The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $Game (D_n({bfSigma}^0_2))$ for some natural number $ngeq 1$, 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 ${bfDelta}^1_2$, 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 ${bfDelta}^1_2$.
We show that there are $Sigma_3^0$-complete languages of infinite words accepted by non-deterministic Petri nets with Buchi acceptance condition, or equivalently by Buchi blind counter automata. This shows that omega-languages accepted by non-deterministic Petri nets are topologically more complex than those accepted by deterministic Petri nets.
We prove that the determinacy of Gale-Stewart games whose winning sets are infinitary rational relations accepted by 2-tape Buchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a large cardinal assumption. Then we prove that winning strategies, when they exist, can be very complex, i.e. highly non-effective, in these games. We prove the same results for Gale-Stewart games with winning sets accepted by real-time 1-counter Buchi automata, then extending previous results obtained about these games. Then we consider the strenghs of determinacy for these games, and we prove that there is a transfinite sequence of 2-tape Buchi automata (respectively, of real-time 1-counter Buchi automata) $A_alpha$, indexed by recursive ordinals, such that the games $G(L(A_alpha))$ have strictly increasing strenghs of determinacy. Moreover there is a 2-tape Buchi automaton (respectively, a real-time 1-counter Buchi automaton) B such that the determinacy of G(L(B)) is equivalent to the (effective) analytic determinacy and thus has the maximal strength of determinacy. We show also that the determinacy of Wadge games between two players in charge of infinitary rational relations accepted by 2-tape Buchi automata is equivalent to the (effective) analytic determinacy, and thus not provable in ZFC.
We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by real-time 1-counter Buchi 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 Buchi 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 Buchi automaton A and a Buchi 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)).
We survey recent results on the topological complexity of context-free omega-languages which form the second level of the Chomsky hierarchy of languages of infinite words. In particular, we consider the Borel hierarchy and the Wadge hierarchy of non-deterministic or deterministic context-free omega-languages. We study also decision problems, the links with the notions of ambiguity and of degrees of ambiguity, and the special case of omega-powers.
An {omega}-language is a set of infinite words over a finite alphabet X. We consider the class of recursive {omega}-languages, i.e. the class of {omega}-languages accepted by Turing machines with a Buchi acceptance condition, which is also the class {Sigma}11 of (effective) analytic subsets of X{omega} for some finite alphabet X. We investigate here the notion of ambiguity for recursive {omega}-languages with regard to acceptance by Buchi Turing machines. We first present in detail essentials on the literature on {omega}-languages accepted by Turing Machines. Then we give a complete and broad view on the notion of ambiguity and unambiguity of Buchi Turing machines and of the {omega}-languages they accept. To obtain our new results, we make use of results and methods of effective descriptive set theory.
We prove that the injectively omega-tree-automatic ordinals are the ordinals smaller than $omega^{omega^omega}$. Then we show that the injectively $omega^n$-automatic ordinals, where $n>0$ is an integer, are the ordinals smaller than $omega^{omega^n}$. This strengthens a recent result of Schlicht and Stephan who considered in [Schlicht-Stephan11] the subclasses of finite word $omega^n$-automatic ordinals. As a by-product we obtain that the hierarchy of injectively $omega^n$-automatic structures, n>0, which was considered in [Finkel-Todorcevic12], is strict.
We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by real-time 1-counter Buchi 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 Buchi 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 Buchi automaton A and a Buchi 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)).
We consider $omega^n$-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length $omega^n$ for some integer $ngeq 1$. We show that all these structures are $omega$-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for $omega^2$-automatic (resp. $omega^n$-automatic for $n>2$) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for $omega^n$-automatic boolean algebras, $n > 1$, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a $Sigma_2^1$-set nor a $Pi_2^1$-set. We obtain that there exist infinitely many $omega^n$-automatic, hence also $omega$-tree-automatic, atomless boolean algebras $B_n$, $ngeq 1$, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [FT10].
We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an omega-language $L(A)$ accepted by a Buchi 1-counter automaton $A$. We prove the following surprising result: there exists a 1-counter Buchi automaton $A$ such that the cardinality of the complement $L(A)^-$ of the omega-language $L(A)$ is not determined by ZFC: (1). There is a model $V_1$ of ZFC in which $L(A)^-$ is countable. (2). There is a model $V_2$ of ZFC in which $L(A)^-$ has cardinal $2^{aleph_0}$. (3). There is a model $V_3$ of ZFC in which $L(A)^-$ has cardinal $aleph_1$ with $aleph_0<aleph_1<2^{aleph_0}$. We prove a very similar result for the complement of an infinitary rational relation accepted by a 2-tape Buchi automaton $B$. As a corollary, this proves that the Continuum Hypothesis may be not satisfied for complements of 1-counter omega-languages and for complements of infinitary rational relations accepted by 2-tape Buchi automata. We infer from the proof of the above results that basic decision problems about 1-counter omega-languages or infinitary rational relations are actually located at the third level of the analytical hierarchy. In particular, the problem to determine whether the complement of a 1-counter omega-language (respectively, infinitary rational relation) is countable is in $Sigma_3^1 setminus (Pi_2^1 cup Sigma_2^1)$. This is rather surprising if compared to the fact that it is decidable whether an infinitary rational relation is countable (respectively, uncountable).
