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67-411: Among HM's more notable properties are its completeness and its ability to infer the most general type of a given program without programmer-supplied type annotations or other hints. Algorithm W is an efficient type inference method in practice and has been successfully applied on large code bases, although it has a high theoretical complexity . HM is preferably used for functional languages . It

134-462: A n → B o o l e a n → B o o l e a n {\displaystyle {\mathsf {Boolean}}\rightarrow {\mathsf {Boolean}}\rightarrow {\mathsf {Boolean}}} ): Note that in the definitions above, B o o l e a n {\displaystyle {\mathsf {Boolean}}} is a type argument to x {\displaystyle x} , specifying that

201-628: A p   ( S e t   s t r i n g )   i n t {\displaystyle {\mathtt {Map\ (Set\ string)\ int}}} . The latter types are examples of applications of type functions , for example, from the set { M a p 2 ,   S e t 1 ,   s t r i n g 0 ,   i n t 0 ,   → 2 } {\displaystyle \{{\mathtt {Map^{2},\ Set^{1},\ string^{0},\ int^{0}}},\ \rightarrow ^{2}\}} , where

268-603: A typing judgment of the form Γ   ⊢   e : σ {\displaystyle \Gamma \ \vdash \ e:\sigma } , stating that under assumptions Γ {\displaystyle \Gamma } , the expression e {\displaystyle e} has type σ {\displaystyle \sigma } . In a type ∀ α 1 … ∀ α n . τ {\displaystyle \forall \alpha _{1}\dots \forall \alpha _{n}.\tau } ,

335-429: A Haskell type a -> a means ∀ α . α → α {\displaystyle \forall \alpha .\alpha \rightarrow \alpha } here. Related and also very uncommon is the binding effect of the right hand side σ {\displaystyle \sigma } of the assignments. Typically, the mixture of both bound and unbound type variables originate from

402-403: A family of systems, where induction is based on the kinds permitted in each system: In the limit, we can define system F ω {\displaystyle F_{\omega }} to be That is, F ω is the system which allows functions from types to types where the argument (and result) may be of any order. Note that although F ω places no restrictions on the order of

469-403: A monomorphic (ground) type for a quantified variable is straight forward, substituting a polytype has some pitfalls caused by the presence of free variables. Most particularly, unbound variables must not be replaced. They are treated as constants here. Additionally, quantifications can only occur top-level. Substituting a parametric type, one has to lift its quantifiers. The table on the right makes

536-576: A natural manner, related to that in Martin-Löf's type theory . Abstract structures ( S ) are created using constructors . These are functions typed as: Recursivity is manifested when S itself appears within one of the types K i {\displaystyle K_{i}} . If you have m of these constructors, you can define the type of S as: For instance, the natural numbers can be defined as an inductive datatype N with constructors The System F type corresponding to this structure

603-400: A polymorphic type for 'id'. As indicated, the expression syntax is extended to make the let-bound variables explicit, and by restricting the type system to allow only let-bound variable to have polymorphic types, while the parameters in lambda-abstractions must get a monomorphic type, type inference becomes decidable. The remainder of this article proceeds as follows: The same description of

670-419: A quantified type variable in τ {\displaystyle \tau } is called bound and all unbound type variables in τ {\displaystyle \tau } are called free . Additionally to the quantification ∀ {\displaystyle \forall } in polytypes, type variables can also be bound by occurring in the context, but with the inverse effect on

737-773: A type expression. The application binds stronger than the infix arrow, which is right-binding. Type variables are admitted as monotypes. Monotypes are not to be confused with monomorphic types, which exclude variables and allow only ground terms. Two monotypes are equal if they have identical terms. Polytypes (or type schemes ) are types containing variables bound by zero or more for-all quantifiers, e.g. ∀ α . α → α {\displaystyle \forall \alpha .\alpha \rightarrow \alpha } . A function with polytype ∀ α . α → α {\displaystyle \forall \alpha .\alpha \rightarrow \alpha } can map any value of

