nLab pure type system

Idea

In type theory, under a pure type system one understand and explicitly typed lambda calculus using dependent product as the type of lambda expressions: the basic idea is that

$$\Gamma, x:A \vdash b:B$$

implies

$$\Gamma \vdash (\lambda x:A . b) : (\prod x:A . B).$$

In other words a pure type system is

1. a system of natural deduction;

2. for dependent types;

3. and with (only) the dependent product type formation rule specified.

Definition

A pure type system is defined by

• a set $S$ of sorts, all of which are constants,

• a set $A$ of axioms of the form $c : s$ with $c$ a constant and $s$ a sort,

• a set $R \hookrightarrow S \times S \times S$ of rules: triples $(s_{1}, s_{2}, s_{3})$ of sorts.

  Rules $(s_{1}, s_{2}, s_{2})$ are abbreviated as $(s_{1}, s_{2})$.

Remark

These rules will appear in the type formation rule for dependent product types below. They will say that for a type of sort $s_2$ depending on a type of sort $s_1$ its dependent product is a type of sort $s_3$.

Remark

In fact all rules $(s_{1}, s_{2}, s_{2})$ appearing in the following have $s_2 = s_3$. So we can just as well regard $R$ as a binary relation $R \hookrightarrow S\times S$ (rather than a ternary one).

From the above input data we derive the following

1. The terms of a pure type system are the following (recursive definition):

   • variables and constants
   • $(\lambda x : A . B)$ (abstraction)
   • $(\prod x : A . B)$ (product)
   • $A B$ (application)

   Here $x$ is a variable and $A$, $B$ are terms. The operators $\lambda$ and $\prod$ bind the variable $x$.

2. The typing of terms is inductively defined by the following rules.

   A typing is of the form

   $$x_{1} : A_{1}, \dots, x_{n} : A_{n} \vdash A : B$$

   meaning that the types of the variables declared on the left induces the term $A$ has type $B$. Note that types are also terms.

   The order of variable declarations is significant: the declaration $x_{i} : A_{i}$ may depend on declarations to its left.

The natural deduction rules are defined to be the following, for all $s \in S$ and where $x$ ranges over the set of variables.

name	rule	condition
(axioms)	$\vdash c : s$	if $(c : s)$ is an axiom;
(start)	$\frac{\Gamma \vdash A:s}{\Gamma, x : A \vdash x : A}$	if $x \notin \Gamma$
(weakening)	$\frac{\Gamma \vdash A:B; \quad \Gamma \vdash C:s}{\Gamma, x : C \vdash A : B}$	if $x \notin \Gamma$
(product)	$\frac{\Gamma \vdash A : s_{1}; \quad \Gamma, x:A \vdash B : s_{2}}{\Gamma \vdash (\prod x:A . B) : s_{3}}$	if $(s_{1}, s_{2}, s_{3}) \in R$;
(application)	$\frac{\Gamma \vdash F : (\prod x:A . B); \quad \Gamma \vdash a : A}{\Gamma \vdash Fa : B [x := a]}$
(abstraction)	$\frac{\Gamma, x:A \vdash b : B; \quad \Gamma \vdash (\prod x:A . B) : s}{\Gamma \vdash (\lambda x:A.b) : (\prod x:A.B)}$
(conversion)	$\frac{\Gamma \vdash A : B; \quad \Gamma \vdash B' : s; \quad B =_{\beta} B'}{\Gamma \vdash A : B'}$

Examples

Strongly normalizing systems

Intuitionistic (higher-order) logic

Lambda cube

The lambda cube (Barendregt 91) consists of eight systems arranged in a cube. The most expressive is given by the following choice of sorts, axioms and rules:

symbol	actual value
$S =$	$\{\ast, \square\}$
$A =$	$\{(\ast : \square)\}$
$R =$	$\{(\ast, \ast), (\ast, \square), (\square, \ast), (\square, \square)\}$

(Here $\{\ast, \Box\}$ denotes the 2-element set, see Barendregt 91, 2.1)

The other systems omit some of the last three rules. Some specific systems are the following:

name	$(\ast, \ast)$	$(\ast, \square)$	$(\square, \ast)$	$(\square, \square)$
$\lambda\rightarrow$ simply typed lambda calculus	yes
${\lambda}P$ logical framework	yes	yes
$\lambda2$ System F	yes		yes
$\lambda\underline{\omega}$	yes			yes
${\lambda}C$ calculus of constructions	yes	yes	yes	yes

Calculus of constructions

For instance for the calculus of constructions we have

• $* =$ Prop, the type of propositions

• $\Box =$ Type, the type of types.

The single axiom $* \colon \Box$ hence says that $Prop \colon Type$, hence that Prop is a type.

The rules express the usual rule for dependent product type:

• a dependent product over a generic dependent type is itself some type;

• but the dependent product of a dependent proposition is itself a proposition, namely the universal quantification.

Inconsistent systems

The most permissive pure type system:

symbol	actual value
$S =$	$\{\ast\}$
$A =$	$\{(\ast : \ast)\}$
$R =$	$\{(\ast, \ast)\}$

But there is an example with even non-circular system of axioms (System $\mathsf{U}^-$):

symbol	actual value
$S =$	$\{ \ast, \square, \triangle\}$
$A =$	$\{(\ast : \square), (\square : \triangle)\}$
$R =$	$\{(\ast, \ast), (\square, \ast), (\square, \square), (\triangle, \square)\}$

Related concepts

• simple type theory

• Pure type systems can be augmented with a cumulativity relation between sorts, so that if $s_1 \preceq s_2$, then any type in $s_1$ is also in $s_2$; see Barras-Gregoire.

• The calculus of inductive constructions can be formulated as a particular pure type system (with a hierarchy of type of types) augmented by rules for introducing inductive types.

References

• Henk Barendregt, Introduction to generalized type systems, J. Funct. Program. 1(2), 1991 (pdf)

• Henk Barendregt (Catholic University Nijmegen), Lambda calculi with types?, To appear in Samson Abramsky, D.M. Gabbay and T.S.E. Maibaum (eds.) Handbook of Logic in Computer Science, Volume II, Oxford University Press. (preprint pdf)

Survey includes

• Frade, Calculus of inductive constructions (pdf)

• Cody Roux, Pure type systems: Dependents when you need them, talk at Boston Haskell Meetup 2015 (slides,video)

A generalization to cumulativity can be found in

• Bruno Barras and Benjamin Gregoire, On the role of type decorations in the Calculus of Inductive Constructions, Lecture Notes in Computer Science Volume 3634, 2005, pp 151-166, PDF