn : int list -> int listNote that the system does not evaluate the expression append_c [1], because it is a function. It only computes its type. The evaluation will be performed only when we provide also the second argument. For instance:
- append_one [5,6]; val it = [1,5,6] : int list
Consider the functions sum_all : int list -> int and product_all : int list -> int (respectively sum and product of all the elements in a list of integers). They can be defined as follows:
- fun sum_all [] = 0 | sum_all (x::l) = x + sum_all l; val sum_all = fn : int list -> int - fun product_all [] = 1 | product_all (x::l) = x * product_all l; val product_all = fn : int list -> intNote that these two function work according to the same scheme: they scan the list (recursively) element by element, and perform a certain operation on every element (and on the result of the recursive call), and give a certain initial result when the list is empty.
This scheme is common to several other functions. We could then think of defining an abstract function (abstract wrt the operation and the initial element), which represent the general scheme. The particular functions (like sum_all and product_all can then be defined by providing the particular operation and initial value. This general function is commonly called reduce, and it is "more natural" to define it by using currying, as follows:
fun reduce f v [] = v | reduce f v (x::l) = f(x, reduce f v l); val reduce = fn : ('a * 'b -> 'b) -> 'b -> 'a list -> 'bHere, f: 'a * 'b -> 'b represents the operation, and v:'b represents the initial value.
The definitions of sum_all and product_all can now be given as follows:
- val sum_all = reduce (op +) 0; val sum_all = fn : int list -> int - val product_all = reduce (op * ) 1; val product_all = fn : int list -> intWe need to use (op +) and (op * ) instead of + and * because the latter are infix, while in the definition of reduce the parameter f is prefix. The operator op changes a function from infix to prefix.
We can now apply these functions to lists of integers, as illustrated in the following examples:
- sum_all []; val it = 0 : int - sum_all [1,2,3,4]; val it = 10 : int - product_all []; val it = 1 : int - product_all [1,2,3,4]; val it = 24 : intAnother example of function that we can define by using reduce is the function forall, which checks whether a certain property p: 'a -> bool holds for all elements of a list (of type 'a list). We can define it as follows:
fun forall p = let fun f(x,b) = p x andalso b in reduce f true end; val forall = fn : ('a -> bool) -> 'a list -> boolExamples of uses (note that (fn x => x>0) represents the property of being positive; (fn x => x mod 2 = 0) represents the property of being even).
- forall (fn x => x>0) [1,2,3]; val it = true : bool - forall (fn x => x>0) [~1,2,3]; val it = false : bool - forall (fn x => x mod 2 = 0) [2,0,4]; val it = true : bool - forall (fn x => x mod 2 = 0) [2,1]; val it = false : boolOther examples can be found in Assignment 8 (CSE 428, Spring 99). Note that in the assignment the type of reduce is restricted. The reason is due to a certain limitation of the type system of the present implementation of SML. We won't go into that because it's a bit complicated, and not so important.
The concept of currying extends naturally to arbitrary tuples of arguments.
Exp ::= Ide (identifiers) | Exp Exp (functional application) | fn Pattern => Exp (functional abstraction) | (Exp,Exp) (pairing) | Exp :: Exp (cons on lists) | hd Exp (head of a list) | tl Exp (tail of a list) | nil (empty list) Pattern ::= Ide | (Ide,Ide)This is a very small subset of ML (both wrt the expressions and the patterns), and we may wish to extend it later, so to include other interesting data types and constants (like numbers). For the moment however we prefer focussing on just few constructs.
The types of this language constitute a language (type expressions) described by the following grammar:
Type ::= TVar (type variables, i.e. parametres for types) | Type * Type (Cartesian product, the type of pairs) | Type -> Type (functional type, the type of functions) | Type list (the type of lists)We will use the Greek letters alpha, beta, gamma,... to represent the type variables. They correspond to the dashed symbols 'a, 'b, 'c, ... used by the ML system.
Convention: * is left associative. -> is right associative. list has precedence wrt *, and * has precedence wrt ->.
Before studying the Type System formally, let us see some examples of type inferences that are done automatically by the type system of ML.
