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Homological Dehn Functions And The Word Problem
, 1999
"... . Homological Dehn functions over R and over Z are introduced to measure minimal fillings of integral 1cycles by (real or integral) 2chains in the Cayley 2complex of a finitely presented group. If the group G is the fundamental group of a finite graph of finitely presented vertexgroups Hv and ..."
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. Homological Dehn functions over R and over Z are introduced to measure minimal fillings of integral 1cycles by (real or integral) 2chains in the Cayley 2complex of a finitely presented group. If the group G is the fundamental group of a finite graph of finitely presented vertexgroups Hv and finitely generated edgegroups, 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 1relator 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 wellknown in Riemannian geometry that under appropr...
Refined Complexity Analysis of Cut Elimination
"... Abstract. In [1, 2] Zhang shows how the complexity of cut elimination depends primarily on the nesting of quantifiers in cut formulas. By studying the role of contractions in cut elimination we can refine that analysis and show how the complexity depends on a combination of contractions and quantifi ..."
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Abstract. In [1, 2] Zhang shows how the complexity of cut elimination depends primarily on the nesting of quantifiers in cut formulas. By studying the role of contractions in cut elimination we can refine that analysis and show how the complexity depends on a combination of contractions and quantifier nesting. With the refined analysis the upper bound on cut elimination coincides with Statman’s lower bound. Every nonelementary growth example must display a combination of nesting of quantifiers and contractions similar to Statman’s lower bound example. The upper and lower bounds on cut elimination immediately translate into bounds on Herbrand’s theorem. Finally we discuss the role of quantifier alternations and show an elementary upper bound for the ∀−∧case (resp. ∃−∨case). 1
Streams and Strings in Formal Proofs
"... 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. ..."
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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.
Pathways of deduction A. Carbone
"... Cyclic structures underlie formal mathematical reasoning, and replication and folding play a crucial role in the complexity of proofs. These two aspects of the geometry of proofs are discussed. 1 Deductions, foldings and the brain Different models of various regions of the brain have been proposed a ..."
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Cyclic structures underlie formal mathematical reasoning, and replication and folding play a crucial role in the complexity of proofs. These two aspects of the geometry of proofs are discussed. 1 Deductions, foldings and the brain Different models of various regions of the brain have been proposed and they stimulated the discussion on the way our mind works. The essential feature of most of these models is the hierarchical structure which is underlying the organization. What we “see ” is nevertheless not necessarily the basic mechanism. Recent studies in computational complexity and proof theory reveal that hierarchical organizations, even though structurally appealing, are computationally inefficient. In fact, our brain seems to be “fast ” in performing certain tasks (such as perceiving the presence of an animal in the landscape, or intuitively grasping a complicated mathematical idea) and extremely “slow ” in performing others (as the construction of a mathematical