Misplaced Pages

#546453

67-526: 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 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 }

134-401: A {\displaystyle b\to a} , a → ( a → a ) {\displaystyle a\to (a\to a)} , ..., ( b → a ) → ( a → b ) {\displaystyle (b\to a)\to (a\to b)} , ... A set of term constants is also fixed for the base types. For example, it might be assumed that one of

201-462: 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

268-445: A mathematical variable ranging over types . Even in programming languages that allow mutable variables , a type variable remains an abstraction, in the sense that it does not correspond to some memory locations. Programming languages that support parametric polymorphism make use of universally quantified type variables. Languages that support existential types make use of existentially quantified type variables. For example,

335-428: A unit term . Given two terms s : σ {\displaystyle s{\mathbin {:}}\sigma } and t : τ {\displaystyle t{\mathbin {:}}\tau } , the term ( s , t ) {\displaystyle (s,t)} has type σ × τ {\displaystyle \sigma \times \tau } . Likewise, if one has

402-457: A category by taking the types as the objects . The morphisms σ → τ {\displaystyle \sigma \to \tau } are equivalence classes of pairs ( x : σ , t : τ ) {\displaystyle (x{\mathbin {:}}\sigma ,t{\mathbin {:}}\tau )} where x is a variable (of type σ {\displaystyle \sigma } ) and t

469-403: 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

536-407: A finite set of primitive symbols, denoting abstraction and scope, as well as four constants: negation, disjunction, universal quantification, and selection respectively; and also, a finite set of rules I to VI. This finite set of rules included rule V modus ponens as well as IV and VI for substitution and generalization respectively. Rules I to III are known as alpha, beta, and eta conversion in

603-453: A form of type theory , is a typed interpretation of the lambda calculus with only one type constructor ( → {\displaystyle \to } ) that builds function types . It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical use of the untyped lambda calculus . The term simple type

670-576: 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

737-509: A term u : τ 1 × τ 2 {\displaystyle u{\mathbin {:}}\tau _{1}\times \tau _{2}} , then there are terms π 1 ( u ) : τ 1 {\displaystyle \pi _{1}(u){\mathbin {:}}\tau _{1}} and π 2 ( u ) : τ 2 {\displaystyle \pi _{2}(u){\mathbin {:}}\tau _{2}} where

SECTION 10

#1732798791547

804-708: 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

871-544: Is variable reference , abstractions , application , or constant : where c {\displaystyle c} is a term constant. A variable reference x {\displaystyle x} is bound if it is inside of an abstraction binding x {\displaystyle x} . A term is closed if there are no unbound variables. In comparison, the syntax of untyped lambda calculus has no such typing or term constants: Whereas in typed lambda calculus every abstraction (i.e. function) must specify

938-475: 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 the ML family. Over time, as

1005-429: 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

1072-495: 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

1139-411: Is a term (of type τ {\displaystyle \tau } ), having no free variables in it, except for (optionally) x . The set of terms in the language is the closure of this set of terms under the operations of abstraction and application . This correspondence can be extended to include "language homomorphisms" and functors between the category of Cartesian closed categories, and

1206-411: Is a term of type σ {\displaystyle \sigma } in context Γ {\displaystyle \Gamma } . In this case e {\displaystyle e} is said to be well-typed (having type σ {\displaystyle \sigma } ). Instances of the typing relation are called typing judgments . The validity of a typing judgment

1273-508: Is also used to refer to extensions of the simply typed lambda calculus with constructs such as products , coproducts or natural numbers ( System T ) or even full recursion (like PCF ). In contrast, systems that introduce polymorphic types (like System F ) or dependent types (like the Logical Framework ) are not considered simply typed . The simple types, except for full recursion, are still considered simple because

1340-413: Is an abbreviation for . By the 1970s stand-alone arrow notation was in use; for example in this article non-subscripted symbols σ {\displaystyle \sigma } and τ {\displaystyle \tau } can range over types. The infinite number of axioms were then seen to be a consequence of applying rules I to VI to the types (see Peano axioms ). Informally,

1407-481: Is assigned an order, a number o ( τ ) {\displaystyle o(\tau )} . For base types, o ( T ) = 0 {\displaystyle o(T)=0} ; for function types, o ( σ → τ ) = max ( o ( σ ) + 1 , o ( τ ) ) {\displaystyle o(\sigma \to \tau )={\mbox{max}}(o(\sigma )+1,o(\tau ))} . That is,

