Research objectives and content
There are two parallel main themes of research I want to explore in Nancy. They pertain to a purely logical understanding of computations with structured state in the framework of abstract logic programming. The two threads are intimately related each other.
The first theme is about compositionality of my LES (labelled event structures) semantics for Forum (and linear logic). How does it behaves wrt linear logic connectives? We most probably can obtain compositionality from the structures I already have studied provided we further constrain the language, and this can be done without losing completeness. Having a strong behavioural semantics for linear logic should be of great help in understanding issues pertaining to the second thread of research I propose. The second line of research I want to pursue is about studying and defining abstract logic programming in variants of linear logic, in the same way Miller did with Forum wrt linear logic. In particular I am interested in pomset logic. It is already clear that LES semantics would be perfectly adequate if we had some way to express causality (sequentiality) directly in the language, so I expect to export results I already obtained to the new setting.
Since the general case of pomset logic can be difficult, I propose, in case, to limit this study to the particular case of pomset logic restricted to series-parallel orders. Series-parallel orders should be perfectly adequate to applications.
I am also interested in the definition of abstract logic programming directly on a proof-net syntax. Achievements in this direction would be particularly important, mainly because proof-nets are a much less redundant syntax than sequents. Moreover, in this setting pomset logic could be immediately usable as a basis for abstract logic programming, without the need of a sequent presentation of it.
Training content (objective, benefit and expected impact)
I expect to bring to mature state results already obtained in linear logic, mainly I expect to present a clear and comprehensive account of proof search in linear logic and its behavioural semantics. I am also confident that major advances may be obtained wrt second research theme above, i.e. I think we will be able to prove cut-elimination at least for the fragment of pomset logic of interest for abstract logic programming. Then I think I shall be able to define abstract logic programming in this new, richer logic.
There is a good possibility to obtain these results on proof nets, as opposed to sequent systems, so having a much more innovative comprehension of the subject, and in my opinion a really enlightening one. The work I propose to carry on is important, in general, because it is aimed to fill the gap between "real" languages, i.e. Ianguages independently studied and defined in many different areas, and logic. With a logical comprehension of languages, several problems in verification, optimization, debugging, abstraction and in planning and computational linguistics would find a rich mathematical tool to be used to confront with them. In particular, representing state is a key problem. Our work should further our comprehension of state in logic.