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Non-uniform (hyper/multi)coherence spaces
Published online by Cambridge University Press: 01 November 2010
Abstract
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In (hyper)coherence semantics, proofs/terms are cliques in (hyper)graphs. Intuitively, the vertices represent the results of computations and the edge relation witnesses the ability to carry out the computation assembled into a single piece of data or a single (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, compared to the relational semantics where there is no edge relation, some vertices are missing. Recovering these vertices is essential if we are to reconstruct proofs/terms from their interpretations. It will also be useful for comparing with other semantics, such as game semantics.
Bucciarelli and Ehrhard (2001) introduced a non-uniform coherence space semantics, where no vertex is missing. By constructing the co-free exponential, we get a new version of this semantics, together with non-uniform versions of hypercoherences and multicoherences. This provides a new semantics in which 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 an anti-clique contains at most one vertex, which is a result of interaction, and they then extensionally collapse onto the corresponding uniform semantics.
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