SECTION 20

#1732798791547

1474-431: Is not the only way of defining the syntax of the simply typed lambda calculus. One alternative is to remove type annotations entirely (so that the syntax is identical to the untyped lambda calculus), while ensuring that terms are well-typed via Hindley–Milner type inference . The inference algorithm is terminating, sound, and complete: whenever a term is typable, the algorithm computes its type. More precisely, it computes

1541-423: 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

1608-467: Is shown by providing a typing derivation , constructed using typing rules (wherein the premises above the line allow us to derive the conclusion below the line). Simply typed lambda calculus uses these rules: In words, Examples of closed terms, i.e. terms typable in the empty context, are: These are the typed lambda calculus representations of the basic combinators of combinatory logic . Each type τ {\displaystyle \tau }

1675-518: Is the internal language of Cartesian closed categories (CCCs), as was first observed by Joachim Lambek . Given any CCC, the basic types of the corresponding lambda calculus are the objects , and the terms are the morphisms . Conversely, the simply typed lambda calculus with product types and pairing operators over a collection of base types and given terms forms a CCC whose objects are the types, and morphisms are equivalence classes of terms. There are typing rules for pairing , projection , and

1742-406: 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 the bound x is of type α {\displaystyle \alpha } ; the expression after the colon is the type of

1809-493: The π i {\displaystyle \pi _{i}} correspond to the projections of the Cartesian product. The unit term , of type 1, written as ( ) {\displaystyle ()} and vocalized as 'nil', is the final object . The equational theory is extended likewise, so that one has This last is read as " if t has type 1, then it reduces to nil ". The above can then be turned into

1876-560: 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

1943-412: The Church encodings of such structures can be done using only → {\displaystyle \to } and suitable type variables, while polymorphism and dependency cannot. In the 1930s Alonzo Church sought to use the logistic method : his lambda calculus , as a formal language based on symbolic expressions, consisted of a denumerably infinite series of axioms and variables, but also

2010-407: The function type σ → τ {\displaystyle \sigma \to \tau } refers to the type of functions that, given an input of type σ {\displaystyle \sigma } , produce an output of type τ {\displaystyle \tau } . By convention, → {\displaystyle \to } associates to

2077-631: The operational semantics of simply typed lambda calculus can be fixed as for the untyped lambda calculus, using call by name , call by value , or other evaluation strategies . As for any typed language, type safety is a fundamental property of all of these evaluation strategies. Additionally, the strong normalization property described below implies that any evaluation strategy will terminate on all simply typed terms. The simply typed lambda calculus enriched with product types, pairing and projection operators (with β η {\displaystyle \beta \eta } -equivalence)

System F - Misplaced Pages Continue

2144-411: The tautologies of this logic. From his logistic method Church 1940 p.58 laid out an axiom schema , p. 60, which Henkin 1949 filled in with type domains (e.g. the natural numbers, the real numbers, etc.). Henkin 1996 p. 146 described how Church's logistic method could seek to provide a foundation for mathematics (Peano arithmetic and real analysis), via model theory . The presentation given above

2211-490: 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 } is a type variable, and α   type {\displaystyle \alpha ~{\text{type}}} in

2278-590: 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

2345-567: 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

2412-515: The base types is nat , and its term constants could be the natural numbers. The syntax of the simply typed lambda calculus is essentially that of the lambda calculus itself. The term x : τ {\displaystyle x{\mathbin {:}}\tau } denotes that the variable x {\displaystyle x} is of type τ {\displaystyle \tau } . The term syntax, in Backus–Naur form ,

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

2546-610: The calculus with only one base type, usually o {\displaystyle o} , is considered. The Greek letter subscripts α , β {\displaystyle \alpha ,\beta } , etc. denote type variables; the parenthesized subscripted ( α β ) {\displaystyle (\alpha \beta )} denotes the function type β → α {\displaystyle \beta \to \alpha } . Church 1940 p.58 used 'arrow or → {\displaystyle \to } ' to denote stands for , or

2613-526: The category of simply typed lambda theories. Part of this correspondence can be extended to closed symmetric monoidal categories by using a linear type system . The simply typed lambda calculus is closely related to the implicational fragment of propositional intuitionistic logic , i.e., the implicational propositional calculus , via the Curry–Howard isomorphism : terms correspond precisely to proofs in natural deduction , and inhabited types are exactly

2680-541: 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 is defined as: ∀ α . α → α → α {\displaystyle \forall \alpha .\alpha \to \alpha \to \alpha } , where α {\displaystyle \alpha }

