Non-Uniform Hypercoherences

In [BE01], Bucciarelli and Ehrhard propose a general tool for building a wide class of models of linear logic where a formula is interpreted as a set (the web) together with a kind of phase valued “coherence relation”. These interpretations are nonuniform in the sense that the semantics of a proof makes no assumption about the behaviour of its possible counter-proofs, unlike e.g. in the usual stable semantics where the argument of a stable functional is always a stable function. However, until now, it was suspected that this non-uniformity necessarily induces a kind of non-determinism, namely that a “clique” and an “anti-clique” could have more than one point in common. We provide a new non-uniform semantics of linear logic where this property of determinism is preserved. This is done by constructing the co-free exponential in the “non-uniform coherence space” framework described at the end of [BE01]. We discuss the issue of sequentiality in this new model. Notations. We use the notation [ ] for multisets while the notation { } is, as usual, for sets. The pairwise union of multisets is denoted by a + sign and following this notation the generalised union is denoted by a ∑ sign. The neutral element for this operation, the empty multiset, is denoted by []. If k ∈ N, k[a] denotes the multiset ∑k 1[a]. If [ai | i ∈ I] is a multiset, its support is the set {ai | i ∈ I}. The cardinality ][ai | i ∈ I] of a multiset [ai | i ∈ I] is the cardinality ]I of the set I. If m is a multiset we denote by supp(m) its support. The disjoint sum operation on sets is defined by setting A+B = {1} ×A ∪ {0} ×B. The categorical composition is denoted by #. 1 Thanks to my supervisor, Thomas Ehrhard, for his support. 2 Email:[email protected] c ©2002 Published by Elsevier Science B. V.