Formal Proof for Gene Clustering

We present a formal proof of Uno and Yagiura’s algorithm [16], using the theorem proving assistant Coq. This is a fundamental combinatorial algorithm which operates over permutation. Direct applications include computational biology (e.g. heuristics in estimating the genealogical distance between two species), text processing (e.g. generating the synchronous context-free grammar of two sequences with many-to-many alignment links [18]), the problem has been generalized to d permutations by Heber and Stoye [10], . . . This text is part of an internship project which took place at LIP6, Universite Pierre & Marie Curie, Paris, under the joint supervision of B.M. BuiXuan and F. Peschanski. Acknowledgment I would like to thank my two supervisors Bin and Frederic. The first one support my repetitive questions about Uno and Yagiura’s algorithm, even if he was not able to concern me about a few other arithmetical questions. I would like to thank also the PhD students, in whose office I worked. They made me discover the work of a PhD student. 2013 ENS Rennes Mathias Fleury 1 L3 Internship: Formal Proof

[1]  Jeremy P. Spinrad,et al.  Linear-time modular decomposition and efficient transitive orientation of comparability graphs , 1994, SODA '94.

[2]  Ross M. McConnell,et al.  Linear-time modular decomposition of directed graphs , 2005, Discret. Appl. Math..

[3]  Jens Gustedt,et al.  Efficient and Practical Algorithms for Sequential Modular Decomposition , 2001, J. Algorithms.

[4]  Jens Gustedt,et al.  Efficient and practical modular decomposition , 1997, SODA '97.

[5]  David A. Bader,et al.  A Linear-Time Algorithm for Computing Inversion Distance between Signed Permutations with an Experimental Study , 2001, J. Comput. Biol..

[6]  Michel Habib,et al.  Revisiting T. Uno and M. Yagiura's Algorithm , 2005, ISAAC.

[7]  Takeaki Uno,et al.  Fast Algorithms to Enumerate All Common Intervals of Two Permutations , 1997, Algorithmica.

[8]  Jens Stoye,et al.  Finding All Common Intervals of k Permutations , 2001, CPM.

[9]  Petr Kolman.  Approximating Reversal Distance for Strings with Bounded Number of Duplicates , 2005, MFCS.

[10]  Daniel Gildea,et al.  Extracting Synchronous Grammar Rules From Word-Level Alignments in Linear Time , 2008, COLING.

[11]  Michel Habib,et al.  Simple, linear-time modular decomposition , 2007, ArXiv.

[12]  Michel Habib,et al.  A Simple Linear-Time Modular Decomposition Algorithm for Graphs, Using Order Extension , 2004, SWAT.

[13]  Georges Gonthier,et al.  The Four Colour Theorem: Engineering of a Formal Proof , 2008, ASCM.

[14]  Georges Gonthier,et al.  Engineering mathematics: the odd order theorem proof , 2013, POPL.