2747-735: The different semantic interpretations discussed below can be seen through either an intrinsic or extrinsic perspective. The simply typed lambda calculus (STLC) has the same equational theory of βη-equivalence as untyped lambda calculus , but subject to type restrictions. The equation for beta reduction holds in context Γ {\displaystyle \Gamma } whenever Γ , x : σ ⊢ t : τ {\displaystyle \Gamma ,x{\mathbin {:}}\sigma \vdash t{\mathbin {:}}\tau } and Γ ⊢ u : σ {\displaystyle \Gamma \vdash u{\mathbin {:}}\sigma } , while

System F - Misplaced Pages Continue

2814-572: The effect that terms differing only by type annotations can nonetheless be assigned different meanings. For example, the identity term λ x : i n t .   x {\displaystyle \lambda x{\mathbin {:}}{\mathtt {int}}.~x} on integers and the identity term λ x : b o o l .   x {\displaystyle \lambda x{\mathbin {:}}{\mathtt {bool}}.~x} on booleans may mean different things. (The classic intended interpretations are

2881-472: The equation for eta reduction holds whenever Γ ⊢ t : σ → τ {\displaystyle \Gamma \vdash t\!:\sigma \to \tau } and x {\displaystyle x} does not appear free in t {\displaystyle t} . The advantage of typed lambda calculus is that STLC allows potentially nonterminating computations to be cut short (that is, reduced ). Likewise,

2948-474: The following OCaml code defines a polymorphic identity function that has a universally quantified type, which is printed by the interpreter on the second line: In mathematical notation, the type of the function id is ∀ a . a → a {\displaystyle \forall a.a\to a} , where a {\displaystyle a} is a type variable. This programming language theory or type theory -related article

3015-555: The identity function on integers and the identity function on boolean values.) In contrast, an extrinsic semantics assigns meaning to terms regardless of typing, as they would be interpreted in an untyped language. In this view, λ x : i n t .   x {\displaystyle \lambda x{\mathbin {:}}{\mathtt {int}}.~x} and λ x : b o o l .   x {\displaystyle \lambda x{\mathbin {:}}{\mathtt {bool}}.~x} mean

3082-475: The lambda abstraction in rule [3], because the type of the bound variable can be deduced from the type at which we check the function. Finally, we explain rules [5] and [6] as follows: Because of these last two rules coercing between synthesis and checking, it is easy to see that any well-typed but unannotated term can be checked in the bidirectional system, so long as we insert "enough" type annotations. And in fact, annotations are needed only at β-redexes. Given

3149-678: The lambda calculus. Church sought to use English only as a syntax language (that is, a metamathematical language) for describing symbolic expressions with no interpretations. In 1940 Church settled on a subscript notation for denoting the type in a symbolic expression. In his presentation, Church used only two base types: o {\displaystyle o} for "the type of propositions" and ι {\displaystyle \iota } for "the type of individuals". The type o {\displaystyle o} has no term constants, whereas ι {\displaystyle \iota } has one term constant. Frequently

3216-520: 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 the lambda cube , together with even more expressive typed lambda calculi, including those with dependent types . According to Girard,

3283-400: The language by either having a special operator f i x α {\displaystyle {\mathtt {fix}}_{\alpha }} of type ( α → α ) → α {\displaystyle (\alpha \to \alpha )\to \alpha } or adding general recursive types , though both eliminate strong normalization. Unlike

3350-572: The level of types). System F <: , pronounced "F-sub", is an extension of system F with subtyping . System F <: has been of central importance to programming language theory since the 1980s because the core of functional programming languages , like those in the ML family, support both parametric polymorphism and record subtyping, which can be expressed in System F <: . Simply typed lambda calculus The simply typed lambda calculus ( λ → {\displaystyle \lambda ^{\to }} ),

3417-552: 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

SECTION 50

#1732798791547

3484-469: The order of a type measures the depth of the most left-nested arrow. Hence: Broadly speaking, there are two different ways of assigning meaning to the simply typed lambda calculus, as to typed languages more generally, variously called intrinsic vs. extrinsic, ontological vs. semantical, or Church-style vs. Curry-style. An intrinsic semantics only assigns meaning to well-typed terms, or more precisely, assigns meaning directly to typing derivations. This has

