Chapter 12

Coherence Semantics of the Sum

Here we consider the denotational semantics of Emp and + (corresponding to ? and _) introduced in chapter 10.

Emp is naturally interpreted as the coherence space Emp whose web is empty, and the interpretation of "U follows immediately1.

The sum, on the other hand, poses some delicate problems. When A and B are two coherence spaces, there is just one obvious notion of sum, namely the direct sum introduced below. Unfortunately, the scheme is not interpreted. This objection also holds for other kinds of semantics, for example Scott domains.

After examining and rejecting a certain number of fudged alternatives, we are led back to the original solution, which would work with linear functions (i.e. preserving unions), and we arrive at a representation of the sum type as:

!A !B

It is this decomposition which is the origin of linear logic: the operations (direct sum) and ! (linearisation) are in fact logical operations in their own right.

1The reader familiar with category theory should notice that Emp is not an initial object. This is to be expected in any reasonable category of domains, because there can be no initial object in a non-degenerate Cartesian closed category where every object is inhabited (as it will be if there are xpoints). With linear logic, the problem vanishes because we do not require a Cartesian closed category.

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