❤️ Type class 🐒

"In computer science, a type class is a type system construct that supports ad hoc polymorphism. This is achieved by adding constraints to type variables in parametrically polymorphic types. Such a constraint typically involves a type class `T` and a type variable `a`, and means that `a` can only be instantiated to a type whose members support the overloaded operations associated with `T`. Type classes were first implemented in the Haskell programming language after first being proposed by Philip Wadler and Stephen Blott as an extension to "eqtypes" in Standard ML, and were originally conceived as a way of implementing overloaded arithmetic and equality operators in a principled fashion. In contrast with the "eqtypes" of Standard ML, overloading the equality operator through the use of type classes in Haskell does not require extensive modification of the compiler frontend or the underlying type system. Since their creation, many other applications of type classes have been discovered. Overview Type classes are defined by specifying a set of function or constant names, together with their respective types, that must exist for every type that belongs to the class. In Haskell, types can be parameterized; a type class `Eq` intended to contain types that admit equality would be declared in the following way: class Eq a where () :: a -> a -> Bool (/=) :: a -> a -> Bool where `a` is one instance of the type class `Eq`, and `a` defines the function signatures for 2 functions (the equality and inequality functions), which each take 2 arguments of type `a` and return a boolean. The type variable `a` has kind * (also known as `Type` in the latest GHC release),`Type` from `Data.Kind` appeared in version 8 of the Glasgow Haskell Compiler meaning that the kind of `Eq` is Eq :: Type -> Constraint The declaration may be read as stating a "type `a` belongs to type class `Eq` if there are functions named `()`, and `(/=)`, of the appropriate types, defined on it." A programmer could then define a function `elem` (which determines if an element is in a list) in the following way: elem :: Eq a => a -> [a] -> Bool elem y [] = False elem y (x:xs) = (x y) elem y xs The function `elem` has the type `a -> [a] -> Bool` with the context `Eq a`, which constrains the types which `a` can range over to those `a` which belong to the `Eq` type class. (Note: Haskell ` => ` can be called a 'class constraint'.) A programmer can make any type `t` a member of a given type class `C` by using an instance declaration that defines implementations of all of `C`'s methods for the particular type `t`. For instance, if a programmer defines a new data type `t`, they may then make this new type an instance of `Eq` by providing an equality function over values of type `t` in whatever way they see fit. Once they have done this, they may use the function `elem` on `[t]`, that is, lists of elements of type `t`. Note that type classes are different from classes in object-oriented programming languages. In particular, `Eq` is not a type: there is no such thing as a value of type `Eq`. Type classes are closely related to parametric polymorphism. For example, note that the type of `elem` as specified above would be the parametrically polymorphic type `a -> [a] -> Bool` were it not for the type class constraint "`Eq a =>`". Higher-kinded polymorphism A type class need not take a type variable of kind `Type` but can take one of any kind. These type classes with higher kinds are sometimes called constructor classes (the constructors referred to are type constructors such as `Maybe`, rather than data constructors such as `Just`). An example is the `Monad` class: class Monad m where return :: a -> m a (>>=) :: m a -> (a -> m b) -> m b The fact that m is applied to a type variable indicates that it has kind `Type -> Type`, i.e. it takes a type and returns a type, the kind of `Monad` is thus: Monad :: (Type -> Type) -> Constraint Multi-parameter type classes Type classes permit multiple type parameters, and so type classes can be seen as relations on types.Haskell' page MultiParamTypeClasses. For example, in the GHC standard library, the class `IArray` expresses a general immutable array interface. In this class, the type class constraint `IArray a e` means that `a` is an array type that contains elements of type `e`. (This restriction on polymorphism is used to implement unboxed array types, for example.) Like multimethods, multi-parameter type classes support calling different implementations of a method depending on the types of multiple arguments, and indeed return types. Multi-parameter type classes do not require searching for the method to call on every call at runtime;In GHC, the C Core uses Girard & Reynold's System F type signatures to identify a typed case for processing in the optimization phases. -- Simon Peyton-Jones "Into the Core - Squeezing Haskell into Nine Constructors" Erlang User Conference, Sep 14, 2016 rather the method to call is first compiled and stored in the dictionary of the type class instance, just as with single- parameter type classes. Haskell code that uses multi-parameter type classes is not portable, as this feature is not part of the Haskell 98 standard. The popular Haskell implementations, GHC and Hugs, support multi-parameter type classes. Functional dependencies In Haskell, type classes have been refined to allow the programmer to declare functional dependencies between type parameters--a concept inspired from relational database theory.Mark Jones. Type Classes with Functional Dependencies. From Proc. 9th European Symposium on Programming. March, 2000.Haskell' page FunctionalDependencies. That is, the programmer can assert that a given assignment of some subset of the type parameters uniquely determines the remaining type parameters. For example, a general monads `m` which carry a state parameter of type `s` satisfy the type class constraint `Monad.State s m`. In this constraint, there is a functional dependency `m -> s`. This means that for a given monad `m` of type class `Monad.State`, the state type accessible from `m` is uniquely determined. This aids the compiler in type inference, as well as aiding the programmer in