Table Of ContentLECTURES ON
LINEAR LOGIC
CSLI
Lecture Notes
No.29
LECTURES ON
LINEAR LOGIC
A. S. Troelstra
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Library of Congress Cataloging-in-Publication Data
Troelstra, A. S. (Anne Sjerp)
Lectures on linear logic / A. S. Troelstra.
p. em. -- (CSLI lecture notes; no. 29)
Includes bibliographical references and index.
ISBN 0-937073-78-4 - ISBN 0-937073-77-6 (pbk.)
1. Logic, Symbolic and mathematical. I. Title. II. Title: Linear logic.
III. Series.
QA9.T76 1991
511.3--dc20 91-38902
CIP
Preface
The present lecture notes originated in a course on linear logic at the
University of Amsterdam, spring semester 1991. A condensed pre
sentation was given at the University of Bern, Switzerland in June
1991. As an appendix to the body of the text, D. Roorda contributed
the last chapter. The notes aim at giving an all-around introduction.
Some very recent work had to be left out, sometimes with regret, e.g
bounded linear logic, most of the work on decidability problems, Gi
rard's work on the "geometry of interaction," etc. Certain syntactical
aspects on the other hand have gotten a bit of extra attention, since
they can play an important role if one wants to familiarize oneself
with linear logic. Of the first seven chapters dealing with syntax,
Chapters 4, 5, 7 are not needed in what follows. If these notes are
well received, and the subject continues to flourish, then perhaps I
will some day find the courage to prepare an updated and expanded
version.
Notwithstanding the fact that already many papers have been us
ing J.-Y. Girard's notation for the operators of linear logic, I have
ventured, with some trepidation, to introduce some changes in the
notation, so as to obtain a better harmony with the notations of lat
tice theory and the relevant parts of category theory. Most of my
changes will not lead to confusion, since I have used different symbols
instead of giving a new meaning to symbols already in use for linear
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vi LECTURES ON LINEAR LOGIC
logic. There is one exception: I have interchanged the role of Girard's
0 and 1..
Prerequisites. These lectures are intended for readers with some so
phistication in logic, such as graduate students in logic or theoretical
computer science, or an experienced researcher new to linear logic.
Some familiarity with classical and possibly intuitionistic predicate
logic is necessary, and it is certainly helpful to have seen Gentzen
style sequent calculi such as LK, LJ and cut elimination for these
systems {see, e.g., the treatment of sequent calculi in Kleene 1952}.
As for category theory, we shall assume familiarity with the material
of the following chapters and sections of MacLane 1971: I, II 1-4;
III 1-5; IV 1-2, 4, 6; V 1-5; VI 1; and optional is VII 1-3, 7 since
the relevant notions are explained in the notes. Good preliminary
reading is Girard, Lafont & Taylor 1988.
Acknowledgments. I am indebted to H. Geuvers, R. Hindley, B. Ja
cobs, J. van Oosten, D. Roorda, R. Stark, and especially H. Schellinx
for discussions, criticisms, providing proofs, and help in finding the
relevant literature.
Author's address: Faculteit Wiskunde en Informatica
Universiteit van Amsterdam
Plantage Muidergracht 24
1018TV AMSTERDAM (NL)
e-mail: [email protected]
Contents
1 Introduction 1
2 Sequent calculus for linear logic 15
3 Some elementary syntactic results 26
4 The calculus of two implications: a digression 41
5 Embeddings and approximations 45
6 Natural deduction systems for linear logic 56
7 Hilbert-type systems 65
8 Algebraic semantics 71
9 Combinatorial linear logic 81
10 Girard domains 92
11 Coherence in symmetric monoidal
categories 102
12 The storage operator as a cofree comonoid 111
13 Evaluation in typed calculi 119
14 Computation by lazy evaluation in CCC's 125
15 Computation by lazy evaluation in SMC's
and ILC's 134
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viii LECTURES ON LINEAR LOGIC
16 The categorical and linear machine 139
17 Proofnets for the multiplicative fragment 144
18 The algorithm of cut elimination for
proof nets 155
19 Multiplicative operators 162
20 The undecidability of linear logic 172
21 Cut elimination and strong normalization 179
References 191
Index 197
List of Tables
1 The sequent calculus for classical linear logic CLL 18
2 The sequent calculus for classical logic CL 19
3 The one-sided calculus for CLL 34
4 The Grishin embedding 48
5 Definition of embedding of IL into ILL and CLL 51
6 The Kolmogorov-style translation 54
7 Natural deduction systems for linear logic 57
8 Term assignment for N-ILLe 63
9 Hilbert type systems for linear logic 67
10 Equations for intuitionistic linear categories 84
11 Equations for CCC's 88
12 Equations for the exponentials in an intuitionistic
linear category 90
13 Interpretation of CLL in Girard domains 101
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