Duplication of Directed Graphs and Exponential Blow Up of Proofs

We develop a combinatorial model to study the evolution of graphs underlying proofs during the process of cut elimination. Proofs are two-dimensional objects and differences in the behavior of their cut elimination can often be accounted for by differences in their twodimensional structure. Our purpose is to determine geometrical conditions on the graphs of proofs to explain the expansion of the size of proofs after cut elimination. We will be concerned with exponential expansion and we give upper and lower bounds which depend on the geometry of the graphs. The lower bound is computed passing through the notion of universal covering for directed graphs