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804-404: A type has the general form ∀ α 1 … ∀ α n . τ {\displaystyle \forall \alpha _{1}\dots \forall \alpha _{n}.\tau } , where n ≥ 0 {\displaystyle n\geq 0} and τ {\displaystyle \tau } is a monotype. Equality of polytypes

871-419: Is ∀ α . α → α {\displaystyle \forall \alpha .\alpha \rightarrow \alpha } , while the others are more specific and can be derived from the general one by consistently replacing another type for the type parameter , i.e. the quantified variable α {\displaystyle \alpha } . The counter-example fails because

938-708: Is ∀ α . α → ( α → α ) → α {\displaystyle \forall \alpha .\alpha \to (\alpha \to \alpha )\to \alpha } . The terms of this type comprise a typed version of the Church numerals , the first few of which are: If we reverse the order of the curried arguments ( i.e., ∀ α . ( α → α ) → α → α {\displaystyle \forall \alpha .(\alpha \rightarrow \alpha )\rightarrow \alpha \rightarrow \alpha } ), then

1005-399: Is undecidable for a Curry-style variant of System F, that is, one that lacks explicit typing annotations. Wells's result implies that type inference for System F is impossible. A restriction of System F known as " Hindley–Milner ", or simply "HM", does have an easy type inference algorithm and is used for many statically typed functional programming languages such as Haskell 98 and

1072-429: Is a type variable . This means: B o o l e a n {\displaystyle {\mathsf {Boolean}}} is the type of all functions which take as input a type α and two expressions of type α, and produce as output an expression of type α (note that we consider → {\displaystyle \to } to be right-associative .) The following two definitions for

1139-592: Is a typed lambda calculus that introduces, to simply typed lambda calculus , a mechanism of universal quantification over types. System F formalizes parametric polymorphism in programming languages , thus forming a theoretical basis for languages such as Haskell and ML . It was discovered independently by logician Jean-Yves Girard (1972) and computer scientist John C. Reynolds . Whereas simply typed lambda calculus has variables ranging over terms, and binders for them, System F additionally has variables ranging over types , and binders for them. As an example,

1206-495: Is a convenient shorthand for ∀ α . α → α → α {\displaystyle \forall \alpha .\alpha \to \alpha \to \alpha } , but it is not a symbol of System F itself, but rather a "meta-symbol". Likewise, T {\displaystyle \mathbf {T} } and F {\displaystyle \mathbf {F} } are also "meta-symbols", convenient shorthands, of System F "assemblies" (in

1273-464: Is available in the HM-based type system of Haskell. When extending the type inference for the simply-typed lambda calculus towards polymorphism, one has to decide whether assigning a polymorphic type not only as type of an expression, but also as the type of a λ-bound variable is admissible. This would allow the generic identity type to be assigned to the variable 'id' in: Allowing this gives rise to

1340-454: Is bound. The first rule is that of application, and the second is that of abstraction. The B o o l e a n {\displaystyle {\mathsf {Boolean}}} type is defined as: ∀ α . α → α → α {\displaystyle \forall \alpha .\alpha \to \alpha \to \alpha } , where α {\displaystyle \alpha }

1407-722: Is consistently substituted such that one gains σ {\displaystyle \sigma } as shown in the side bar. This order is part of the type definition of the type system. In our previous example, applying the substitution S = { α ↦ string } {\displaystyle S=\left\{\alpha \mapsto {\texttt {string}}\right\}} would result in ∀ α . α → α ⊑ string → string {\displaystyle \forall \alpha .\alpha \rightarrow \alpha \sqsubseteq {\texttt {string}}\rightarrow {\texttt {string}}} . While substituting

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1474-409: Is its smallest element. While specialization of a type scheme is one use of the order, it plays a crucial second role in the type system. Type inference with polymorphism faces the challenge of summarizing all possible types an expression may have. The order guarantees that such a summary exists as the most general type of the expression. The type order defined above can be extended to typings because

1541-583: Is left-binding and binds stronger than abstraction or the let-in construct. Types are syntactically split into two groups, monotypes and polytypes. Monotypes always designate a particular type. Monotypes τ {\displaystyle \tau } are syntactically represented as terms . Examples of monotypes include type constants like i n t {\displaystyle {\mathtt {int}}} or s t r i n g {\displaystyle {\mathtt {string}}} , and parametric types like M

