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A contribution to formal methods : games, logic and automata

Abstract : In the first chapter, we review some standard notations and concepts that are used all through this report. Words and word automata, transition systems, bisimulation relations, trees, MSO and the mu-calculus are presented there. Some original technicalities such as graph κ-expansions also defined and studied. Two player games are presented in the second chapter. We first review some classical definitions and results on discrete two player games from regular, to parity and safety games. An algorithm that solves parity games while computing what we call permissive strategies is presented. To some extent, this algorithm is very close to Jurdzinski's small progress measure algorithm. However, the emphasis made on permissiveness not only shed a new light on this algorithm but also gives new results and raises several new open problems. A presentation of this algorithm has already been published in a joint work with Bernet and Walukiewicz We conclude this chapter by describing several notions of game simulations. These simulations are used in the sequel to compare and prove game (or automata) equivalence in a more uniform way. A generalization of Muller and Schupp alternating tree automata, with transition specifications expressed by means of arbitrary fixed point formulas, is defined in the third chapter. Automata runs are defined in terms of strategies in model checking games. Several operations and automata transformations such as boolean compositions and general substitutions are presented and semantically characterized. This leads to relate the expressive power of alternating automata with the expressive power of the mu-calculus in an inductive way. The mu-calculus invariance properties under bisimulation are obtained as corollaries. This chapter is a detailed presentation and a generalization of techniques and concepts that have been first used during a collaboration with Walukiewicz and latter extended and partially published by the author himself. The relationship between the fixed point calculus and MSO is studied in the fourth chapter. An original semantical notion of non alternating tree automata is defined and characterized syntactically by means of - an extension of - non deterministic automata. A simulation theorem, à la Muller and Schupp, is then established to prove expressiveness equivalence be- tween alternating and non alternating automata. Further proving that non deterministic tree languages are closed under projection, this series of results shows that on trees (resp. on κ-expanded trees) MSO is as expressive as the counting mu-calculus (resp. the modal mu-calculus). On arbitrary graphs, these results provide a characterization of the expressive power of the counting bisimulation invariant (resp. bisimulation invariant) of MSO by the counting mu-calculus (resp. the modal mu-calculus). The bisimulation invariance result has been established in a collaboration with Walukiewicz. Several applications are given at the end of this chapter. The fifth chapter shows that there is even a finer correspondence between the first levels of the quantifier alternation depth hierarchy of monadic second order logic and the fixed point alternation depth of the mu-calculus. More precisely, up to the monadic Σ2 level, the bisimulation invariance correspondence still holds level by level. Classical model theoretical notions such as ultraproducts are reviewed and then used in order to achieve these finer characterizations. The case of levels above the monadic Σ2 level are considered at the end of the chapter. These results have been established in a joint work with Lenzi. The finite model case, that is quite distinct from the general case, is investigated in the sixth chapter. A Büchi like characterization theorem of MSO on finite representations of infinite (ultimately periodic) words, obtained in a joint work with Dawar, is established. Extended to arbitrary finite graphs, this result provides counter examples to properties of bisimulation invariance one could have expected to hold in the finite. Graph acceptors, which generalizes to arbitrary finite graphs the notion of finite state automata on finite words or trees, are reviewed. We study bisimulation invariant properties which are definable by graph acceptors. Various separation results within the first order closure of levels of the monadic hierarchy are also proved. These results were obtained with Marcinkowski. System modeling by means of game is proposed in the seventh chapter. For this purpose, a notion of distributed games is defined and studied. In particular, we show that many distributed synthesis problems can be en- coded and solved in this setting. Complexity issues are also addressed. This work is partially the result of a collaboration with Bernet, Mohalik and Walukiewicz. It is also the core of Bernet's PhD thesis under the author's supervision. Precise open problems, or more general research directions to be developed are presented in the last chapter.
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David Janin. A contribution to formal methods : games, logic and automata. Computer Science [cs]. Université Sciences et Technologies - Bordeaux I, 2005. ⟨tel-00659990v2⟩

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