Table Of ContentLecture Notes in Computer Science 902
Edited by G. Goos, J. Hartmanis and J. van Leeuwen
Advisory Board: W. Brauer D. Gries J. Stoer
Mariangiola Dezani-Ciancaglini
Gordon Plotkin (Eds.)
Typed Lambda Calculi
dna Applications
Second International Conference
on Typed Lambda Calculi and Applications, TLCA '95
Edinburgh, United Kingdom, April 10-12, 5991
sgnideecorP
r e g n~ i r p S
Series Editors
Gerhard Goos
Universit~it Karlsruhe
Vincenz-Priessnitz-Strale 3, D-76128 Karlsruhe, Germany
Jufis Hartmanis
Department of Computer Science, Cornell University
4130 Upson Hall, Ithaca, NY 14853, USA
Jan van Leeuwen
Department of Computer Science, Utrecht University
Padualaan 14, 3584 CH Utrecht, The Netherlands
Volume Editors
Mariangiola Dezani-Ciancaglini
Department of Computer Science, University of Turin
Corso Svizzera, 185, 1-10149 Turin, Italy
Gordon Plotkin
Department of Computer Science, University of Edinburgh
The King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom
CR Subject Classification (1991): E4.1, E3.0, D.I.1
ISBN 3-540-59048-X Springer-Verlag Berlin Heidelberg New York
CIP data applied for
This work is subject to copyright. All rights are reserved, whether the whole or part
of the material is concerned, specifically the fights of translation, reprinting, re-use
of illustrations, recitation, broadcasting, reproduction on microfilms or in any other
way, and storage in data banks. Duplication of this publication or parts thereof is
permitted only under the provisions of the German Copyright Law of September 9,
1965, in its current version, and permission for use must always be obtained from
Springer-Verlag. Violations are liable for prosecution under the German Copyright
Law.
(cid:14)9 Springer-Vedag Berlin Heidelberg 1995
Printed in Germany
Typesetting: Camera-ready by author
SPIN: 10485422 45/3140-543210 - Printed on acid-free paper
Preface
This volume is the proceedings of the Second International Confer-
ence on Typed Lambda Calculi and Applications, TLCA'95, held in Ed-
inburgh, Scotland, from April 10 to April 12, 1995. There are 29 papers
concerning the following topics:
- Proof theory of type systems;
- Logic and type systems;
- Typed A-calculi as models of (higher-order) computation;
- Semantics of type systems;
- Proof verification via type systems;
- Type systems of programming languages;
- Typed term rewriting systems.
These were selected from a total of 58 submissions, of generally high qual-
ity.
We wish to express our gratitude to all the members of the Program
Committee, and to the many referees who assisted them. Moreover, we
would like to thank the members of the Organizing Committee and all
those who submitted papers. Finally, we greatly appreciate the excellent
cooperation with Springer-Verlag.
The lambda calculus has served as a source of ideas, problems and ap-
plications in computer science for over thirty years. This volume demon-
strates i~s continuing vitality.
Turin, January 1995
Mariangiola Dezani-Ciancaglini Gordon Plotkin
Program Committee
H. Barendregt (Catholic University of Nijmegen)
M. Dezani (Chairperson, University of Turin)
J-Y. Girard (University of Marseilles)
R. Hindley (University of Swansea)
F. Honsell (University of Udine)
J. W. Klop (CWI)
G. Longo (ENS)
A. Meyer (MIT)
G. Plotkin (University of Edinburgh)
P. Scott (University of Ottawa)
J. Smith (University of Gothenburg/Chalmers)
J. Tiuryn 'University of Warsaw)
Referees
F. Alessi T. Altenkirch G. Amiot
F. Barbanera E. Barendsen G. Barthe
L.S. van Benthem Jutting S. Berardi C. Berline
M. Bezem M. Boffa G. Boudol
R. Blute F. Cardone G. Castagna
J.R.B. Cockett M. Coppo T. Coquand
A. Compagnoni R.L. Crole D. Cubric
P:L. Curien W. Dekkers R. Di Cosmo
P. Di Gianantonio G. Dowek P. Dybjer
T. Ehrhard D. Fridlender H. Geuvers
P. Giannini G. Ghelli M. Grabowski
P. de Groote T. Hurkens B. Intrigila
B. Jacobs J.P. Jouannaud R. Hasegawa
M. Hofmann D. Kesner Y. Lafont
M. Lenisa U. de' Liguoro P. Lincoln
P. Malacaria S. Martini J. Mason
E. Meijer M. Miculan E. Moggi
R. Nederpelt V. van Oostrom J. Otto
C. Paulin-Mohring A. Piperno E. Poll
R. Pollack F. van Raamsdonk L. Regnier
J.G. Rieeke S. Ronchi della Rocca P. Rosolini
L. Roversi M. Ruys A. Salibra
A. Schubert R.A.G. Seedy R. Statman
P. Urzyczyn R.C. de Vrijer B. Werner
Organizing Committee
G. Cleland, P. Gardner, M. Lekuse, G. Plotkin (University of Edinburgh)
Table of Contents
Comparing ),-calculus translations in sharing graphs
A. Asperti, C. Laneve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Extensions of pure type systems
G. Barthe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
A model for formal parametric polymorphism: a PER interpre-
tation for system
R. Bellucci, M. Abadi, P.-L. Curien . . . . . . . . . . . . . . . . . . . . . 23
A realization of the negative interpretation of the Axiom of
Choice
S. Berardi, M. Bezem, T. Coquand . . . . . . . . . . . . . . . . . . . . . . 47
Using subtyping in program optimization
S. Berardi, L. Boerio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
What is a categorical model of intuitionistic linear logic?
