|logic||category theory||type theory|
|true||terminal object/(-2)-truncated object||h-level 0-type/unit type|
|false||initial object||empty type|
|proposition||(-1)-truncated object||h-proposition, mere proposition|
|cut rule||composition of classifying morphisms / pullback of display maps||substitution|
|cut elimination for implication||counit for hom-tensor adjunction||beta reduction|
|introduction rule for implication||unit for hom-tensor adjunction||eta conversion|
|disjunction||coproduct ((-1)-truncation of)||sum type (bracket type of)|
|implication||internal hom||function type|
|negation||internal hom into initial object||function type into empty type|
|universal quantification||dependent product||dependent product type|
|existential quantification||dependent sum ((-1)-truncation of)||dependent sum type (bracket type of)|
|equivalence||path space object||identity type|
|equivalence class||quotient||quotient type|
|induction||colimit||inductive type, W-type, M-type|
|higher induction||higher colimit||higher inductive type|
|completely presented set||discrete object/0-truncated object||h-level 2-type/preset/h-set|
|set||internal 0-groupoid||Bishop set/setoid|
|universe||object classifier||type of types|
|modality||closure operator, (idemponent) monad||modal type theory, monad (in computer science)|
|linear logic||(symmetric, closed) monoidal category||linear type theory/quantum computation|
|proof net||string diagram||quantum circuit|
|(absence of) contraction rule||(absence of) diagonal||no-cloning theorem|
|synthetic mathematics||domain specific embedded programming language|
The lambda calculus is:
a simple programming language;
It comes in both typed and untyped versions.
The basic constructs of lambda calculus are lambda abstractions and applications. If is an expression of some sort involving a free variable (possibly vacuously), then the lambda abstraction
is intended to represent the function which takes one input and returns the result of substituting that input for in . Thus, for instance, is the function which adds one to its argument. Lambda expressions are often convenient, even outside lambda calculus proper, for referring to a function without giving it a name.
Application is how we “undo” abstraction, by applying a function to an argument. The application of the function to the argument is generally denoted simply by . Applications can be parenthesized, so for instance and and all denote the same thing as .
Application is generally considered to associate to the left. Thus denotes the application of to , followed by application of the result (assumed to again be a function) to . This allows the representation of functions of multiple variables in terms of functions of one variable via “currying” (named for Haskell Curry, although it was invented by Moses Schönfinkel): after being applied to the first argument, we return a function which, applied to the next argument, returns a function which, when applied to the next argument, … , returns a value. For instance, the “addition” function of two variables can be denoted : when applied to an argument , it returns a function which, when applied to an argument , returns . This is so common that it is generally abbreviated .
Evaluation or reduction is the process of “computing” the “value” of a lambda term. The most basic operation is called beta reduction and consists in taking a lambda abstraction at its word about what it is supposed to do when applied to an input. For instance, the application reduces to (and thereby, presuming appropriate rules for , to ). In general, the beta reduction of a term is defined as the capture-avoiding substitution of for in ,
Terms which can be connected by a zigzag of beta reductions (in either direction) are said to be beta-equivalent.
Another basic operation often assumed in the lambda calculus is eta reduction/expansion, which consists of identifying a function, with the lambda abstraction which does nothing other than apply to its argument. (It is called “reduction” or “expansion” depending on which “direction” it goes in, from to or vice versa.)
A more basic operation than either of these, which is often not even mentioned at all, is alpha equivalence; this consists of the renaming of bound variables, e.g. .
More complicated systems that build on the lambda calculus, such as various type theories, will often have other rules of evaluation as well.
In good situations, lambda calculus reduction is confluent (the Church-Rosser theorem), so that every term has at most one normal form?, and two terms are equivalent precisely when they have the same normal form. A term is said to be normalizing if its reduction is also terminating—more precisely, is said to be weakly-normalizing if there exists a finite sequence of reductions
terminating in a normal form , and strongly-normalizing if all sequences of reductions lead to a (by confluence, necessarily unique) normal form. In general, there exist terms which do not normalize by any reduction sequence, although simply-typed lambda calculus and many other typed variants of lambda calculus are strongly-normalizing.
In the untyped (or “pure”) lambda calculus, every term can be applied to any other term. As an example of the sort of freedom this allows, we can form the term which applies its argument to itself, and then define
as the self-application of . Performing beta-reduction on yields
thus giving the classic example of a non-terminating program.
In pure untyped lambda calculus, we can define natural numbers using the Church numerals?: the number is represented by the operation of -fold iteration. Thus for instance we have , the function which takes a function as input and returns a function that applies twice. Similarly is the identity on functions, while takes any function to the identity function (the 0th iterate of ). We can then construct (very inefficiently) all of arithmetic, and prove that the arithmetic functions expressible by lambda terms are exactly the same as those computable by Turing machines or (total) recursive functions.
The most natural sort of model of pure lambda calculus is a reflexive object in a cartesian closed category, that is, an object together with a pair of maps
representing pplication and ambda, such that (to validate the equation). Such data thereby witnesses the exponential object as a retract of . Of course there are no nontrivial such models in sets, but they do exist in other categories, such as domains. It is worth remarking that a necessary condition on such is that every term have a fixed-point; see fixed-point combinator.
From the point of view of type theory and in particular the “types à la Church” perspective, such a reflexive object can also be seen as the intrinsic type of every lambda term, and so the untyped lambda calculus is sometimes referred to (a bit cheekily) as really being “uni-typed”. On the other hand, from the “types à la Curry” perspective, it is also possible to begin with the pure lambda calculus as a foundation, and then try to ascribe individual terms more precise types. For example, the normalizing terms of pure lambda calculus can be characterized precisely as those which are typable in an intersection type? system.
In simply typed lambda calculus, each variable and term has a type, and we can only form the application if is of some type while is of a function type whose domain is ; the type of is then . Similarly, if is a variable of type and is a term of type involving , then has type .
Without some further type and term constructors, there is not much that can be done, but if we add a natural numbers object (that is, a type with constants of type and of type , along with a “definition-by-recursion” operator), then we can express many recursive functions. (We cannot by this means express all computable functions, although we can go beyond primitive recursive function?s; for instance we can define the Ackermann function?. One way to increase the expressiveness to all partial recursive functions is to add a fixpoint? combinator?, or an unbounded search operator).
Every cartesian closed category gives rise to a simply typed lambda calculus whose basic types are its objects, and whose basic terms are its morphisms, while
Every simply typed lambda calculus “generates” a cartesian closed category whose objects are its types and whose morphisms are its equivalence classes of terms.
These two operations are adjoint in an appropriate sense.
The idea that untyped lambda calculus can be modeled internally to a cartesian closed category as a reflexive object (an object such that the exponential is a retract of ) was formulated explicitly by Dana Scott in
This followed his earlier work constructing an explicit such model in the category of domains:
Dana Scott. Outline of a Mathematical Theory of Computation. Proc. 4th Princeton Conf. on Information Sciences and Systems, 169–176, 1970. pdf
Dana Scott. Data types as lattices. SIAM Journal of Computing, 5(3):522–587, September 1976.