Perfect matchings and series-parallel graphs: multiplicatives proof nets as R&B-graphs

Perfect matchings and series-parallel graphs: multiplicatives proof nets as R&B-graphs Extended Abstract] Abstract A graph-theoretical look at multiplicative proof nets lead us to two new descriptions of a proof net, both as a graph endowed with a perfect matching. The rst one is a rather conventional encoding of the connectives which nevertheless allows us to unify various sequentialisation techniques as the corollaries of a single graph theoretical result. The second one is more exciting: a proof net simply consists in the set of its axioms the perfect matching plus one single series-parallel graph which encodes the whole syntactical forest of the sequent. We thus identify proof nets which only diier because of the commutativity or associativity of the connectives, or because nal par have been performed or not. We thus push further the program of proof net theory which is to get closer to the proof itself, ignoring as much as possible the syntactical "bureaucracy". 1 Presentation This paper introduces two new ways of looking at proof structures and nets, and their correctness criteria. Our basic tool for describing proof nets is edge-bicoloured graph, that we call R&B-graphs: one of the colours, B, deenes a perfect matching or 1-factor of the graph, a standard topic in graph theory: a matching B is a set of pairwise non-adjacent edges, and it is said to be perfect whenever each vertex is incident to an edge of B. An edge not in B is in R. We then consider-cycles alternate elementary cycles i.e. the even cycles with edges alternatively in B and in R, which does not use twice the same edge. We prove a theorem related to one by Kotzig 7] which characterises the R&B-graph without-cycles as an inductively deened class of R&B-graphs which recursively contain a B-isthmus. In the rst of our two approaches, the connectives are directly encoded in the R&B-graph. The criterion is the absence of-cycle. Using our theorem, c 1996 Elsevier Science B. V.