Linearity and PCF: a semantic insight!, by Marco Gaboardi, Luca Paolini, and Mauro Piccolo.

Linearity is a multi-faceted and ubiquitous notion in the analysis and the development of programming language concepts. We study linearity in a denotational perspective by picking out programs that correspond to linear functions between coherence spaces.

We introduce a language, named SlPCF*, that increases the higher-order expressivity of a linear core of PCF by means of new operators related to exception handling and parallel evaluation. SlPCF* allows us to program all the finite elements of the model and, consequently, it entails a full abstraction result that makes the reasoning on the equivalence between programs simpler.

Denotational linearity provides also crucial information for the operational evaluation of programs. We formalize two evaluation machineries for the language. The first one is an abstract and concise operational semantics designed with the aim of explaining the new operators, and is based on an infinite-branching search of the evaluation space. The second one is more concrete and it prunes such a space, by exploiting the linear assumptions. This can also be regarded as a base for an implementation.

In this paper, the authors considered a really simple language based on coherence spaces, consisting of the flat coherence space of natural numbers and the linear function space. The nice thing about this model is that tokens of function spaces are just trees with natural numbers at the leaves, with branching determined by the parenthesization of the function type. This really shows off how concrete, simple, and easy-to-use coherence spaces are. Then they found the extension to PCF for which this model was fully abstract, as semanticists are prone to doing.

But: the additional operator and its semantics are

*really bizarre and ugly*. I don't mean this as a criticism -- in fact it is exactly why I liked their paper so much! It implies that there are fundamental facts about linear types which we don't understand. I asked Marco Gaboardi about it after his talk, and he told me that he thinks the issue is that the properties of flat domains in coherence spaces are not well understood.

Anyway, this was a great paper, in a "heightening the contradictions" sort of way.

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