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. Homological Dehn functions over R and over Z are introduced to measure minimal fillings of integral 1-cycles by (real or integral) 2-chains in the Cayley 2complex of a finitely presented group. If the group G is the fundamental group of a finite graph of finitely presented vertex-groups Hv and finitely generated edge-groups, then there is a formula for an isoperimetric function (for genus 0 fillings) for G in terms of the real homological Dehn function for G and the (genus 0) Dehn functions for the Hv . Hierarchies of groups are introduced in which an isoperimetic function is determined by a formula in terms of the real homological Dehn function. In such a hierarchy the word problem is one of homological algebra. All 1-relator groups are in such a hierarchy. Applications are given to the generalized word problem (a.k.a. membership or Magnus problem) and to a homolgical determination of distortion. 1. Introduction. It is well-known in Riemannian geometry that under appropr...

Streams are acyclic directed subgraphs of the logical flow graph of a proof and represent bundles of paths with the same origin and the same end. Streams can be described with a natural algebraic formalism which allows to explain in algebraic terms the evolution of proofs during cutelimination. In our approach, "logic" is often forgotten and combinatorial properties of graphs are taken into account to explain logical phenomena.