type-directed programming. Simon Peyton-Jones has objected to the introduction of functional dependencies in Haskell on grounds of complexity.http://www.haskell.org/pipermail/haskell- prime/2006-February/000289.html Type classes and implicit parameters Type classes and implicit parameters are very similar in nature, although not quite the same. A polymorphic function with a type class constraint such as: sum :: Num a => [a] -> a can be intuitively treated as a function that implicitly accepts an instance of `Num`: sum_ :: Num_ a -> [a] -> a The instance `Num_ a` is essentially a record that contains the instance definition of `Num a`. (This is in fact how type classes are implemented under the hood by the Glasgow Haskell Compiler.) However, there is a crucial difference: implicit parameters are more flexible – you can pass different instances of `Num Int`. In contrast, type classes enforce the so-called coherence property, which requires that there should only be one unique choice of instance for any given type. The coherence property makes type classes somewhat antimodular, which is why orphan instances (instances that are defined in a module that neither contains the class nor the type of interest) are strongly discouraged. On the other hand, coherence adds an additional level of safety to the language, providing the programmer a guarantee that two disjoint parts of the same code will share the same instance.Edward Kmett, Type Classes vs. the World, Boston Haskell Meetup. As an example, an ordered set (of type `Set a`) requires a total ordering on the elements (of type `a`) in order to function. This can be evidenced by a constraint `Ord a`, which defines a comparison operator on the elements. However, there can be numerous ways to impose a total order. Since set algorithms are generally intolerant of changes in the ordering once a set has been constructed, passing an incompatible instance of `Ord a` to functions that operate on the set may lead to incorrect results (or crashes). Thus, enforcing coherence of `Ord a` in this particular scenario is crucial. Instances (or "dictionaries") in Scala type classes are just ordinary values in the language, rather than a completely separate kind of entity. While these instances are by default supplied by finding appropriate instances in scope to be used as the implicit actual parameters for explicitly-declared implicit formal parameters, the fact that they are ordinary values means that they can be supplied explicitly, to resolve ambiguity. As a result, Scala type classes do not satisfy the coherence property and are effectively a syntactic sugar for implicit parameters. This is an example taken from the Cats typelevel.org, Scala Cats documentation: // A type class to provide textual representation trait Show[A] { def show(f: A): String } // A polymorphic function that works only when there is an implicit // instance of Show[A] available def log[A](a: A)(implicit s: Show[A]) = println(s.show(a)) // An instance for String implicit val stringShow = new Show[String] { def show(s: String) = s } // The parameter stringShow was inserted by the compiler. scala> log("a string") a string Coq (version 8.2 onwards) also supports type classes by inferring the appropriate instances.A Gentle Introduction to Type Classes and Relations in Coq Recent versions of Agda 2 also provide a similar feature, called "instance arguments"."Modelling Type Classes With Instance Arguments". Other approaches to operator overloading In Standard ML, the mechanism of "equality types" corresponds roughly to Haskell's built-in type class `Eq`, but all equality operators are derived automatically by the compiler. The programmer's control of the process is limited to designating which type components in a structure are equality types and which type variables in a polymorphic type range over equality types. SML's and OCaml's modules and functors can play a role similar to that of Haskell's type classes, the principal difference being the role of type inference, which makes type classes suitable for ad hoc polymorphism. The object oriented subset of OCaml is yet another approach which is somewhat comparable to the one of type classes. Related notions An analogous notion for overloaded data (implemented in GHC) is that of type family. In Clean typeclasses are similar to Haskell, but have a slightly different syntax. Rust supports traits, which are a limited form of type classes with coherence. Mercury has typeclasses, although they are not exactly the same as in Haskell. In Scala, type classes are a programming idiom which can be implemented with existing language features such as implicit parameters, not a separate language feature per se. Because of the way they are implemented in Scala, it is possible to explicitly specify which type class instance to use for a type at a particular place in the code, in case of ambiguity. However, this is not necessarily a benefit as ambiguous type class instances can be error-prone. The proof assistant Coq has also supported type classes in recent versions. Unlike in ordinary programming languages, in Coq, any laws of a type class (such as the monad laws) that are stated within the type class definition, must be mathematically proved of each type class instance before using them. See also * Polymorphism (computer science) (other kinds of polymorphism) * Haskell programming language (the language in which type classes were first designed) * Operator overloading (one application of type classes) * Monad (functional programming) (`Monad` is an example of a type class) * Concepts (C++) (since C++20) * Rust (programming language) References * Simon Peyton Jones, Mark Jones, Erik Meijer. Type classes: an exploration of the design space. From Proc. ACM SIGPLAN Haskell Workshop. May, 1997. External links * A Gentle Introduction to Haskell, Version 98, chapter 5. Type Classes and Overloading. June 2000. * Advanced Functional Programming course at Utrecht University, 74 lecture slides on Advanced Type Classes. 2005-06-07. * Implementing, and Understanding Type Classes. 2014-11-13. Category:Functional programming Category:Type theory Category:Data types Category:Articles with example Haskell code "

