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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
The Complexity of the Four Colour Theorem
, 2009
"... The four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a nontrivial computer verifica ..."
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The four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a nontrivial computer verification. Recently, a formal proof of the theorem was obtained with the equational logic program Coq. In this paper we use the computational method for evaluating (in a uniform way) the complexity of mathematical problems presented in [8, 6] to evaluate the complexity of the four colour theorem. Our method uses a Diophantine equational representation of the theorem. We show that the four colour theorem has roughly the same complexity as the Riemann hypothesis and almost four times the complexity of Fermat’s last theorem. 1
Evaluating the Complexity of Mathematical Problems. Part 1
, 2009
"... In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method, which is inspired by NKS1, is based on the possibility to completely describe complex mathematical problems, like the Riemann hypothesis, in terms of ..."
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In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method, which is inspired by NKS1, is based on the possibility to completely describe complex mathematical problems, like the Riemann hypothesis, in terms of (very) simple programs. The method is illustrated on a variety of examples coming from different areas of mathematics and its power and limits are studied.
Mathematical Problems. Part 1 ∗
, 2008
"... In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method is illustrated on a variety of examples coming from different areas of mathematics and its power and limits are studied. 1 ..."
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In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method is illustrated on a variety of examples coming from different areas of mathematics and its power and limits are studied. 1
Theoretical Computer Science Proving and Programming
, 2007
"... There is a strong analogy between proving theorems in mathematics and writing programs in computer science. This paper is devoted to an analysis, from the perspective of this analogy, of proof in mathematics. We will argue that while the Hilbertian notion of proof has few chances to change, future p ..."
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There is a strong analogy between proving theorems in mathematics and writing programs in computer science. This paper is devoted to an analysis, from the perspective of this analogy, of proof in mathematics. We will argue that while the Hilbertian notion of proof has few chances to change, future proofs will be of various types, will play different roles, and their truth will be checked differently. Programming gives mathematics a new form of understanding. The computer is the driving force behind these changes. 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
Modified Harmony Search Algorithm for Solving the FourColor Mapping Problem
"... The FourColor Mapping Problem has been solved using different optimization algorithms. Harmony Search (HS) is one of those algorithms, which is based on the imitation of the behavior of musicians when composing their music. The HS algorithm can be summarized in three steps... initialization, improv ..."
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The FourColor Mapping Problem has been solved using different optimization algorithms. Harmony Search (HS) is one of those algorithms, which is based on the imitation of the behavior of musicians when composing their music. The HS algorithm can be summarized in three steps... initialization, improvisation, and selection. We introduced in this paper an approach to enhance the performance of HS algorithm, in solving the FourColor Mapping Problem. A modification has been applied to the initialization section of the HS algorithm, which affects the improvisation process, resulting in a boost in the performance of the improvisation process, and consequently, reducing the time and number of cycles taken to solve the FourColor Mapping Problem compared to the HS algorithm. In this paper, tests have been carried out on maps with different numbers of regions, using both HS and Modified Harmony Search (MHS) algorithms. The obtained results of the MHS algorithm are better than those of the original HS one.