Induction Schemes : From Language Separation to Graph Colorings

Abstract : In this thesis, we present results obtained in two fields: formal language theory and graph theory. In formal language theory, we consider some problems of characterization of classes of regular languages. The generic problem consists in determining whether a given regular language can be defined in a fixed formalism. The current approaches use a more general problem called separation. We present here two types of contributions: a generalization of a decidability result to the setting of infinite words, together with lower bounds for the complexity of the separation problem. In graph theory, we consider the classical problem of graph coloring, where we assign colors to vertices of a graph in such a way that two adjacent vertices receive different colors. The goal is to use the fewest colors. When the graphs are sparse, a crucial tool for this is the discharging method. It is most notably decisive in the proof of the Four-Color Theorem. This method can be seen as an unconventional construction of an inductive proof scheme, specific to the considered problem and graph class, where arguing the validity of the scheme is rarely immediate. We use variants of the discharging method to study two types of coloring problems.
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Théo Pierron. Induction Schemes : From Language Separation to Graph Colorings. Programming Languages [cs.PL]. Université de Bordeaux, 2019. English. ⟨NNT : 2019BORD0119⟩. ⟨tel-02303422⟩

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