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77-411: 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

154-628: 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

231-538: A } ⊨ a ∨ b {\displaystyle \{a\}\models a\lor b} holds even in the propositional subset of first-order logic, but a ∨ b {\displaystyle a\lor b} cannot be derived from { a } {\displaystyle \{a\}} by resolution. However, { a , ¬ ( a ∨ b ) } ⊢ ⊥ {\displaystyle \{a,\lnot (a\lor b)\}\vdash \bot } can be derived. A formal system S

308-603: 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 } ,

385-429: 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

462-403: 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

539-457: A formula set Γ {\displaystyle \Gamma } , it is possible to compute every semantical consequence φ {\displaystyle \varphi } of Γ {\displaystyle \Gamma } , while refutation completeness means that, given a formula set Γ {\displaystyle \Gamma } and a formula φ {\displaystyle \varphi } , it

616-403: 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

693-400: 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

770-432: A property (mostly semantical validity) if each of its theorems has that property. A formal language is expressively complete if it can express the subject matter for which it is intended. A set of logical connectives associated with a formal system is functionally complete if it can express all propositional functions . Semantic completeness is the converse of soundness for formal systems. A formal system

847-419: 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

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924-427: 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, the fact that the identity function can have any type of

1001-773: 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

1078-404: 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

1155-521: 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 the Church numeral for n

1232-419: 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

1309-487: 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 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

1386-410: Is strongly complete or complete in the strong sense if for every set of premises Γ, any formula that semantically follows from Γ is derivable from Γ. That is: A formal system S is refutation-complete if it is able to derive false from every unsatisfiable set of formulas. That is: Every strongly complete system is also refutation complete. Intuitively, strong completeness means that, given

1463-580: Is syntactically complete or deductively complete or maximally complete if for each sentence (closed formula) φ of the language of the system either φ or ¬φ is a theorem of S . This is also called negation completeness , and is stronger than semantic completeness. In another sense, a formal system is syntactically complete if and only if no unprovable sentence can be added to it without introducing an inconsistency . Truth-functional propositional logic and first-order predicate logic are semantically complete, but not syntactically complete (for example,

1540-399: 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

1617-511: 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

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1694-410: Is a type variable, and α   type {\displaystyle \alpha ~{\text{type}}} in the context indicates that α {\displaystyle \alpha } 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

1771-464: 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

1848-542: Is complete with respect to tautologousness or "semantically complete" when all its tautologies are theorems , whereas a formal system is "sound" when all theorems are tautologies (that is, they are semantically valid formulas: formulas that are true under every interpretation of the language of the system that is consistent with the rules of the system). That is, a formal system is semantically complete if: For example, Gödel's completeness theorem establishes semantic completeness for first-order logic . A formal system S

1925-722: 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

2002-400: Is defined as: ∀ α . α → α → α {\displaystyle \forall \alpha .\alpha \to \alpha \to \alpha } , where α {\displaystyle \alpha } is a type variable . This means: B o o l e a n {\displaystyle {\mathsf {Boolean}}} is

2079-444: Is derivable. A theory is model complete if and only if every embedding of its models is an elementary embedding . System F System F (also polymorphic lambda calculus or second-order lambda calculus ) 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

2156-409: 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

2233-583: 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

2310-432: 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 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

2387-587: 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 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

Hindley–Milner type system - Misplaced Pages Continue

2464-425: Is possible to check whether φ {\displaystyle \varphi } is a semantical consequence of Γ {\displaystyle \Gamma } . Examples of refutation-complete systems include: SLD resolution on Horn clauses , superposition on equational clausal first-order logic, Robinson's resolution on clause sets. The latter is not strongly complete: e.g. {

2541-405: Is said to be incomplete . The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantical validity . Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true. The property converse to completeness is called soundness : a system is sound with respect to

2618-738: 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

2695-478: 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

2772-540: 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)

2849-405: 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

2926-439: 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

3003-505: 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 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

3080-499: 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 the addition of the following: where σ , τ {\displaystyle \sigma ,\tau } are types, α {\displaystyle \alpha }

3157-420: 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

Hindley–Milner type system - Misplaced Pages Continue

3234-567: 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

3311-620: 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) 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

3388-399: 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

3465-460: 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 the type of these expressions is ∀ α . α → α → α {\displaystyle \forall \alpha .\alpha \to \alpha \to \alpha } ;

3542-443: The definitions above, B o o l e a n {\displaystyle {\mathsf {Boolean}}} is a type argument to x {\displaystyle x} , specifying that 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

3619-627: 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 '}

3696-509: 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 is used for value-level functions. (The superscripted α {\displaystyle \alpha } means that

3773-505: 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

3850-401: 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

3927-554: 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

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4004-559: 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

4081-782: 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

4158-499: 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

4235-420: The propositional logic statement consisting of a single propositional variable A is not a theorem, and neither is its negation). Gödel's incompleteness theorem shows that any computable system that is sufficiently powerful, such as Peano arithmetic , cannot be both consistent and syntactically complete. In superintuitionistic and modal logics , a logic is structurally complete if every admissible rule

4312-493: 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

4389-681: 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.

4466-439: 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

4543-822: 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

4620-484: 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 }

4697-399: 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) In mathematical logic and metalogic , a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems ; otherwise the system

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4774-420: 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

4851-419: 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

4928-628: 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

5005-405: 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

5082-578: 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

5159-419: 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

5236-417: 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

5313-455: 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 the boolean values T {\displaystyle \mathbf {T} } and F {\displaystyle \mathbf {F} } are used, extending

5390-484: 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

5467-563: 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 is ∀ α . α → ( α → α ) → α {\displaystyle \forall \alpha .\alpha \to (\alpha \to \alpha )\to \alpha } . The terms of this type comprise

5544-533: 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

5621-446: 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,

5698-457: 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}}} is a convenient shorthand for ∀ α . α → α → α {\displaystyle \forall \alpha .\alpha \to \alpha \to \alpha } , but it

5775-817: 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

5852-609: 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

5929-474: 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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