After a brief historical overview over linear logic and its applications,
we discussed a presentation of the sequent calculus for intuitionistic logic
(Gentzen's LJ) and some of its properties.
We discussed a version of the sequent calculus for intuitionistic logic
without explicit structural rules (Kleene's G3), gave properties, and showed
it to be equivalent with LJ. We also presented classical logic and
motivated the differences with intuitionistic logic.
Lecture (1/15 - B):
Linear Sequent Calculi (Iliano Cervesato)
We progressively introduce the connectives of linear logic and discuss some
of their properties. We give two formalization of both the classical and
the intuitionistic variants of this logic: first we consider as a
"traditional" two sided sequent, and then a less commonly seen but more
malleable multizone presentation.
Frank Pfenning, Linear
Logic, unpublished course notes
I will describe work at Xerox PARC on applying linear logic to
problems of natural-language syntax and semantics. Basically, a
fragment of (multiplicative) linear logic is used to "glue together"
semantic representations from the syntax, and to ensure that all
semantic contributions (resources) are accounted for.
Mary Dalrymple, John Lamping, Fernando Pereira, and Vijay Saraswat,
Linear
Logic for Meaning Assembly, in the Proceedings of CLNLP,
Edinburgh, UK,
1995.
Mary Dalrymple, John Lamping, Fernando Pereira, and Vijay Saraswat,
Quantifiers,
anaphora, and intensionality,
Journal of Logic, Language and Information,
6(3):219-273, 1997.
I will motivate the idea of performing computation as search for a proof. I
will start with traditional intuitionistic logic and introduce the notion
of uniform provability as the definition of abstract logic programming. I
will also introduce and motivate resolution.
Dov M. Gabbay, C. J. Hogger, and J. A. Robinson,
Logic Programming, Handbook of Logic in Artificial Intelligence and
Logic Programming,
volume 5:1-67, 1998
In this second half of the lecture, I will apply the above concepts to
intuitionistic linear logic and explain the fundamentals of the linear
logic programming language Lolli.
I will complete Clark's presentation of (intuitionistic) linear logic
programming by discussing the new form of non-determinisms that rises
out of linearity: context management.
Iliano Cervesato, Joshua S. Hodas and Frank Pfenning,
Efficient Resource
Management for Linear Logic Proof Search,
in the Proceedings of the International Workshop on Extensions of Logic
Programming - ELP'96 (R. Dyckhoff, H. Herre, P. Schröder-Heister
editors), pp. 67-81, Springer-Verlag LNAI 1050, Leipzig, Germany, 1996.
I will present an alternate formulation of the meaning of the operators of
intuitionistic logic as a natural deduction system. I will describe how
natural deduction applies to intuitionistic linear logic. Time
permitting, I will discuss the relation between logic and type systems
known as the Curry-Howard isomorphism.
Frank Pfenning, Linear
Logic, unpublished course notes
I will be presenting two papers of Patrick Lincoln's on decidability and
complexity results for various fragments of linear logic. I expect the
meat of the presentation to be the proof that constant-only MLL is
NP-complete.
Patrick Lincoln,
Deciding provability of linear logic formulas.
In J.-Y. Girard, Y. Lafont, and L. Regnier, editors, Advances in
Linear Logic, pages 109-122. Cambridge University Press,
1995. Proceedings of the Workshop on Linear Logic, Ithaca, New York,
June 1993.
Boolean logic can be axiomatized equally well in the Hilbert style or with
Gentzen sequents. The same is true of linear logic. I'll give a Hilbert
style axiomatization of multiplicative linear logic with modus ponens, and
then replace modus ponens by linear distributivity, the equivalence of
which is proved in essence by cut elimination. Time permitting, I will
then treat the Danos-Regnier criterion for cut-free proofhood.
I will recall basic notions about typed lambda-calculi and discuss the
Curry-Howard isomorphism, that relates proofs in a logic to typing
derivations in a lamba-calculus. We will apply these notions to
intuitionistic logic and, time permitting, to its linear refinement.
H. B. Curry and R. Fey, Combinatory Logic. North-Holland,
1958, Chapter 9, Section E.
W. A. Howard, The Formulae-as-Types Notion of Construction. In To
H. B. Curry: Essays on Combatory Logic, Lambda Calculus and
Formalism (J. P. Seldin and J. R. Hindley Editors), Academic Press
1980, pp. 479-490.
I will present connections between Linear Logic and PI Calculus, including
an introduction to Pi-Calculus and a discussion of formalizing it in
Linear Logic.
Robin Milner, Communication and Concurrency. Prentice
hall, 1995.
Functional programming is elegant but not very efficient. Common
issues include dealing with impure functions, garbage collection,
and allowing different evaluation mechanisms to co-exist
(strict/lazy). These can all be handled in a unified setting, by
means of the Linear Abstract Machine, which is based on a
categorical model of Linear Logic.
Yves Lafont, The linear abstract machine. Theoretical Computer
Science, 59:157-180, 1988. Some corrections in volume 62 (1988),
pp. 327-328.
Explicit substitutions provide a theoretical framework within which
the implementation of functional programming languages can be
studied. Linear functional programming, despite its earlier promise,
has been difficult to implement efficiently. In this talk I will
describe how to combine the technologies of explicit substitutions
and linearity in a mathematically coherent way. I start by
describing these technologies and explaining the problems which
arise when we try to mix them. Then I discuss the xSLAM project and
its proposed solutions.
(This is joint work with Eike Ritter and Neil Ghani)
Neil Ghani, Valeria de Paiva, and Eike Ritter, Linear Explicit Substitutions, Technical report CSR-98-02, University of Birmingham, UK.
I will describe an extension of the Curry-Howard isomorphism to
intuitionistic linear logic. The background is any presentation of CH
isomorphism for usual intutionistic logic, for example in the book by
Girard, Lafont and Taylor.
Jean-Yves Girard, Yves Lafont, and P. Taylor. Proofs and
Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge
University Press, 1988.