❤️ Mackworth Island 🐒

"Aerial view of Mackworth Island Sate Park Mackworth Island State Park is an approximately island in Falmouth, Maine, USA, adjacent to its border with Portland, Maine. In 1631, Sir Ferdinando Gorges gave the island to Arthur Mackworth, his deputy in Casco Bay, and the island has retained his name. There is a causeway connecting the island to the mainland in Falmouth. Visitors to the island must pass a State park entrance tollhouse and pay an entrance fee; cars can enter the island but parking is limited. There is a footpath around the perimeter of the island with views of Falmouth, Portland, and other islands surrounding the bay. The island is heavily wooded. One portion of the woods is dedicated to “Fairy Houses” which are constructed by visitors using natural materials found on the island. https://www.fairyhouses.com/news/mackworth-island-permanent-fairy-houses- village/ Mackworth Island has a beautiful rocky shore and offers ample opportunities for surf fisherman of Bluefish and Stripers.Features Why I hate Mackworth Island There is a nominal 3 dollar entrance fee per person. Although most areas of the island are State Park lands, there is a school area that is not open to the public. The island is also home to the Governor Baxter Dog Memorial, a pet cemetery where fourteen of the former governor's Irish Setters and one of his horses were laid to rest, accompanied by two bronze markers and a gravestone, all enclosed by a circular stone wall. Governor Baxter School for The Deaf In 1943, Maine's governor Percival P. Baxter deeded the island and causeway, including his summer home, to the State of Maine. In 1957, the state created the Governor Baxter School for The Deaf (formerly known as the Maine School for The Deaf) on the island.Gannon, Jack. 1981. Deaf Heritage–A Narrative History of Deaf America, Silver Spring, MD: National Association of the Deaf, p. 45 (PDF ) The school is infamous for a sexual abuse scandal which became public in 1981 and involved the school's principal and superintendent. There is currently still a school operating at this site. Friends School of Portland Friends School of Portland, an independent Quaker preschool through eighth grade school, was founded in 2006 and was located on Mackworth Island until 2015, when it moved to new facility in nearby Cumberland, Maine. See also * List of islands of Maine References *Soares, Liz. All for Maine: The Story of Governor Percival P. Baxter. Windswept House Publishers (1996). External links * Mackworth Island Maine Office of Tourism Category:Islands of Portland, Maine Category:Falmouth, Maine Category:Islands of Casco Bay "

❤️ Protasius 🐒

"Protasius may refer to: *Protasius (martyr) *Protasius (bishop of Milan), reigned 328–343, saint *Protasius, an archbishop of Tarragona in 637–646, who assisted the Sixth (638) and Seventh (646) Councils of Toledo "