The $omega$-power of a finitary language L over a finite alphabet $Sigma$ is the language of infinite words over $Sigma$ defined by L $infty$ := {w 0 w 1. .. $in$ $Sigma$ $omega$ | $forall$i $in$ $omega$ w i $in$ L}. The $omega$-powers appear very na
turally in Theoretical Computer Science in the characterization of several classes of languages of infinite words accepted by various kinds of automata, like B{u}chi automata or B{u}chi pushdown automata. We survey some recent results about the links relating Descriptive Set Theory and $omega$-powers.
We prove that $omega$-regular languages accepted by Buchi or Muller automata satisfy an effective automata-theoretic version of the Baire property. Then we use this result to obtain a new effective property of rational functions over infinite words w
hich are realized by finite state Buchi transducers: for each such function $F: Sigma^omega rightarrow Gamma^omega$, one can construct a deterministic Buchi automaton $mathcal{A}$ accepting a dense ${bf Pi}^0_2$-subset of $Sigma^omega$ such that the restriction of $F$ to $L(mathcal{A})$ is continuous.
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-determinist
ic) 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].
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-determi
nistic 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 cardin
al 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)).