- fn f => fn x => fn y => f x y; val it = fn : ('a -> 'b -> 'c) -> 'a -> 'b -> 'c
- fn f => fn (x,y) => f(x,y); val it = fn : ('a * 'b -> 'c) -> 'a * 'b -> 'c
- fn f => fn x => fn y => (f x,f y); val it = fn : ('a -> 'b) -> 'a -> 'a -> 'b * 'b
- fn l => fn f => (f (hd l)) :: (tl l); val it = fn : 'a list -> ('a -> 'a) -> 'a list
alpha -> beta -> gamma -> deltawhere alpha is the type of f, beta is the type of x, gamma is the type of y, and delta is the type of the result.
Now, let us analyze how these types are related. In the resulting expression, f is applied to x and the result is applied to y (remember that f x y = (f x) y because application is left-associative). Hence alpha = beta -> phi where phi is the type of (f x). Since we then apply (f x) to y, and we have called delta the type of the result, we must have phi = gamma -> delta.
In conclusion the type is:
(beta -> gamma -> delta) -> beta -> gamma -> delta(names are not important, only their relation is)
We need to put parentheses around the first beta -> gamma -> delta because if we don't do that then the type becomes
beta -> gamma -> delta -> beta -> gamma -> deltawhich is interpreted as
beta -> (gamma -> (delta -> (beta -> (gamma -> delta))))since -> is right-associative.
fn f => fn (x,y) => f(x,y) is a function, where the second parameter is a pair, hence its type must be of the form
alpha -> (beta * gamma) -> deltawhere f: alpha, x: beta, y: gamma, and (f(x,y)): delta
Since f is applied to (x,y) and the result is of type delta, then it must be
alpha = (beta * gamma) -> deltahence the type is
((beta * gamma) -> delta) -> (beta * gamma) -> delta(the parentheses around (beta * gamma) are not necessary because * has priority wrt ->)
alpha -> beta -> gamma -> deltawhere f: alpha, x: beta, y: gamma, and (f x, f y): delta
Now, since the result is a pair, we must have delta = phi * psi, where (f x): phi and (f y): psi. Since f is applied to x and to y, we must have alpha = beta -> phi, and also alpha = gamma -> psi. Hence we obtain beta = gamma, and phi = psi. Thus the type is (again, names are not important):
(beta -> phi) -> beta -> beta -> phi * phi
alpha -> beta -> gammawhere l: alpha, f: beta, and ((f (hd l)) :: (tl l)) : gamma.
Since the result is constructed with a cons operation, it must be a list. Hence we have gamma = delta list, where delta is the type of (f (hd l)). Furthermore, the rest of the list is given by (tl l), hence also l must have type delta list (l and (tl l) have the same type). Therefore we have alpha = delta list. Finally, observe that (hd l): delta and (f (hd l)): delta, hence we can deduce f: delta -> delta. In conclusion, the resulting type is:
delta list -> (delta -> delta) -> delta list
alpha -> (alpha -> beta) -> betaLet us try to construct an expression with this type. Clearly, it must be a (curried) function with two arguments, i.e. must have the following structure.
fn ... => fn ... => ...Let us now fill in the dots. Let us call x and f, respectively, the first and second argument of the function. Clearly x: alpha and f: alpha -> beta. Now we have only to construct an expression with type beta, and use it as result.
Let us use an analogy: think of alpha as milk, and of beta as yogurt. Then f represents a machine that transforms milk into yogurt. Now, we have some milk (x), we have the machine, and we want to obtain yougurt. What shall we do? The answer is, of course,
turn on the machine and put the milk in iti.e. use f with input x. Hence, the resulting expression is:
fn x => fn f => f x
(alpha * beta -> gamma) -> alpha -> beta -> gammaAn expression with this type must have the following structure:
fn f => fn x => fn y => ...where f: alpha * beta -> gamma, x: alpha, and x: beta. In order to fill in the result, let us use again an analogy similar to the above.
Look at f as a machine that, when provided with flour (alpha) and yeast (beta) together, it makes bread (gamma). We have some flour (x), we have some yeast (y), how to obtain bread? The answer is, of course,
use f and give it in input x and y together.Hence the solution is:
fn f => fn x => fn y => f(x,y)
fn f => f fLet us see why it does not have a type. If it did, it should be something like
(alpha -> beta) -> betawhere f: alpha -> beta and (f f): beta. However, since f is also the argument in (f f), we should have f: alpha. But the equation alpha = alpha -> beta is unsatisfiable (at least for finite types, as it is the case in the Type Theory of ML. Such equation admits only an infinite solution alpha = ((...) -> beta) -> beta) -> beta.)