G.M. Bierman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
An explicit Eta rewrite rule
D. Briaud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Extracting text from proofs
Y. Coscoy, G. Kahn, L. Thdry . . . . . . . . . . . . . . . . . . . . . . . . 901
Higher-order abstract syntax in Coq
J. Despeyroux, A. Felty, A. Hirschowitz ................... 421
Expanding extensional polymorphism
R. Di Cosmo, A. Piperno . . . . . . . . . . . . . . . . . . . . . . . . . . . 931
Lambda-calculus, combinators and the comprehension scheme
G. Dowek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
~/-equality for coproducts
N. Ghani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Typed operational semantics
H. Goguen ................ .................... 681
A simple calculus of exception handling
P. de Groote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A simple model for quotient types
M. Hofmann 612
1IIV
Untyped )~-calculus with relative typing
M.R. Holmes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
Final semantics for untyped )~-calculus
F. Honselt, M. Lenisa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942
A simplification of Girard's paradox
A. J. C. Hurkens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662
Basic properties of data types with inequational refinements
1t. Kondoh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972
Decidable properties of intersection type systems
T. Kurata, M. Takahashi . . . . . . . . . . . . . . . . . . . . . . . . . . . 792
Termination proof of term rewriting system with the multiset
path ordering. A complete development in the system Coq
F. Leelerc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Typed )~-calculi with explicit substitutions may not terminate
P.A. Mellies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823
On equivalence classes of interpolation equations
V. Padovani ........................... : ....... 533
Strict functionals for termination proofs
J. van de Pol, H. Schwichtenberg . . . . . . . . . . . . . . . . . . . . . . . 053
A verified typechecker
R. Pollack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
Categorical semantics of the call-by-value )~-calculus
A. Pravato, S. Ronchi della Rocca, L. Roversi . . . . . . . . . . . . . . . . 183
A fully abstract translation between a )~-calculus with reference
types and standard ML
E. R}tter, A.M. Pitts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793
Categorical completeness results for the simply-typed )~-calculus
A.K. Simpson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
Third-order matching in the presence of type constructors
J. Springiutveld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824
Author Index . . ............................... 344
Comparing A-calculus translations in
Sharing Graphs *
Andrea Asperti 1 and Cosimo Laneve ~
1 Dip. di Matematica, P.za di Porta .S Donato, ,5 40127 Bologna, Italy.
INRIA Sophia Antipolis, 2004 Route des Lucioles BP ,39 06902 Valbonne, France.
Abstract. Since Lamping's seminal work Lam90 on optimal graph re-
duction techniques for the A-calculus, several different translations based
on the same set of control operators (sharing graphs) have been proposed
in the literature GAL92a, GAL92b, AL93a, As94. In this paper we clar-
ify the correspondence between all these translations, passing through
the so called bus-notation GAL92a. Indeed all the sharing graph en-
codings turn out to be equivalent modulo the way of counting bus levels.
1 Introduction
In Lam90, Lamping proposed a complex graph reduction technique for the
A-calculus that was optimal in the sense of L6vy Le78. Lamping's approach
was revisited in GAL92a, where a restricted set of control nodes and reduction
rules was proved sufficient for the implementation. In GAL92b Gonthier-Abadi-
L6vy pointed out a strong analogy between optimal reductions and Linear Logic
Gi86. In particular, the optimal implementation of A-calculus is actually a re-
finement of Girard's proof-net representation. This refinement just provides a
local implementation of the non-linear operations over shared data (nets into
bozes), which in proof-nets are performed as a single, global step. By their en-
coding of Linear Logic in sharing graphs, and some encoding of A-calculus in
Linear Logic, a third (slightly different) translation was implicitly obtained. This
encoding has been used (and explicited) in AL93a, in the much more general
case of Interaction Systems.
