Coherent Banach Spaces : a Continuous Denotational Semantics Extended 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`1-like norms and negative connectives yield`1-like norms. The problem of non-reeexivity of Banach spaces is handled by A specifying the dual in advance B, 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 simple solution to the question of a topological cartesian closed category : our analytical maps send an open ball into a closed one and therefore do not compose. However a slight modiication of the logical system allowing to multiply a function by a scalar of modulus < 1 is enough to cope with this problem. The logical status of the new system should be clariied. We shall not discuss the general issue of topology and logic (e.g. logical approach to topology as in-say-formal topologies), but the restricted question of adding topological features to logic. .1 Topology in logic .1.1 Scott domains Logic is by nature discrete ; in many situations we would like to connect its rules with analysis, i.e. with real or complex numbers. Na ve attempts at introducing some A fuzziness B in logic eventually ended in fuzzy: : : methodology and notorious parascience. The most important attempt at reconciling continuity and logic amounts to the works of Dana Scott (and independently Ershov), around 1970, see e.g. 8]. The problem at stake was to give a concrete model of the Heyting-Kolmogoroo paradigm of A proofs as functions B, c 1996 Elsevier Science B. V.