1608-423: Is requested: will do. A predicate is a function which returns a B o o l e a n {\displaystyle {\mathsf {Boolean}}} -typed value. The most fundamental predicate is ISZERO which returns T {\displaystyle \mathbf {T} } if and only if its argument is the Church numeral 0 : System F allows recursive constructions to be embedded in

1675-570: Is undecidable. Under the Curry–Howard isomorphism , System F corresponds to the fragment of second-order intuitionistic logic that uses only universal quantification. System F can be seen as part of the lambda cube , together with even more expressive typed lambda calculi, including those with dependent types . According to Girard, the "F" in System F was picked by chance. The typing rules of System F are those of simply typed lambda calculus with

1742-738: Is up to reordering the quantification and renaming the quantified variables ( α {\displaystyle \alpha } -conversion). Further, quantified variables not occurring in the monotype can be dropped. To meaningfully bring together the still disjoint parts (syntax expressions and types) a third part is needed: context. Syntactically, a context is a list of pairs x : σ {\displaystyle x:\sigma } , called assignments , assumptions or bindings , each pair stating that value variable x i {\displaystyle x_{i}} has type σ i . {\displaystyle \sigma _{i}.} All three parts combined give

1809-488: Is used for value-level functions. (The superscripted α {\displaystyle \alpha } means that the bound x is of type α {\displaystyle \alpha } ; the expression after the colon is the type of the lambda expression preceding it.) As a term rewriting system , System F is strongly normalizing . However, type inference in System F (without explicit type annotations)

1876-478: The ∀ α {\displaystyle \forall \alpha } in the type of k {\displaystyle k} . But such a scoping cannot be expressed in HM. Rather, the binding is realized by the context. Polymorphism means that one and the same expression can have (perhaps infinitely) many types. But in this type system, these types are not completely unrelated, but rather orchestrated by

1943-560: The Bourbaki sense ); otherwise, if such functions could be named (within System F), then there would be no need for the lambda-expressive apparatus capable of defining functions anonymously and for the fixed-point combinator , which works around that restriction.) Then, with these two λ {\displaystyle \lambda } -terms, we can define some logic operators (which are of type B o o l e

2010-540: The ML family. Over time, as the restrictions of HM-style type systems have become apparent, languages have steadily moved to more expressive logics for their type systems. GHC , a Haskell compiler, goes beyond HM (as of 2008) and uses System F extended with non-syntactic type equality; non-HM features in OCaml 's type system include GADT . In second-order intuitionistic logic , the second-order polymorphic lambda calculus (F2)

2077-405: The cardinality of a set would be a value of this type. Quantifiers can only appear top level. For instance, a type ∀ α . α → ∀ α . α {\displaystyle \forall \alpha .\alpha \rightarrow \forall \alpha .\alpha } is excluded by the syntax of types. Also monotypes are included in the polytypes, thus

Hindley–Milner type system - Misplaced Pages Continue

2144-439: The depth grammar , and leaves some syntactical details open. This form of presentation is usual. Building on this, typing rules are used to define how expressions and types are related. As before, the form used is a bit liberal. The expressions to be typed are exactly those of the lambda calculus extended with a let-expression as shown in the adjacent table. Parentheses can be used to disambiguate an expression. The application

2211-420: The polymorphic lambda calculus ; however, unfortunately, type inference in this system is not decidable. Instead, HM distinguishes variables that are immediately bound to an expression from more general λ-bound variables, calling the former let-bound variables, and allows polymorphic types to be assigned only to these. This leads to let-polymorphism where the above example takes the form which can be typed with

2278-591: The Church numeral for n is a function that takes a function f as argument and returns the n power of f . That is to say, a Church numeral is a higher-order function – it takes a single-argument function f , and returns another single-argument function. The version of System F used in this article is as an explicitly typed, or Church-style, calculus. The typing information contained in λ-terms makes type-checking straightforward. Joe Wells (1994) settled an "embarrassing open problem" by proving that type checking

