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simply typed lambda calculus

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**simply typed lambda calculus**(lambda^to), a formof type theory, is a typed interpretation of the lambda calculus with only one type constructor: 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 uses of the untyped lambda calculus, and it exhibits many desirable and interesting properties.The term

*simple type*is also used to refer to extensions of the simply typed lambda calculus such as products, coproducts or natural numbers (System T) or even full recursion (like PCF). In contrast, systems which introduce polymorphic types (like System F) or dependent types (like the Logical Framework) are not considered

*simply typed*. The former, except full recursion, are still considered

*simple*because the Church encodings of such structures can be done using only to and suitable type variables, while polymorphism and dependency cannot.

## Syntax

In this article, we use sigma and tau to range over types. Informally, the*function type*sigma to tau refers to the type of functions that, given an input of type sigma, produce an output of type tau.By convention, to associates to the right: we read sigmatotautorho as sigmato(tautorho).To define the types, we begin by fixing a set of

*base types*, B. These are sometimes called

*atomic types*or

*type constants*. With this fixed, the syntax of types is:

tau ::= tau to tau mid T quad mathrm{where} quad T in B.

For example, B = {a, b}, generates an infinite set of types starting with a,b,a to a,a to b,bto b,bto a, a to (a to a),ldots,(bto a) to (ato b), ldotsWe also fix a set of *term constants*for the base types. For example, we might assume a base type nat, and the term constants could be the natural numbers. In the original presentation, Church used only two base types: o for "the type of propositions" and iota for "the type of individuals". The type o has no term constants, whereas iota has one term constant. Frequently the calculus with only one base type, usually o, is considered.The syntax of the simply typed lambda calculus is essentially that of the lambda calculus itself. We write xmathbin{:}tau to denote that the variable x is of type tau. The term syntax, in BNF, is then:

e ::= x mid lambda xmathbin{:}tau.e mid e , e mid c

where c is a term constant.That is, *variable reference*,

*abstractions*,

*application*, and

*constant*. A variable reference x is

*bound*if it is inside of an abstraction binding x. A term is

*closed*if there are no unbound variables.Compare this to the syntax of untyped lambda calculus:

e ::= x mid lambda x.e mid e , e

We see that in typed lambda calculus every function (*abstraction*) must specify the type of its argument.

## Typing rules

To define the set of well typed lambda terms of a given type, we will define a typing relation between terms and types. First, we introduce*typing contexts*or

*typing environments*Gamma,Delta,dots, which are sets of typing assumptions. A

*typing assumption*has the form xmathbin{:}sigma, meaning x has type sigma.The

*typing relation*Gammavdash emathbin{:}sigma indicates that e is a term of type sigma in context Gamma. In this case e is said to be

*well-typed*(having type sigma). Instances of the typing relation are called

*typing judgements*. The validity of a typing judgement 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:{| align="center" cellpadding="9"

{frac{xmathbin{:}sigma in Gamma}{Gamma vdash xmathbin{:}sigma} } (1) |

{frac{c text{ is a constant of type } T}{Gammavdash cmathbin{:}T}} (2) |

{frac{Gamma,xmathbin{:}sigmavdash emathbin{:}tau}{Gammavdash (lambda xmathbin{:}sigma.~e)mathbin{:}(sigma to tau)}} (3) |

{frac{Gammavdash e_1mathbin{:}sigmatotauquadGammavdash e_2mathbin{:}sigma}{Gammavdash e_1~e_2mathbin{:}tau}} (4) |

- If x has type sigma in the context, we know that x has type sigma.
- Term constants have the appropriate base types.
- If, in a certain context with x having type sigma, e has type tau, then, in the same context without x, lambda xmathbin{:}sigma.~e has type sigma to tau.
- If, in a certain context, e_1 has type sigma to tau, and e_2 has type sigma, then e_1~e_2 has type tau.