For a more intuitive explanation: Think again of f as a machine that transforms milk into yogurt. We can clone (make a copy) of this machine and try to feed the first machine with the second, but it won't work.
fn f => fn x => (f x, f(x,x))Intuitively, the type of f should be compatible with both the inputs x and (x,x). Namely, we should have at the same time f: alpha -> beta and f: alpha * alpha -> gamma. This would be possible only if the equation alpha = alpha * alpha were solvable, but this is not the case for finite types (a set cannot be equal to the cartesian product of the same set with itself). Note the analogy with the numeric equation x = x + 1, which is unsolvable (for finite numbers) for the same reason.
Note that if we write in ML the above expressions, we get a type error.
(alpha -> beta) -> beta (alpha -> beta) -> alpha beta -> (alpha -> beta) -> alpha alpha -> alpha * betaIf we try to construct an expression for any of these types, we won't succeed. Consider for instance the first type. We have a machine that tranforms milk into yogurt, but we don't have any milk. How can we obtain yogurt? The answer is "We can't". (Feeding the machine with (a copy of) itself does not work, as seen before. Selling the machine and using the money to buy yogurt is not allowed :-)
For instance, the type (alpha -> beta) -> beta above is not a tautology and in fact it is empty. The type alpha -> (alpha -> beta) -> beta, on the contrary, is inhabited, and in fact it is a tautology.Theorem
A type is inhabited (i.e. not empty) only if it corresponds to a logically valid formula (i.e. a tautology).
Hence we can give the following criterion for the emptyness of a type t:
if t is not a tautology, then we know that t is emptyThe reverse does not hold: there are types which are tautologies, but still are empty. One example of such a type is
((alpha -> alpha) -> alpha) -> alphaIn the sub-theory of Classical Logic called Intuitionistic Logic, however, the correspondence is complete. We have in fact that:
A type is inhabited (i.e. not empty) if and only if it corresponds to a formula intuitionistically valid.Sub-theory here means that less formulae are valid.
fn x => xThe type of this expression is
alpha -> alphaLet us call f the function represented by the above expression. Clearly, we can use f also with less general (i.e. more instantiated) types, like for instance pairs, or functions. This means that f can be regarded also as a function of the follwing types:
beta * gamma -> beta * gamma (beta -> gamma) -> beta ->gamma beta list -> beta list ...The difference between alpha -> alpha and the types above is that alpha -> alpha is more general: each type above can be obtained by replacing (instantiating) alpha with a more specific type (respectively, with beta * gamma, beta -> gamma and beta list). It is possible to prove that alpha -> alpha is the most general (or principal) type of the above expression. Any type for that expression can be obtained by intantiating the type alpha -> alpha.
This result is nice because it allows us to consider only the most general type t of an expression: if later, when we use the expression in a certain context, we need an other type, we know we can just derive it from t. The ML type system always derives the most general type of an expression.Theorem
If an expression has a type, then it has a most general type.
In general, when we construct the type of an expression, we derive first the structure s of the type, and then, by analysing the expression, a set of equations E between type variables. The final type is then obtained by instantiating all the type variables in s so to reflect the constraints imposed by the equations.
More precisely, in order to make sure that we do not miss any constraint, and that we do not impose constraints that aren't necessary, we should instantiate s by using the "most general solution" of the set of equations E. A solution for E is an association between type variables and types which validates all the equations.
A solution for E can be expressed as a system of equations E' in solved form, i.e. such that the left hand sides of each equation is a distinct type variable, and the right hand sides are type expressions containing none of the variables of the lhs. We say that a solution E' for E is most general if E' is equivalent to E, i.e. E and E' have exactly the same solutions.
The most general solution for E can be obtained by repeatedly performing the following operations on E:
alpha -> beta = gamma -> delta * epsilonshould be replaced by the two equations
alpha = gamma beta = delta * epsilon
alpha = gamma -> delta * epsilonthen we should replace alpha by gamma -> delta * epsilon in every other equations
fn f => fn x => (f (f x))Its type will have the following structure s:
alpha -> beta -> gammaWith f: alpha, x: beta, and ((f (f x)): gamma. The constraints, which we derive from the form of the result ((f (f x)), are expressed by the following system of equations E:
alpha = beta -> epsilon, where (f x): epsilon, and alpha = epsilon -> gammaThe most general solution of this set of equations can be obtained by first replacing alpha by beta -> epsilon in the other equation:
alpha = beta -> epsilon beta -> epsilon = epsilon -> gammathen simplify the second equation:
alpha = beta -> epsilon beta = epsilon epsilon = gammathen, replace epsilon by gamma (third equation) in the other equations:
alpha = beta -> gamma beta = gamma epsilon = gammaFinally, replace beta by gamma (second equation) in the other equations:
alpha = gamma -> gamma beta = gamma epsilon = gammaThis final system is in solved form, hence it is the most general solution of E. By applying this solution to s we obtain the most general type, i.e.
