natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
basic constructions:
strong axioms
A pure type system is an explicitly typed lambda calculus using dependent product as the type of lambda expressions: the basic idea is that
implies
In other words a pure type system is
a system of natural deduction;
for dependent types;
and with (only) the dependent product type formation rule specified.
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 relations: triples $(s_{1}, s_{2}, s_{3})$ of sorts.
Relations $(s_{1}, s_{2}, s_{2})$ are abbreviated as $(s_{1}, s_{2})$.
These relations 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 $s_3$.
In fact all such triples appearing in the following have $s_2 = s_3$. So we can just as well regard them as binary relations $R \hookrightarrow S\times S$ (rather than ternary ones).
From the above input data we derive the following
The terms of a pure type system are the following (recursive definition):
Here $x$ is a variable and $A$, $B$ are terms. The operators $\lambda$ and $\prod$ bind the variable $x$.
The typing of terms is inductively defined by the following rules.
A typing is of the form
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) | $\Gamma \vdash c : s$ | 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) | $\frac{\Gamma \vdash A : B; \quad \Gamma \vdash B' : s; \quad B =_{\beta} B'}{\Gamma \vdash A : B'}$ |
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 relations:
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 relations. 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 |
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 relations express the usual rule for dependent product type:
a dependent product over a generic dependent type is itself some type;
but if the dependent product is over a proposition then it is in fact itself a proposition, namely the universal quantification of the given definition.
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:
symbol | actual value |
---|---|
$S =$ | $\{ \ast, \square, \triangle\}$ |
$A =$ | $\{(\ast : \square), (\square : \triangle)\}$ |
$R =$ | $\{(\ast, \ast), (\square, \ast), (\square, \square), (\triangle, \square)\}$ |
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.
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)
A quick survey is in