2345-395: The addition of the following: where σ , τ {\displaystyle \sigma ,\tau } are types, α {\displaystyle \alpha } is a type variable, and α   type {\displaystyle \alpha ~{\text{type}}} in the context indicates that α {\displaystyle \alpha }

2412-567: The arguments in these mappings, it does restrict the universe of the arguments for these mappings: they must be types rather than values. System F ω does not permit mappings from values to types ( dependent types ), though it does permit mappings from values to values ( λ {\displaystyle \lambda } abstraction), mappings from types to values ( Λ {\displaystyle \Lambda } abstraction), and mappings from types to types ( λ {\displaystyle \lambda } abstraction at

2479-410: The boolean values T {\displaystyle \mathbf {T} } and F {\displaystyle \mathbf {F} } are used, extending the definition of Church booleans : (Note that the above two functions require three — not two — arguments. The latter two should be lambda expressions, but the first one should be a type. This fact is reflected in the fact that

2546-399: The deduction system is used throughout, even for the two algorithms, to make the various forms in which the HM method is presented directly comparable. The type system can be formally described by syntax rules that fix a language for the expressions, types, etc. The presentation here of such a syntax is not too formal, in that it is written down not to study the surface grammar , but rather

2613-627: The example suggests, substitution is not only strongly related to an order, that expresses that a type is more or less special, but also with the all-quantification which allows the substitution to be applied. Formally, in HM, a type σ ′ {\displaystyle \sigma '} is more general than σ {\displaystyle \sigma } , formally σ ′ ⊑ σ {\displaystyle \sigma '\sqsubseteq \sigma } , if some quantified variable in σ ′ {\displaystyle \sigma '}

2680-441: The fact that the identity function can have any type of the form A → A would be formalized in System F as the judgement where α {\displaystyle \alpha } is a type variable . The upper-case Λ {\displaystyle \Lambda } is traditionally used to denote type-level functions, as opposed to the lower-case λ {\displaystyle \lambda } which

2747-505: The implied all-quantification of typings enables consistent replacement: Contrary to the specialisation rule, this is not part of the definition, but like the implicit all-quantification rather a consequence of the type rules defined next. Free type variables in a typing serve as placeholders for possible refinement. The binding effect of the environment to free type variables on the right hand side of ⊢ {\displaystyle \vdash } that prohibits their substitution in

Hindley–Milner type system - Misplaced Pages Continue

2814-401: The judgments, some extra conditions introduced above might be used as premises, too. A proof using the rules is a sequence of judgments such that all premises are listed before a conclusion. The examples below show a possible format of proofs. From left to right, each line shows the conclusion, the [ N a m e ] {\displaystyle [{\mathtt {Name}}]} of

2881-554: The logical relations P2. Reynolds proved that a Girard projection followed by a Reynolds embedding form the identity, i.e., the Girard-Reynolds Isomorphism . While System F corresponds to the first axis of Barendregt's lambda cube , System F ω or the higher-order polymorphic lambda calculus combines the first axis (polymorphism) with the second axis ( type operators ); it is a different, more complex system. System F ω can be defined inductively on

2948-559: The most general type. In 1978, Robin Milner , independently of Hindley's work, provided an equivalent algorithm, Algorithm W . In 1982, Luis Damas finally proved that Milner's algorithm is complete and extended it to support systems with polymorphic references. In the simply typed lambda calculus , types T are either atomic type constants or function types of form T → T {\displaystyle T\rightarrow T} . Such types are monomorphic . Typical examples are

3015-451: The other two parameters that are given to x {\displaystyle x} are of type B o o l e a n {\displaystyle {\mathsf {Boolean}}} . As in Church encodings, there is no need for an IFTHENELSE function as one can just use raw B o o l e a n {\displaystyle {\mathsf {Boolean}}} -typed terms as decision functions. However, if one

3082-782: The parametric polymorphism. As an example, the identity λ x . x {\displaystyle \lambda x.x} can have ∀ α . α → α {\displaystyle \forall \alpha .\alpha \rightarrow \alpha } as its type as well as string → string {\displaystyle {\texttt {string}}\rightarrow {\texttt {string}}} or int → int {\displaystyle {\texttt {int}}\rightarrow {\texttt {int}}} and many others, but not int → string {\displaystyle {\texttt {int}}\rightarrow {\texttt {string}}} . The most general type for this function

