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Mathematical proofs at a crossroad
 Theory Is Forever, Lectures Notes in Comput. Sci. 3113
, 2004
"... Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimen ..."
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Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimental, psychological and social aspects, yesterday only marginal, but now changing radically the very essence of proof. In this paper, we try to organize this evolution, to distinguish its different steps and aspects, and to evaluate its advantages and shortcomings. Axiomaticdeductive proofs are not a posteriori work, a luxury we can marginalize nor are computerassisted proofs bad mathematics. There is hope for integration! 1
Centre for Discrete Mathematics and Theoretical Computer ScienceSearching for Spanning kCaterpillars and kTrees
, 2008
"... We consider the problems of finding spanning kcaterpillars and ktrees in graphs. We first show that the problem of whether a graph has a spanning kcaterpillar is NPcomplete, for all k ≥ 1. Then we give a linear time algorithm for finding a spanning 1caterpillar in a graph with treewidth k. Also ..."
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We consider the problems of finding spanning kcaterpillars and ktrees in graphs. We first show that the problem of whether a graph has a spanning kcaterpillar is NPcomplete, for all k ≥ 1. Then we give a linear time algorithm for finding a spanning 1caterpillar in a graph with treewidth k. Also, as a generalized versions of the depthfirst search and the breadthfirst search algorithms, we introduce the ktree search (KTS) algorithm and we use it in a heuristic algorithm for finding a large kcaterpillar in a graph. 1