Coherent Banach spaces: a continuous denotational semantics

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 category: our analytical maps send an open ball into a closed one and therefore do not compose. However, a slight modification 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 clarified.