3149-499: The polymorphism used here is parametric. One finds the notation of type schemes in the literature, too, emphasizing the parametric nature of the polymorphism. Additionally, constants may be typed with (quantified) type variables. E.g.: Polymorphic types can become monomorphic by consistent substitution of their variables. Examples of monomorphic instances are: More generally, types are polymorphic when they contain type variables, while types without them are monomorphic. Contrary to

3216-493: The replacement is not consistent. The consistent replacement can be made formal by applying a substitution S = {   a i ↦ τ i ,   …   } {\displaystyle S=\left\{\ a_{i}\mapsto \tau _{i},\ \dots \ \right\}} to the term of a type τ {\displaystyle \tau } , written S τ {\displaystyle S\tau } . As

3283-681: The right hand side of the ⊢ {\displaystyle \vdash } . Such variables then behave like type constants there. Finally, a type variable may legally occur unbound in a typing, in which case they are implicitly all-quantified. The presence of both bound and unbound type variables is a bit uncommon in programming languages. Often, all type variables are implicitly treated all-quantified. For instance, one does not have clauses with free variables in Prolog . Likewise in Haskell, where all type variables implicitly occur quantified, i.e.

3350-439: The rule [ I n s t ] {\displaystyle [{\mathtt {Inst}}]} should be clear from the section on specialization above , [ G e n ] {\displaystyle [{\mathtt {Gen}}]} complements the former, working in the opposite direction. It allows generalization, i.e. to quantify monotype variables not bound in the context. The following two examples exercise

3417-822: The rule applied and the premises, either by referring to an earlier line (number) if the premise is a judgment or by making the predicate explicit. The side box shows the deduction rules of the HM type system. One can roughly divide the rules into two groups: The first four rules [ V a r ] {\displaystyle [{\mathtt {Var}}]} (variable or function access), [ A p p ] {\displaystyle [{\mathtt {App}}]} ( application , i.e. function call with one parameter), [ A b s ] {\displaystyle [{\mathtt {Abs}}]} ( abstraction , i.e. function declaration) and [ L e t ] {\displaystyle [{\mathtt {Let}}]} (variable declaration) are centered around

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3484-484: The rule precise. Alternatively, consider an equivalent notation for the polytypes without quantifiers in which quantified variables are represented by a different set of symbols. In such a notation, the specialization reduces to plain consistent replacement of such variables. The relation ⊑ {\displaystyle \sqsubseteq } is a partial order and ∀ α . α {\displaystyle \forall \alpha .\alpha }

3551-628: The rule system in action. Since both the expression and the type are given, they are a type-checking use of the rules. Completeness (logic) Too Many Requests If you report this error to the Wikimedia System Administrators, please include the details below. Request from 172.68.168.226 via cp1108 cp1108, Varnish XID 214908462 Upstream caches: cp1108 int Error: 429, Too Many Requests at Thu, 28 Nov 2024 10:58:46 GMT System F System F (also polymorphic lambda calculus or second-order lambda calculus )

3618-420: The same type to itself, and the identity function is a value for this type. As another example, ∀ α . ( S e t   α ) → i n t {\displaystyle \forall \alpha .({\mathtt {Set}}\ \alpha )\rightarrow {\mathtt {int}}} is the type of a function mapping all finite sets to integers. A function which returns

3685-419: The specialisation rule is again that a replacement has to be consistent and would need to include the whole typing. This article will discuss four different rule sets: The syntax of HM is carried forward to the syntax of the inference rules that form the body of the formal system , by using the typings as judgments . Each of the rules define what conclusion could be drawn from what premises. Additionally to

3752-628: The superscript indicates the number of type parameters. The complete set of type functions C {\displaystyle C} is arbitrary in HM, except that it must contain at least → 2 {\displaystyle \rightarrow ^{2}} , the type of functions. It is often written in infix notation for convenience. For example, a function mapping integers to strings has type i n t → s t r i n g {\displaystyle {\mathtt {int}}\rightarrow {\mathtt {string}}} . Again, parentheses can be used to disambiguate

