Abstract

We present a denotational semantics based on Banach spaces; it is inspired from the familiar coherent semantics of linear logic, the role of coherence being played by the norm: coherence is rendered by a supremum, whereas incoherence is rendered by a sum, and cliques are rendered by vectors of norm at most 1. The basic constructs of linear (and therefore intuitionistic) logic are implemented in this framework: positive connectives yield ℓ 1-like norms and negative connectives yield ℓ ∞-like norms. The problem of non-reflexivity of Banach spaces is handled by specifying the dual in ¡ advance, whereas the exponential connectives (i.e. intuitionistic implication) are handled by means of analytical functions on the open unit ball. The fact that this ball is open (and not closed) explains the absence of a simple solution to the question of a topological cartesian closed

Citations

541	Linear Logic - Girard - 1987
202	Categories For the Working Mathematician. Graduate Texts in Mathematics - Lane, Saunders - 1971
199	Data types as lattices - Scott - 1976
142	Linear Logic: Its syntax and semantics - Girard - 1995
55	Hypercoherences: a strongly stable model of linear logic - Ehrhard - 1993
17	Linear L"auchli semantics - Blute, Scott - 1996
8	The shuffle hopf algebra and noncommutative full completeness - Blute, Scott - 1998
5	Normal functors� power series� and ��calculus - Girard - 1988
5	On denotational completeness - Girard - 1997
1	The Chu construction. preprint, ftp : triples@math.mcgill.ca /pub/barr/asymm.dvi - Barr - 1996