*i.e.*terms typable in the empty context, are:

- For every type tau, a term lambda xmathbin{:}tau.xmathbin{:}tautotau (identity function/I-combinator),
- For types sigma,tau, a term lambda xmathbin{:}sigma.lambda ymathbin{:}tau.xmathbin{:}sigma to tau to sigma (the K-combinator), and
- For types tau,tau',tau
*, a term lambda xmathbin{:}tautotau'totau*.lambda ymathbin{:}tautotau'.lambda zmathbin{:}tau.x z (y z) : (tautotau'totau*)to(tautotau')totautotau*(the S-combinator).

o(iota to iota to iota) = 1

o((iota to iota) to iota) = 2

## Semantics

### Intrinsic vs. extrinsic interpretations

Broadly speaking, there are two different ways of assigning meaning to the simply typed lambda calculus, as to typed languages more generally, sometimes called*intrinsic*vs.

*extrinsic*, or

*Church-style*vs.

*Curry-style*.JOHN > LAST = REYNOLDS, John_C._Reynolds, Cambridge University Press, Cambridge, England, Theories of Programming Languages, 1998, An intrinsic/Church-style semantics only assigns meaning to well-typed terms, or more precisely, assigns meaning directly to typing derivations. This has the effect that terms differing only by type annotations can nonetheless be assigned different meanings. For example, the identity term lambda xmathbin{:}mathtt{int}.~x on integers and the identity term lambda xmathbin{:}mathtt{bool}.~x on booleans may mean different things. (The classic intended interpretationsare the identity function on integers and the identity function on boolean values.)In contrast, an extrinsic/Curry-style semantics assigns meaning to terms regardless of typing, as they would be interpreted in an untyped language. In this view, lambda xmathbin{:}mathtt{int}.~x and lambda xmathbin{:}mathtt{bool}.~x mean the same thing (

*i.e.*, the same thing as 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 a Curry-style semantics on annotated terms simply by ignoring the types (

*i.e.*, through type erasure), as it is possible to give a Church-style 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 the different semantic interpretations discussed below can be seen through either a Church or Curry perspective.

### Equational theory

The simply typed lambda calculus has the same equational theory of Î²Î·-equivalence as untyped lambda calculus, but subject to type restrictions. The equation for beta reduction
(lambda xmathbin{:}sigma.~t),u =_{beta} t[x:=u]

holds in context Gamma whenever Gamma,xmathbin{:}sigma vdash tmathbin{:}tau and Gammavdash umathbin{:}sigma, while the equation for eta reduction
lambda xmathbin{:}sigma.~t,x =_eta t

holds whenever Gammavdash t!:sigma to tau and x does not appear free in t.### Operational semantics

Likewise, 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.### Categorical semantics

The simply typed lambda calculus (with betaeta-equivalence) is the internal language of Cartesian closed categories (CCCs), as was first observed by Lambek. Given any specific CCC, the basic types of the corresponding lambda calculus are just the objects, and the terms are the morphisms. Conversely, every simply typed lambda calculus gives a CCC whose objects are the types, and morphisms are equivalence classes of terms.To make the correspondence clear, a type constructor for the Cartesian product is typically added to the above. To preserve the categoricity of the Cartesian product, one adds type rules for*pairing*,

*projection*, and a

*unit term*. Given two terms smathbin{:}sigma and tmathbin{:}tau, the term (s,t) has type sigmatimestau. Likewise, if one has a term umathbin{:}tau_1timestau_2, then there are terms pi_1(u)mathbin{:}tau_1 and pi_2(u)mathbin{:}tau_2 where the pi_i correspond to the projections of the Cartesian product. The

*unit term*, of type 1, is written as () and vocalized as 'nil', is the final object. The equational theory is extended likewise, so that one has

pi_1(smathbin{:}sigma,tmathbin{:}tau) = smathbin{:}sigma
pi_2(smathbin{:}sigma,tmathbin{:}tau) = tmathbin{:}tau
(pi_1(umathbin{:}sigmatimestau) , pi_2(umathbin{:}sigmatimestau)) =umathbin{:}sigmatimestau
tmathbin{:}1 = ()

This last is read as "*if t has type 1, then it reduces to nil*".The above can then be turned into a category by taking the types as the objects. The morphisms sigmatotau are equivalence classes of pairs (xmathbin{:}sigma, tmathbin{:}tau) where

*x*is a variable (of type sigma) and

*t*is a term (of type tau), having no free variables in it, except for (optionally)

*x*. Closure is obtained from currying and application, as usual.More precisely, there exist functors between the category of Cartesian closed categories, and the category of simply-typed lambda theories.It is common to extend this case to closed symmetric monoidal categories by using a linear type system. The reason for this is that the CCC is a special case of the closed symmetric monoidal category, which is typically taken to be the category of sets. This is fine for laying the foundations of set theory, but the more general topos seems to provide a superior foundation.

