The Determinacy of Context-Free Games

We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by real-time 1-counter B\"uchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a large cardinal assumption. We show also that the determinacy of Wadge games between two players in charge of omega-languages accepted by 1-counter B\"uchi automata is equivalent to the (effective) analytic Wadge determinacy. Using some results of set theory we prove that one can effectively construct a 1-counter B\"uchi automaton A and a B\"uchi automaton B such that: (1) There exists a model of ZFC in which Player 2 has a winning strategy in the Wadge game W(L(A), L(B)); (2) There exists a model of ZFC in which the Wadge game W(L(A), L(B)) is not determined. Moreover these are the only two possibilities, i.e. there are no models of ZFC in which Player 1 has a winning strategy in the Wadge game W(L(A), L(B)).Comment: To appear in the Proceedings of the 29 th International Symposium on Theoretical Aspects of Computer Science, STACS 201

Unveiling Dynamics and Complexity

We introduce generalized Wadge games and show that each lower cone in the Weihrauch degrees is characterized by such a game. These generalized Wadge games subsume (a variant of) the original Wadge game, the eraser and backtrack games as well as Semmes’s tree games. In particular, we propose that the lower cones in the Weihrauch degrees are the answer to Andretta’s question on which classes of functions admit game characterizations. We then discuss some applications of such generalized Wadge games.SCOPUS: cp.kinfo:eu-repo/semantics/publishe

Wadge Degrees of ω\omega-Languages of Petri Nets

We prove that $\omega$-languages of (non-deterministic) Petri nets and $\omega$-languages of (non-deterministic) Turing machines have the same topological complexity: the Borel and Wadge hierarchies of the class of $\omega$-languages of (non-deterministic) Petri nets are equal to the Borel and Wadge hierarchies of the class of $\omega$-languages of (non-deterministic) Turing machines which also form the class of effective analytic sets. In particular, for each non-null recursive ordinal $\alpha < \omega\_1^{{\rm CK}}$ there exist some ${\bf \Sigma}^0\_\alpha$-complete and some ${\bf \Pi}^0\_\alpha$-complete $\omega$-languages of Petri nets, and the supremum of the set of Borel ranks of $\omega$-languages of Petri nets is the ordinal $\gamma\_2^1$, which is strictly greater than the first non-recursive ordinal $\omega\_1^{{\rm CK}}$. We also prove that there are some ${\bf \Sigma}\_1^1$-complete, hence non-Borel, $\omega$-languages of Petri nets, and that it is consistent with ZFC that there exist some $\omega$-languages of Petri nets which are neither Borel nor ${\bf \Sigma}\_1^1$-complete. This answers the question of the topological complexity of $\omega$-languages of (non-deterministic) Petri nets which was left open in [DFR14,FS14].Comment: arXiv admin note: text overlap with arXiv:0712.1359, arXiv:0804.326

The Wadge Hierarchy of Deterministic Tree Languages

We provide a complete description of the Wadge hierarchy for deterministically recognisable sets of infinite trees. In particular we give an elementary procedure to decide if one deterministic tree language is continuously reducible to another. This extends Wagner's results on the hierarchy of omega-regular languages of words to the case of trees.Comment: 44 pages, 8 figures; extended abstract presented at ICALP 2006, Venice, Italy; full version appears in LMCS special issu

Semilinear order property and infinite games

Falta resumen y palabras claveEn este trabajo se analiza la determinación de juegos de Lipschitz y Wadge, junto con la propiedad de ordenación semilineal, estrechamente relacionada con estos juegos, en el contexto de la Aritmética de segundo orden y el programa de la Matemática inversa (Reverse Mathematics). En primer lugar, se obtienen pruebas directas, formalizables en la Aritmética de segundo orden, de la determinación de los juegos de Lipschitz y Wadge para los primeros niveles de la Jerarquía de diferencias de Hausdorff. A continuación, se determinan los axiomas de existencia suficientes para la formalización de dichas pruebas dentro de los subsistemas clásicos de la Aritmética de segundo orden (fórmula). Finalmente, en algunos casos se muestra que dichos axiomas de existencia son óptimos, probando que resultan ser equivalentes (sobre un subsistema débil adecuado, como RCA0 o ACA0) a las correspondientes formalizaciones de los principios de determinación o de ordenación semilineal. Los principales resultados obtenidos son los siguientes: Teorema A. Sobre RCA0 son equivalentes: (fórmula) (el principio de determinación para juegos de Lipschitz entre subconjuntos del espacio de Cantor que son diferencia de dos cerrados). (fórmula) (la propiedad de ordenación semilineal de la reducibilidadLipschitz entre subconjuntos del espacio de Cantor que son diferencia de dos cerrados). Teorema B. Sobre RCA0 son equivalentes: (fórmula) (el principio de determinación para juegos de Lipschitz entre subconjuntos abiertos o cerrados del espacio de Baire). Teorema C. Sobre ACA0 son equivalentes: (fórmula) (el principio de determinación para juegos de Lipschitz entre subconjuntos del espacio de Baire que son simultáneamente abiertos y cerrados). (fórmula) (la propiedad de ordenación semilineal de la reducibilidadLipschitzentre subconjuntos del espacio de Baire simultáneamente abiertos y cerrados).In this thesis we analyze the determinacy of the Lipschitz and Wadge games, as well as the tightly related semilinear ordering principle, in the setting of second order arithmetic and of the program of Reverse Mathematics. Firstly, we obtain direct proofs, formalizable in second order arithmetic, of the determinacy of Lipschitz and Wadge games for the first levels of the Hausdorff's hierarchy of differences. Then we determine the set existence axioms needed to formalize such proofs within the classical subsystems of second order arithmetic (fórmula). Finally, in some cases we show that these axioms of existence are optimal, proving that they turn out to be equivalent (over a suitable weak subsystem asRCA0 orACA0) to the corresponding formalization of the principles of determinacy or semilinear ordering. The main results are: Theorem A.The following assertions are pairwise equivalent over RCA0: (fórmula) (determinacy of Lipschitz games for subsets of the Cantor space which are differences of closed sets). (fórmula) (Lipschitz semilinear ordering for subsets of the Cantor space which are differences of closed sets). Theorem B.The following assertions are pairwise equivalent over RCA0: (fórmula) (determinacy of Lipschitz games for open or closed subsets of the Baire space). Theorem C.The following assertions are pairwise equivalent over ACA0: (fórmula) (determinacy of Lipschitz games for clopen subsets of the Baire space). (fórmula) (Lipschitz semilinear ordering for clopen subsets of the Baire space)

An Upper Bound on the Complexity of Recognizable Tree Languages

The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $\Game (D\_n({\bf\Sigma}^0\_2))$ for some natural number $n\geq 1$, where $\Game$ is the game quantifier. We first give a detailed exposition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space $2^\omega$ into the class ${\bf\Delta}^1\_2$, and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual ${\bf\Delta}^1\_2$