In mathematical logic and type theory, the λ-cube is a framework introduced by Henk Barendregt to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Each dimension of the cube corresponds to a new kind of dependency between terms and types. Here, "dependency" refers to the capacity of a term or type to bind a term or type. The respective dimensions of the λ-cube correspond to:
- y-axis (): terms that can bind types, corresponding to polymorphism.
- x-axis (): types that can bind terms, corresponding to dependent types.
- z-axis (): types that can bind types, corresponding to (binding) type operators.
The different ways to combine these three dimension yield the 8 vertices of the cube, each corresponding to a different kind of typed system. The λ-cube can be generalized into the concept of a pure type system.
Examples of Systems
(λ→) Simply typed lambda calculus
The simplest system found in the λ-cube is the simply typed lambda calculus, also called λ→. In this system, the only way to construct an abstraction is by making a term depend on a term, with the typing rule
(λ2) System F
In System F (also named λ2 for the "second order typed lambda calculus") there is another type of abstraction, written with a , that allows terms to depend on types, with the following rule:
The terms beginning with a are called polymorphic, as they can be applied to different types to get different functions, similarly to polymorphic functions in ML-like languages. For instance, the polymorphic identity
fun x -> x
of OCaml has type
'a -> 'a
meaning it can take an argument of any type
'a and return an element of that type. This type corresponds in λ2 to the type .
(F) System F
In System F a construction is introduced to supply types that depend on other types. This is called a type constructor and provides a way to build "a function with a type as a value". An example of such a type constructor is , where "" informally means " is a type". This is a function that takes a type parameter as an argument and returns the type of s of values of type . In concrete programming, this feature corresponds to the ability to define type constructors inside the language, rather than considering them as primitives. The previous type constructor roughly corresponds to the following definition of a tree with labeled leaves in OCaml:
type 'a tree = | Leaf of 'a | Node of 'a tree * 'a tree
This type constructor can be applied to other types to obtain new types. E.g., to obtain type of trees of integers:
type int_tree = int tree
In the λP system, also named ΛΠ, and closely related to the LF Logical Framework, one has so called dependent types. These are types that are allowed to depend on terms. The crucial introduction rule of the system is
where represents valid types. The new type constructor corresponds via the Curry-Howard isomorphism to a universal quantifier, and the system λP as a whole corresponds to first order logic with implication as only connective. An example of these dependent types in concrete programming is the type of vectors on a certain length: the length is a term, on which the type depends.
(Fω) System Fω
System Fω combines both the constructor of System F and the type constructors from System F. Thus System Fω provides both terms that depend on types and types that depend on types.
(λC) Calculus of constructions
In the calculus of constructions, denoted as λC in the cube, these three features cohabit, so that both types and terms can depend on types and terms. The clear border that exists in λ→ between terms and types is somewhat abolished, as all types except the universal are themselves terms with a type.
As for all systems based upon the simply typed lambda calculus, all systems in the cube are given in two steps: first, raw terms, together with a notion of β-reduction, and then typing rules that allow to type those terms.
The set of sorts is defined as , sorts are represented with the letter . There is also a set of variables, represented by the letters . The raw terms of the eight systems of the cube are given by the following syntax:
and denoting when does not occur free in .
The environment, as is usual in typed systems, are given by
The notion of β-reduction is common to all systems in the cube. It is written and given by the rules
Its reflexive, transitive closure is written .
The following typing rules are also common to all systems in the cube:
The difference between the systems is in the pairs of sorts that are allowed in the following two typing rules:
The correspondence between the systems and the pairs allowed in the rules is the following:
Each direction of the cube corresponds to one pair (excluding the pair shared by all systems), and in turn each pair corresponds to one possibility of dependency between terms and types:
- allows terms to depend on terms.
- allows types to depend on terms.
- allows terms to depend on types.
- allows types to depend on types.
Comparison between the systems
A typical derivation that can be obtained is
or with the arrow shortcut
closely resembling the identity (of type ) of the usual λ→. Note that all types used must appear in the context, because the only derivation that can be done in an empty context is .
In λ2, such terms can be obtained as
with . If one reads as a universal quantification, via the Curry-Howard isomorphism, this can be seen as a proof of the principle of explosion. In general, λ2 adds the possibility to have impredicative types such as , that is terms quantifying over all types including themselves.
The polymorphism also allows the construction of functions that were not constructible in λ→. More precisely, the function definable in System F are those provably total in second order Peano arithmetic. In particular, all primitive recursive functions are definable.
In λP, the ability to have types depending on terms means one can express logical predicates. For instance, the following is derivable:
which corresponds, via the Curry-Howard isomorphism, to a proof of .
From the computational point of view, however, having dependent types does not enhance computational power, only the possibility to express more precise type properties.
The conversion rule is strongly needed when dealing with dependent types, because it allows to perform computation on the terms in the type. For instance, if you have and , you need to apply the conversion rule to obtain to be able to type .