3819-405: The symbol ∀ {\displaystyle \forall } is the quantifier binding the type variables α i {\displaystyle \alpha _{i}} in the monotype τ {\displaystyle \tau } . The variables α i {\displaystyle \alpha _{i}} are called quantified and any occurrence of

3886-578: The syntax, presenting one rule for each of the expression forms. Their meaning is obvious at the first glance, as they decompose each expression, prove their sub-expressions and finally combine the individual types found in the premises to the type in the conclusion. The second group is formed by the remaining two rules [ I n s t ] {\displaystyle [{\mathtt {Inst}}]} and [ G e n ] {\displaystyle [{\mathtt {Gen}}]} . They handle specialization and generalization of types. While

3953-419: The type ∀ α ∀ β . α → β → α {\displaystyle \forall \alpha \forall \beta .\alpha \rightarrow \beta \rightarrow \alpha } . One could imagine the free type variable α {\displaystyle \alpha } in the type of f {\displaystyle f} be bound by

4020-417: The type ∀ γ . γ → α {\displaystyle \forall \gamma .\gamma \rightarrow \alpha } . The free monotype variable α {\displaystyle \alpha } originates from the type of the variable x {\displaystyle x} bound in the surrounding scope. k {\displaystyle k} has

4087-445: The type of these expressions is ∀ α . α → α → α {\displaystyle \forall \alpha .\alpha \to \alpha \to \alpha } ; the universal quantifier binding the α corresponds to the Λ binding the alpha in the lambda expression itself. Also, note that B o o l e a n {\displaystyle {\mathsf {Boolean}}}

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4154-484: The type systems used for example in Pascal (1970) or C (1972), which only support monomorphic types, HM is designed with emphasis on parametric polymorphism. The successors of the languages mentioned, like C++ (1985), focused on different types of polymorphism, namely subtyping in connection with object-oriented programming and overloading . While subtyping is incompatible with HM, a variant of systematic overloading

4221-533: The types only from a small portion of source code, but rather from complete programs or modules. Being able to cope with parametric types , too, it is core to the type systems of many functional programming languages. It was first applied in this manner in the ML programming language. The origin is the type inference algorithm for the simply typed lambda calculus that was devised by Haskell Curry and Robert Feys in 1958. In 1969, J. Roger Hindley extended this work and proved that their algorithm always inferred

4288-446: The types used in arithmetic values: Contrary to this, the untyped lambda calculus is neutral to typing at all, and many of its functions can be meaningfully applied to all type of arguments. The trivial example is the identity function which simply returns whatever value it is applied to. Less trivial examples include parametric types like lists . While polymorphism in general means that operations accept values of more than one type,

4355-817: The use of free variables in an expression. The constant function K = λ x . λ y . x {\displaystyle \lambda x.\lambda y.x} provides an example. It has the monotype α → β → α {\displaystyle \alpha \rightarrow \beta \rightarrow \alpha } . One can force polymorphism by l e t   k = λ x . ( l e t   f = λ y . x   i n   f )   i n   k {\displaystyle \mathbf {let} \ k=\lambda x.(\mathbf {let} \ f=\lambda y.x\ \mathbf {in} \ f)\ \mathbf {in} \ k} . Herein, f {\displaystyle f} has

4422-609: Was discovered by Girard (1972) and independently by Reynolds (1974). Girard proved the Representation Theorem : that in second-order intuitionistic predicate logic (P2), functions from the natural numbers to the natural numbers that can be proved total, form a projection from P2 into F2. Reynolds proved the Abstraction Theorem : that every term in F2 satisfies a logical relation, which can be embedded into

4489-474: Was first implemented as part of the type system of the programming language ML . Since then, HM has been extended in various ways, most notably with type class constraints like those in Haskell . As a type inference method, Hindley–Milner is able to deduce the types of variables, expressions and functions from programs written in an entirely untyped style. Being scope sensitive, it is not limited to deriving

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