At the same time of GAL92b, Asperti proposed yet another and sensibly
different translation of A-calculus into sharing graphs As94. This latter transla-
tion is strongly connected to the dynamic algebra interpreta,tion of Linear Logic
IRe92, DR93 (a relation that has been recently formalized in ADLR94).
Up to now, no correspondence has been established in the literature between
all these different translations. In this paper we prove that such differences may
be explained, up to minor syntactic details, by the way of counting the nesting of
boxes into proof-nets. Precisely, the innermost box into a proof-net may be con-
sidered as a box at "depth 0" or a box at ~depth n", if there are n outer boxes.
In order to formalize this intuition, we shall use a further sharing graph imple-
mentation of :~-calculus: the so-called bus-notation GAL92a. The bus notation
* Partially supported by the ESPRIT Basic Research Project 6454 - CONFER.
was introduced with the aim of reducing the number of interaction rules between
control nodes. This is achieved by considering edges as set of wires (buses) and
an n-in(texed control node as affecting the n-th wire only. The above conventions
for counting nestings of boxes are reflected into the two ways of counting wires:
"from the left" or "from the right". So, if we have h + 1 wires in the bus, and
a node on a wire, this node can be seen as an operator at level n (assuming rt
wires on the right) or h - rt (when counting from the left).
The two different choices underlie respectively the translations in As94 and
GAL92b, which therefore are mapped onto the same sharing graph in the bus
notation. This also explains why exactly the same set of rewriting rules works
in both cases.
We remark that this work could have been written in the framework of proof
nets Gi86. Indeed the results presented here could be transposed to proof nets
without any difficulty. We choose to stick to S-calculus for simplicity sake.
We warn the reader that some general knowledge of the literature (in partic-
ular GAL92a and As94) is a prerequisite for understanding this paper.
2 Sharing Graphs
In GAL92a it was proved that Lamping's original set of .operators (nodes)
needed for the optimal implementation of h-calculus could be reduced to only
five (indexed) control ,sedo~r described in the following picture:
)toor( (erasing) (croissant) (bracket) )naf(
Actually, only three of them (fan, croissant and bracket) are really important.
The fan node represents sharing (or duplication), while croissant and brackets
are needed for the correct interaction of fans during the reduction. In Lamping's
original intuition, brackets and croissants must surround fans, implicitly defining
their scope. Two different kinds of "brackets" are needed, since they propagate in
opposite directions. This is essentially equivalent to consider brackets and crois-
sants as delimiting the extent of the box of Linear Logic (recall that the box
is just a sort of global bracket surrounding a datum that can be duplicated or
erased) (see GAL92a, GAL92b). This relation was furtherly stressed in ,49sA
where a tight relation between croissant, bracket and the two categorical oper-
ations associated with the comonad "!" of Linear Logic was proved.
Erasing nodes are needed in the case of lambda abstractions over variables
not appearing in the body of the function; until ew do not consider garbage
collection rules, these nodes have no operational effect.
In AL93a it was shown that all these control nodes could be considered as
an abstract set of operators for implementing sharing in virtually every class of
higher order rewriting systems. This suggested the name of "sharing graphs",
that we shall adopt in the following.
The rules governing the interactions between control operators are drawn in
Figure 1. We shall also explicitly consider t~vo "proper nodes", for application
and lambda abstraction.
G |
A~
:Z
ix7 Vi
(cid:12)9 o (cid:12)9 o
$ I I I
- S' ,--L
i
j i
' I I
? oJ i
Vi
x. ef
i I i iTyX
Fig. 1. The control rules (0 _< i < j)
The abstraction and application nodes are ternary nodes, i.e. they are connected
to three edges. In particular, in the case of abstraction nodes, one edge points
to the root, one to the body of the function, and the third one to the bound
variable. Thus, bound variables are supposed to be explicitly connected to their
binders (an idea that goes back 7to Bourbaki).
As shown in GAL92a, application and lambda nodes are not actually nec-
essary, since from the operational point of view they can be assimilated to fan.
Note in particular that the interaction between an application and an abstrac-
tion amounts to connect the root of the application to the root of the body of
the function, and the bound variable to the argument of the application.
For the sake of clarity, we prefer to keep the distinction; however we shall
not explicitly draw the interaction rules between proper operators and control
nodes, that are exactly the same as for fans.
