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Simply typed lambda calculus

The simply typed lambda calculus (

lambda^to

) is a typed interpretation of the lambda calculus with only one type combinator:

to

(function type). 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 word simple types 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 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

The syntax of the simply typed lambda calculus is essentially that of the lambda calculus itself. The term syntax used in this article is as follows:

$e = x mid lambda x:tau.~e mid e~e$

That is, variable reference, abstractions, and application. 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.

It is possible to define the syntax as in the untyped lambda calculus, with unannotated $lambda$ abstractions. Since type inference or reconstruction is tractable for the simply typed lambda calculus, the decision doesn't matter from a "typeability" perspective. The type system for the annotated version, however, has an algorithmic interpretation, while the algorithm for type-checking unannotated and partially annotated terms is more complex. For more information, see the article on Algorithm W.

The syntax of types is:

$tau = alpha mid tau to tau$

We take $alpha$ and $beta$ to be type variables drawn from a predetermined set, while we reserve $sigma$ and $tau$ as metavariables, meaning "some type". The function type $sigma to tau$ refers to the set 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 $sigmato(tautorho)$ .

Each type $sigma$ is assigned an order, a number $o(sigma)$ . For base types, $o(alpha)=0$ ; for function types, $o(sigmatotau)=mbox{max}(o(sigma)+1,o(tau))$ . That is, the order of a type measures the depth of the most left-nested arrow.

Semantics

There are two distinct presentations of simply typed lambda calculus. In Curry-style semantics, the operational semantics comes first, and the static semantics (typing rules) comes second. In Church-style semantics, the static semantics is given first; the operational semantics apply only to terms that are accepted by the static semantics.

Note that this distinction is not precisely the same as that between unannotated and annotated abstractions. It is possible to give a Curry-style semantics using annotated lambdas: the operational semantics simply ignore types altogether.

Beyond these distinctions, there are then further variations based on calling conventions and evaluation strategy (call by name, call by value, etc.) and presentation (big-step, small-step, etc.).

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 $Gamma,Delta,dots$ , which are sets of typing assumptions. A typing assumption has the form $x:sigma$ , meaning $x$ has type $sigma$ .

The typing relation $Gammavdash e : sigma$ indicates that $e$ is a term of type $sigma$ in context $Gamma$ . It is therefore said that " $e$ is well-typed (at $sigma$ )". A typing judgment is constructed via of the following rules, wherein the premises above the line allow us to derive the conclusion below the line:

{x:sigma in Gamma}over{Gamma vdash x : sigma }

(1)

{Gamma,x:sigmavdash e:tau}over{Gammavdash lambda x : sigma.~e : sigma to tau}

(2)

{Gammavdash e_1:sigmatotauquadGammavdash e_2:sigma}over{Gammavdash e_1~e_2 : tau}

(3)

In other words,

If $x$ has type $sigma$ in the context, we know that $x$ has type $sigma$ .
If, in a certain context with $x$ having type $sigma$ , $e$ has type $tau$ , then, in the same context without $x$ , $lambda x : sigma.~e$ 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 $tau$ .

One can construe these judgments not merely as relations between terms and types, but as the construction of terms themselves. This coincides with Church-style semantics, as it is then only possible to construct well-typed terms.

Examples of closed terms, i.e. terms typable in the empty context, are:

$lambda x:alpha.x : alphatoalpha$ (the I-combinator/identity function),
$lambda x:alpha.lambda y:beta.x:alpha to beta to alpha$ (the K-combinator), and
$lambda x:alphatobetatogamma.lambda y:alphatobeta.lambda z:alpha.x z (y z) : (alphatobetatogamma)to(alphatobeta)toalphatogamma$ (the S-combinator).

These are the typed lambda calculus representations of the basic combinators of combinatory logic.

In the original presentation, Church used only two base types: $o$ for the type of propositions and $iota$ for the type of individuals. Frequently the calculus with only one base type, usually $o$ , is considered.

Terms of the same type are identified via $betaeta$ -equivalence, which is generated by the equations $(lambda x:sigma.t),u =_{beta} t[x:=u]$ , where $t[x:=u]$ stands for $t$ with all free occurrences of $x$ replaced by $u$ , and $lambda x:sigma.t,x =_eta t$ , if $x$ does not appear free in $t$ . The simply typed lambda calculus (with $betaeta$ -equivalence) is the internal language of Cartesian Closed Categories (CCCs), this was first observed by Lambek.

Typing without annotations

In the unannotated syntax, lambda abstractions carry no type information. As a consequence, terms can have more than one type. If we consider the type variables to be implicitly universally quantified, then it is possible to find a type more general than the others, called the principal type.

For example, the identity function $I = lambda x.~x$ can have many types: $alpha to alpha$ , say, or $(alpha to alpha) to (alpha to alpha)$ , and so on. Without quantifiers in our type language (as in System F), it is impossible to say what type $I$ "really" has. Extensions of simply typed lambda calculus can add support for polymorphism: terms that can have multiple types. System F, for example, adds general abstractions over types: $I_F = Lambda alpha.lambda x:alpha.~x$ takes a type $alpha$ and returns the identity function at that type. The term $I_F$ has type $forall alpha.~alpha to alpha$ .

If one considers the unannotated lambda calculus extended with constants, such as integers, booleans, and tuples, the problem of typing becomes easier: $lambda x.~x + 1$ might be easily determined to have type $texttt{int} to texttt{int}$ . The problem above does not go away entirely, though: what type does $lambda x.~1$ have? There are an infinite number ( $texttt{bool} to texttt{int}$ , $texttt{int} to texttt{int}$ , etc.). The principal type of the term is $forall alpha.~alpha to texttt{int}$ .

General observations

The simply typed lambda calculus is closely related to propositional intuitionistic logic using only implication ( $to$ ) as a connective (minimal logic) via the Curry-Howard isomorphism: the types inhabited by closed terms are precisely the tautologies of minimal logic.

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 $texttt{fix}_alpha$ of type $(alpha to alpha) to alpha$ or adding general recursive types, though both eliminate strong normalization.

Since it is strongly normalizing, it is decidable whether or not a simply typed lambda calculus program halts: it does! We can therefore conclude that the language is not Turing complete, which has an undecidable halting problem.

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 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. Whether higher order matching (unification where only one term contains existential variables) is decidable is still open. [2006: Colin Stirling, Edinburgh, has published a proof-sketch in which he claims that the problem is decidable; however, the complete version of the proof is still unpublished]
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 (Plotkin-Statman-conjecture). The conjecture was shown to be false by Loader in 1993.

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, 23:21--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
Jung, A.,Tiuryn, J.:A New Characterization of Lambda Definability, TLCA 1993
R. Loader: The Undecidability of λ-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 0198537611.
L. Baxter: The undecidability of the third order dyadic unification problem, Information and Control 38(2), 170-178 (1978)