### Proof-theoretic semantics

The simply typed lambda calculus is closely related to the implicational fragment of propositional intuitionistic logic, i.e., minimal logic, via the Curryâ€“Howard isomorphism: terms correspond precisely to proofs in natural deduction, and inhabited types are exactly the tautologies of minimal logic.## Alternative syntaxes

The presentation given above 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 Hindleyâ€“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 the term's principal type, since often an unannotated term (such as lambda x.~x) may have more than one type (mathtt{int} to mathtt{int}, mathtt{bool} to mathtt{bool}, etc., which are all instances of the principal type alpha to alpha).Another alternative presentation of simply typed lambda calculus is based on**bidirectional type checking**, which requires more type annotations than Hindleyâ€“Milner inference but is easier to describe. The type system is divided into two judgments, representing both

*checking*and

*synthesis*, written Gamma vdash e Leftarrow tau and Gamma vdash e Rightarrow tau respectively. Operationally, the three components Gamma, e, and tau are all

*inputs*to the checking judgment Gamma vdash e Leftarrow tau, whereas the synthesis judgment Gamma vdash e Rightarrow tau only takes Gamma and e as inputs, producing the type tau as output. These judgments are derived via the following rules:{| align="center" cellpadding="9"

{frac{xmathbin{:}sigma in Gamma}{Gamma vdash x Rightarrow sigma} } [1] |

{frac{c text{ is a constant of type } T}{Gammavdash c Rightarrow T}} [2] |

{frac{Gamma,xmathbin{:}sigmavdash eLeftarrow tau}{Gammavdash lambda x.~e Leftarrow sigma to tau}} [3] |

{frac{Gammavdash e_1Rightarrow sigmatotauquadGammavdash e_2Leftarrowsigma}{Gammavdash e_1~e_2 Rightarrow tau}} [4] |

{frac{Gammavdash e Rightarrow tau}{Gammavdash eLeftarrow tau}} [5] |

{frac{Gammavdash e Leftarrow tau}{Gammavdash (emathbin{:}tau)Rightarrow tau}} [6] |

- If xmathbin{:}sigma is in the context, we can synthesize type sigma for x.
- The types of term constants are fixed and can be synthesized.
- To check that lambda x.~e has type sigma to tau in some context, we extend the context with xmathbin{:}sigma and check that e has type tau.
- If e_1 synthesizes type sigma to tau (in some context), and e_2 checks against type sigma (in the same context), then e_1~e_2 synthesizes type tau.

**not**need any annotation on 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:To check that e has type tau, it suffices to synthesize type tau.If e checks against type tau, then the explicitly annotated term (emathbin{:}tau) synthesizes tau.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 Î²-redexes.

## General observations

Given the standard semantics, the simply typed lambda calculus is strongly normalizing: that is, well-typed terms always reduce to a value, i.e., a lambda abstraction. 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 Omega = (lambda x.~x~x) (lambda x.~x~x). Recursion can be added to the language by either having a special operator mathtt{fix}_alphaof type (alpha to alpha) to alpha or adding general recursive types, though both eliminate strong normalization.Since it is strongly normalising, it is decidable whether or not a simply typed lambda calculus program halts: in fact, it*always*halts. We can therefore conclude that the language is

*not*Turing complete.

