Masahiro Hamano, Japan Society for the Promotion of Science fellow,
                 University of Ottawa.        

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Title:"$Z$-modules and Full Completeness of Multiplicative Linear Logic"

Abstract: We prove that the full completeness theorem for MLL+Mix
holds by the simple interpretation 
via formulas as objects and proofs as $Z$-invariant morphisms
in the *-autonomous category of topologized vector spaces.
We shall do this by generalizing the recent work of
R. F. Blute and P. J. Scott (APAL '96) where they used
the semantical framework of dinatural transformation
introduced by Girard-Scedrov-Scott ('92).
By omitting the use of dinatural transformation,
our semantics evidently allows the interpretation of the cut-rule,
while the original Blute-Scott's does not.
Moreover, our interpretation for proofs
is preserved automatically under the cut elimination procedure. (In this sense,
our semantics is considered as a denotational semantics.)
In our semantics proofs themselves are characterized by the concrete
algebraic notion ``Z-invariance'',
and our denotational semantics provides the full completeness.
Our semantics is naturally extended to the full completeness semantics
for $CyLL+Mix$ thanks to an elegant method of       
Blute-Scott (JSL `98) (which is their sequel of (APAL '96)). 

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Title:"Pontrjagin Duality and a Full Completeness of Multiplicative
Linear Logic Without the Mix Rule " 

Abstract: While a semantical characterization of "provability" is expressed
by the traditional notion of "completeness", a semantical
characterization of "proof" is expressed by the notion of "full-completeness".
Various versions of full completeness theorem for MLL+Mix
were proved by Abramsky-Jagadeesan(JSL'94), Loader(LICS'94),
Blute-Scott(APAL'96), etc.. However, the Mix rule destroys
the usual logical meaning of syntax since the distinction
between  truth and the falsehood collapses with the presence of the Mix
rule.
As for MLL without the Mix rule, a smaller number of
full completeness theorems (Hyland-Ong's ('94) and A.Tan's ('97))
are known.  
In this talk, by using a mathematically more concrete framework
than theirs, we prove the full completeness theorem for MLL 
WITHOUT the Mix rule. Our framework uses Barr's category of
the topological abelian groups, and an MLL-proof is interpreted as 
a dinatural transformation of the category.
We first prove the unique interpretation theorem
for a binary provable sequent by a purely category-theoretical setting 
and method. Then we prove the completeness theorem for a binary
sequent by investigating a mathematically concrete structure of the category, 
which is derived by the classical Pontrjagin duality theorem.
Combining these theorems, we achieve to prove the full completeness theorem
under the semantical framework ``associated binary space''.