(gamma -> gamma) -> gamma -> gammaNote that, when we solve a system of equations, there are in general several ways to proceed, depending on the choice of the particular equation considered at every step. Different choices bring to different (but equivalent) formulation of the result, and to different names in the final type. This does not matter, since the names of type variables are not important.
fun f x = ewithout recursion. In order to derive the type of f we can transform this declaration into the equivalent val declaration:
val f = fn x => enow we can derive the type of fn x => e in the way seen before. This will be also the type of f.
With a little bit of fantasy, we can apply the reasoning directly, without the need of transforming the fun declaration into a val declaration.
fun f v [] = v | f v (x::l) = (x,x) :: []f is curried and takes two arguments, hence the structure s of its type is
alpha -> beta -> gammaWe have the following constraints:
gamma = alpha, from the result in the fst line beta = delta list, from the pattern of the snd arg in the fst line gamma = (phi * phi) list, from the result in the snd line (where x: phi) delta = phi, from the pattern of the snd arg in the snd lineA solved form of this system is
beta = phi list, gamma = (phi * phi) list, alpha = (phi * phi) list, delta = phi,Which, applied to s, gives the type
(phi * phi) list -> phi list -> (phi * phi) list
fun f v [] = [v] | f v (x::l) = f x lf is curried and takes two arguments, hence the structure s of its type is
alpha -> beta -> gammaWe have the following constraints:
gamma = alpha list, from the result in the fst line beta = delta list, from the pattern of the snd arg in the fst line delta = alpha, from the recursive call in the snd lineA solution is
gamma = alpha list beta = alpha list delta = alphawhich, applied to s, gives the type
alpha -> alpha list -> alpha list
The type system consists in a set of rules which define, inductively, the relation
r |- e : t (e has type t under the assumptions r)where:
Note the analogy of the type statement r |- e : t and the evaluation statement env |- e eval t in past lecture notes. Due to this analogy, the type system is sometimes called "static semantics", and r is called "static environment". The roles of r and env are very similar, and their behaviors define the same scoping rules.
Exp ::= Ide (identifiers) | Exp Exp (functional application) | fn Ide => Exp (functional abstraction 1) | fn (Ide,Ide) => Exp (functional abstraction 2) | (Exp,Exp) (pairing) | Exp :: Exp (cons on lists) | hd Exp (head of a list) | tl Exp (tail of a list) | nil (empty list) Type ::= TVar (type variables, i.e. parametres for types) | Type * Type (Cartesian product, the type of pairs) | Type -> Type (functional type, the type of functions) | Type list (the type of lists)
(ide) ------------------ (where x is an identifier) r |- x : r(x) r |- e1 : alpha -> beta r |- e2 : alpha (app) ---------------------------------------------- r |- (e1 e2) : beta r[x:alpha] |- e : beta (abs1) ---------------------------------- r |- (fn x => e) : alpha -> beta
r[x:alpha][y:beta] |- e : gamma (abs2) ----------------------------------------------- r |- fn (x,y) => e : (alpha * beta) -> gamma r |- e1 : alpha r |- e2 : beta (pair) --------------------------------------- r |- (e1,e2) : alpha * beta
r |- e1 : alpha r |- e2 : alpha list (cons) ------------------------------------------- r |- (e1 :: e2) : alpha list r |- e : alpha list (head) ----------------------- r |- (hd e) : alpha r |- e : alpha list (tail) ---------------------------- r |- (tl e) : alpha list (nil) ---------------------------- r |- nil : alpha list
We say that e has type t if we can derive the statement emptyset |- e : t.
In order to derive the most general type of an expression, we need to construct a proof for the statement emptyset |- e : alpha, and then instantiate alpha with the solution of the constraint that we find in this proof (i.e. the constraints imposed by the particular form of the types in the conclusion of the rules) from the var rule. A similar process allows us to derive an expression for a given type t.