3551-451: 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

3618-716: The principal type α → α {\displaystyle \alpha \to \alpha } ). Another alternative presentation of simply typed lambda calculus is based on bidirectional type checking , which requires more type annotations than Hindley–Milner inference but is easier to describe. The type system is divided into two judgments, representing both checking and synthesis , written Γ ⊢ e ⇐ τ {\displaystyle \Gamma \vdash e\Leftarrow \tau } and Γ ⊢ e ⇒ τ {\displaystyle \Gamma \vdash e\Rightarrow \tau } respectively. Operationally,

3685-459: 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)

3752-492: The right: σ → τ → ρ {\displaystyle \sigma \to \tau \to \rho } is read as σ → ( τ → ρ ) {\displaystyle \sigma \to (\tau \to \rho )} . To define the types, a set of base types , B {\displaystyle B} , must first be defined. These are sometimes called atomic types or type constants . With this fixed,

3819-419: The same thing ( i.e. , the same thing as λ x .   x {\displaystyle \lambda x.~x} ). The distinction between intrinsic and extrinsic semantics is sometimes associated with the presence or absence of annotations on lambda abstractions, but strictly speaking this usage is imprecise. It is possible to define an extrinsic semantics on annotated terms simply by ignoring

3886-532: The standard semantics, the simply typed lambda calculus is strongly normalizing : every sequence of reductions eventually terminates. This is because recursion is not allowed by the typing rules: it is impossible to find types for fixed-point combinators and the looping term Ω = ( λ x .   x   x ) ( λ x .   x   x ) {\displaystyle \Omega =(\lambda x.~x~x)(\lambda x.~x~x)} . Recursion can be added to

3953-501: The syntax of types is: For example, B = { a , b } {\displaystyle B=\{a,b\}} , generates an infinite set of types starting with a {\displaystyle a} , b {\displaystyle b} , a → a {\displaystyle a\to a} , a → b {\displaystyle a\to b} , b → b {\displaystyle b\to b} , b →

4020-502: The term's principal type , since often an unannotated term (such as λ x .   x {\displaystyle \lambda x.~x} ) may have more than one type ( i n t → i n t {\displaystyle {\mathtt {int}}\to {\mathtt {int}}} , b o o l → b o o l {\displaystyle {\mathtt {bool}}\to {\mathtt {bool}}} , etc., which are all instances of

4087-663: The three components Γ {\displaystyle \Gamma } , e {\displaystyle e} , and τ {\displaystyle \tau } are all inputs to the checking judgment Γ ⊢ e ⇐ τ {\displaystyle \Gamma \vdash e\Leftarrow \tau } , whereas the synthesis judgment Γ ⊢ e ⇒ τ {\displaystyle \Gamma \vdash e\Rightarrow \tau } only takes Γ {\displaystyle \Gamma } and e {\displaystyle e} as inputs, producing

SECTION 60

#1732798791547

4154-492: The type τ {\displaystyle \tau } as output. These judgments are derived via the following rules: Observe that rules [1]–[4] are nearly identical to rules (1)–(4) above , except for the careful choice of checking or synthesis judgments. These choices can be explained like so: Observe that the rules for synthesis are read top-to-bottom, whereas the rules for checking are read bottom-to-top. Note in particular that we do not need any annotation on

4221-836: The type of its argument. To define the set of well-typed lambda terms of a given type, one defines a typing relation between terms and types. First, one introduces typing contexts , or typing environments Γ , Δ , … {\displaystyle \Gamma ,\Delta ,\dots } , which are sets of typing assumptions. A typing assumption has the form x : σ {\displaystyle x{\mathbin {:}}\sigma } , meaning variable x {\displaystyle x} has type σ {\displaystyle \sigma } . The typing relation Γ ⊢ e : σ {\displaystyle \Gamma \vdash e{\mathbin {:}}\sigma } indicates that e {\displaystyle e}

4288-445: 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}}}

4355-424: The types ( i.e. , through type erasure ), as it is possible to give an intrinsic semantics on unannotated terms when the types can be deduced from context ( i.e. , through type inference ). The essential difference between intrinsic and extrinsic approaches is just whether the typing rules are viewed as defining the language, or as a formalism for verifying properties of a more primitive underlying language. Most of

4422-400: The untyped lambda calculus, the simply typed lambda calculus is not Turing complete . All programs in the simply typed lambda calculus halt. For the untyped lambda calculus, there are programs that do not halt, and moreover there is no general decision procedure that can determine whether a program halts. Type variable In type theory and programming languages , a type variable is

4489-607: 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

#546453