In λω, the following operator
is definable, that is . The derivation
can be obtained already in λω, however the polymorphic can only be defined if the rule is also present.
The calculus of constructions has both the predicate expressiveness of λP and the computational power of λω, so it is very powerful, both on the logical side and on the computational side.
Relation to other systems
The system Automath is similar to λ2 from a logical point of view. The ML-like languages, from a typing point of view, lie somewhere between λ→ and λ2, as they admit a restricted kind of polymorphic types, that is the types in prenex normal form. However, because they feature some recursion operators, their computing power is greater than that of λ2. The Coq system is based on an extension of λC with a linear hierarchy of universes, rather than only one untypable , and the ability to construct inductive types.
Pure Type Systems can be seen as a generalization of the cube, with an arbitrary set of sorts, axiom, product and abstraction rules. Conversely, the systems of the lambda cube can be expressed as Pure Type Systems with two sorts , the only axiom , and a set of rules such that .
|System of the cube||Logical System|
|λ→||(First Order) Propositional Calculus|
|λ2||Second Order Propositional Caculus|
|λω||Weakly Higher Order Propositional Calculus|
|λω||Higher Order Propositional Calculus|
|λP||(First Order) Predicate Logic|
|λP2||Second Order Predicate Calculus|
|λPω||Weak Higher Order Predicate Calculus|
|λC||Calculus of Constructions|
All the logics are implicative (i.e. the only connectives are and ), however one can define other connectives such as or in an impredicative way in second and higher order logics. In the weak higher order logics, there are variables for higher order predicates, but no quantification on those can be done.
All systems in the cube enjoy
- the Church-Rosser property: if and then there exists such that and ;
- the subject reduction property: if and then ;
- the uniqueness of types: if and then .
Subtyping however is not represented in the cube, even though systems like , known as higher-order bounded quantification, which combines subtyping and polymorphism are of practical interest, and can be further generalized to bounded type operators. Further extensions to allow the definition of purely functional objects; these systems were generally developed after the lambda cube paper was published.
The idea of the cube is due to the mathematician Henk Barendregt (1991). The framework of pure type systems generalizes the lambda cube in the sense that all corners of the cube, as well as many other systems can be represented as instances of this general framework. This framework predates the lambda cube by a couple of years. In his 1991 paper, Barendregt also defines the corners of the cube in this framework.
- In his Habilitation à diriger les recherches, Olivier Ridoux gives a cut-out template of the lambda cube and also a dual representation of the cube as an octahedron, where the 8 vertices are replaced by faces, as well as a dual representation as a dodecahedron, where the 12 edges are replaced by faces.
- Homotopy type theory
- Barendregt, Henk (1991). "Introduction to generalized type systems". Journal of Functional Programming. 1 (2): 125–154. doi:10.1017/s0956796800020025. hdl:2066/17240. ISSN 0956-7968.
- Nederpelt, Rob; Geuvers, Herman (2014). Type Theory and Formal Proof (1st ed.). United Kingdom: Cambridge University Press. p. 69. ISBN 9781107036505.
- Nederpelt, Rob; Geuvers, Herman (2014). Type Theory and Formal Proof (1st ed.). United Kingdom: Cambridge University Press. p. 85. ISBN 9781107036505.
- Nederpelt, Rob; Geuvers, Herman (2014). Type Theory and Formal Proof (1st ed.). United Kingdom: Cambridge University Press. p. 100. ISBN 9781107036505.
- Schwichtenberg, Helmut (1975). "Definierbare Funktionen imλ-Kalkül mit Typen". Archiv für Mathematische Logik und Grundlagenforschung (in German). 17 (3–4): 113–114. doi:10.1007/bf02276799. ISSN 0933-5846.
- Jean-Yves., Girard (1989). Proofs and types. Cambridge University Press. OCLC 504959682.
- Ridoux, Olivier (1998). Lambda-Prolog de A à Z ... ou presque. [s.n.] OCLC 494473554.
- C., Pierce, Benjamin (1989). Programming in higher-order typed lambda-calculi. Carnegie Mellon University, Computer Science Dept. OCLC 20442222.
- Sørensen, Morten Heine; Urzyczyin, Pawel (2006), "Pure type systems and the λ-cube", Lectures on the Curry-Howard Isomorphism, Elsevier, pp. 343–359, doi:10.1016/s0049-237x(06)80015-7, ISBN 9780444520777
- Pierce, Benjamin (2002). Types and programming languages. MIT Press. pp. 467–490. ISBN 978-0262162098. OCLC 300712077.
- Pierce, Benjamin (2002). Types and programming languages. MIT Press. p. 466. ISBN 978-0262162098. OCLC 300712077.
- Ridoux, Olivier (1998). Lambda-Prolog de A à Z ... ou presque. [s.n.] p. 70. OCLC 494473554.
- Simon Peyton Jones and Erik Meijer, 1997. Henk: A Typed Intermediate Language