## Important results

- Tait showed in 1967 that beta-reduction is strongly normalizing. As a corollary betaeta-equivalence is decidable. Statman showed in 1977 that the normalisation problem is not elementary recursive, a proof which was later simplified by Mairson (1992). A purely semantic normalisation proof (see normalisation by evaluation) was given by Berger and Schwichtenberg in 1991.
- The unification problem for betaeta-equivalence is undecidable. Huet showed in 1973 that 3rd order unification is undecidable and this was improved upon by Baxter in 1978 then by Goldfarb in 1981 by showing that 2nd order unification is already undecidable. A proof that higher order matching (unification where only one term contains existential variables) is decidable was announced by Colin Stirling in 2006, and a full proof was published in 2009.JOURNAL, Stirling, Colin, Decidability of higher-order matching, Logical Methods in Computer Science, 22 July 2009, 5, 3, 1â€“52, 10.2168/LMCS-5(3:2)2009, 0907.3804,
- We can encode natural numbers by terms of the type (oto o)to(o to o) (Church numerals). Schwichtenberg showed in 1976 that in lambda^to exactly the extended polynomials are representable as functions over Church numerals; these are roughly the polynomials closed up under a conditional operator.
- A
*full model*of lambda^to is given by interpreting base types as sets and function types by the set-theoretic function space. Friedman showed in 1975 that this interpretation is complete for betaeta-equivalence, if the base types are interpreted by infinite sets. Statman showed in 1983 that betaeta-equivalence is the maximal equivalence which is*typically ambiguous*, i.e. closed under type substitutions (*Statman's Typical Ambiguity Theorem*). A corollary of this is that the*finite model property*holds, i.e. finite sets are sufficient to distinguish terms which are not identified by betaeta-equivalence. - Plotkin introduced logical relations in 1973 to characterize the elements of a model which are definable by lambda terms. In 1993 Jung and Tiuryn showed that a general form of logical relation (Kripke logical relations with varying arity) exactly characterizes lambda definability. Plotkin and Statman conjectured that it is decidable whether a given element of a model generated from finite sets is definable by a lambda term (
*Plotkinâ€“Statman conjecture*). The conjecture was shown to be false by Loader in 1993.

## Notes

{{reflist}}## References

- A. Church: A Formulation of the Simple Theory of Types, JSL 5, 1940
- W.W.Tait: Intensional Interpretations of Functionals of Finite Type I, JSL 32(2), 1967
- G.D. Plotkin: Lambda-definability and logical relations, Technical report, 1973
- G.P. Huet: The Undecidability of Unification in Third Order Logic Information and Control 22(3): 257-267 (1973)
- H. Friedman: Equality between functionals. LogicColl. '73, pages 22-37, LNM 453, 1975.
- H. Schwichtenberg: Functions definable in the simply-typed lambda calculus, Arch. Math Logik 17 (1976) 113-114.
- R. Statman: The Typed lambda-Calculus Is not Elementary Recursive FOCS 1977: 90-94
- W. D. Goldfarb: The undecidability of the 2nd order unification problem, TCS (1981), no. 13, 225- 230.
- R. Statman. lambda-definable functionals and betaeta conversion. Arch. Math. Logik, 23:21â€“26, 1983.
- J. Lambek: Cartesian Closed Categories and Typed Lambda-calculi. Combinators and Functional Programming Languages 1985: 136-175
- U. Berger, H. Schwichtenberg: An Inverse of the Evaluation Functional for Typed lambda-calculus LICS 1991: 203-211
- H. Mairson: A simple proof of a theorem of Statman, TCS 103(2):387-394, 1992.
- Jung, A.,Tiuryn, J.:A New Characterization of Lambda Definability, TLCA 1993
- R. Loader: The Undecidability of Î»-definability, appeared in the Church Festschrift, 2001
- H. Barendregt, Lambda Calculi with Types, Handbook of Logic in Computer Science, Volume II, Oxford University Press, 1993. {{isbn|0-19-853761-1}}.
- L. Baxter: The undecidability of the third order dyadic unification problem, Information and Control 38(2), 170-178 (1978)

## External links

- WEB, Loader, Ralph, Notes on Simply Typed Lambda Calculus, February 1998,weblink
- {{SEP|type-theory-church|Church's Type Theory}}

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