Propositional Proofs via Combinatorial Geometry and the Search for Symmetry

This paper is motivated by questions of complexity and com-binatorics of proofs in the sequent calculus. We shall pay particular attention to the role that symmetry plays in these questions. We want to have combinatorial models for proofs, models that will reeect complexity phenomena as well as accommodate other kinds of mathematical structures or relationships. For interpolation in proofs such a model was given in 3], and indeed this model could adapt itself to all sorts of structures (concerning graphs, polygons, surfaces...). However this was a model for cut-free proofs, and we would like to deal with the combinato-rial structure of proofs with cuts. Proofs with cuts can have cycles, and the structure of these cycles is considered in 6], where they are analyzed through associated nitely presented groups. In some cases these groups are strongly distorted, and this can be seen as a reeection of the processes underlying the proof. The model discussed here is related to the concept of the universal covering surface in topology. We shall deal only with phenomena connected to propositional proofs, without the cycling and substitutions and recur-sion to be found with quantiiers. We shall be working in the range of exactly one level of exponential expansion, and we shall investigate the role of symmetry.