In (hyper)coherence semantics, proofs/terms are cliques in (hyper)graphs.
Intuitively, vertices represent results of computations and the edge relation
witnesses the ability of being assembled into a same piece of data or a same
(strongly) stable function, at arrow types. In (hyper)coherence semantics, the
argument of a (strongly) stable functional is always a (strongly) stable
function. As a consequence, comparatively to the relational semantics, where
there is no edge relation, some vertices are missing. Recovering these vertices
is essential for the purpose of reconstructing proofs/terms from their
interpretations. It shall also be useful for the comparison with other
semantics, like game semantics. In [BE01], Bucciarelli and Ehrhard introduced a
so called non uniform coherence space semantics where no vertex is missing. By
constructing the co-free exponential we set a new version of this last
semantics, together with non uniform versions of hypercoherences and
multicoherences, a new semantics where an edge is a finite multiset. Thanks to
the co-free construction, these non uniform semantics are deterministic in the
sense that the intersection of a clique and of an anti-clique contains at most
one vertex, a result of interaction, and extensionally collapse onto the
corresponding uniform semantics.Comment: 32 page