Bridging Curry and Church's typing style

There are two versions of type assignment in the -calculus: Church-style, in which the type of each variable is fixed, and Curry-style (also called domain free), in which it is not. As an example, in Church-style typing, x:A.x is the identity function on type A, and it has type AA but not BB for a type B different from A. In Curry-style typing, x.x is a general identity function with type CC for every type C. In this paper, we will show how to interpret in a Curry-style system every Pure Type System (PTS) in the Church-style without losing any typing information. We will also prove a kind of conservative extension result for this interpretation, a result which implies that for most consistent PTSs of the Church-style, the corresponding Curry-style system is consistent. We will then show how to interpret in a system of the Church-style (a modified PTS, stronger than a PTS) every PTS-like system in the Curry style.

[1]  H. B. Curry,et al.  Combinatory Logic, Volume I. , 1961 .

[2]  Morten Heine Sørensen,et al.  Domain-free pure type systems , 2000, J. Funct. Program..

[3]  J. Roger Hindley,et al.  Lambda-Calculus and Combinators: An Introduction , 2008 .

[4]  Fairouz Kamareddine,et al.  Refining the Barendregt Cube Using Parameters , 2001, FLOPS.

[5]  Henk Barendregt,et al.  The Lambda Calculus: Its Syntax and Semantics , 1985 .

[6]  Enno Ohlebusch,et al.  Term Rewriting Systems , 2002 .

[7]  Zhaohui Luo.  An extended calculus of constructions , 1990 .

[8]  J. Y. Girard,et al.  Interpretation fonctionelle et elimination des coupures dans l'aritmetique d'ordre superieur , 1972 .

[9]  J. Seldin.  Progress report on generalized functionality , 1979 .

[10]  John C. Reynolds,et al.  Towards a theory of type structure , 1974, Symposium on Programming.

[12]  William A. Howard,et al.  The formulae-as-types notion of construction , 1969 .

[13]  Zhaohui Luo,et al.  ECC, an extended calculus of constructions , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[14]  Richard Statman,et al.  Lambda Calculus with Types , 2013, Perspectives in logic.

[15]  Wil Dekkers,et al.  Pure type systems with more liberal rules , 2001, Journal of Symbolic Logic.

[16]  Samson Abramsky,et al.  Handbook of logic in computer science. , 1992 .

[17]  Thierry Coquand,et al.  The Calculus of Constructions , 1988, Inf. Comput..

[18]  G. Pottinger,et al.  A Tour of the Multivariate Lambda Calculus , 1990 .

[19]  Fairouz Kamareddine,et al.  A Modern Perspective on Type Theory: From Its Origins Until Today , 2004 .

[20]  Luigi Liquori,et al.  Comparing Cubes of Typed and Type Assignment Systems , 1997, Ann. Pure Appl. Log..

[21]  G. Pottinger,et al.  Interpreting Church-Style Typed -Calculus in Curry-Style Type Assignment , 1998 .

[22]  Alonzo Church,et al.  A formulation of the simple theory of types , 1940, Journal of Symbolic Logic.