An Equivalence between the Kaminski Hierarchy and the Barua Hierarchy

In this paper, we argue several decompositions of ω-regular sets into rational G_δ sets. We measure the complexity of ω-regular sets by the number of rational G_δ sets obtained by the decompositions. Barua (1992) studied a hierarchy R_n (n=1, 2, 3,…), where R_n is a class of ω-regular sets which are decomposed into n rational G_δ sets forming a decreasing sequence. On the other hand, Kaminski (1985) defined a hierarchy B_m (m=1, 2, 3,…), where B_m is a class of ω-regular sets which are decomposed into 2m rational G_δ sets not necessarily forming a decreasing sequence. As a main result, we claim that R_=B_n in spite of the differences of defining conditions

Four Hierarchies of ω-Regular Languages

We argue several decompositions of ω-regular sets into rational G_δ sets. We measure the complexity of ω-regular sets by the number of rational G_δ sets obtained by the decompositions. Barua (1992) studied a hierarchy R_n(n=1, 2, 3,…), where R_n is a class of ω-regular sets which are decomposed into n rational G_δ sets forming a decreasing sequence. On the other hand, Kaminski (1985) defined a hierarchy B_m(m=1, 2, 3,…), where B_m is a class of ω-regular sets which are decomposed into 2m rational G_δ sets not necessarily forming a decreasing sequence. Already it is reported that B_n=R_ by Takahashi (1995). And besides we show B_n=R_, where B_n is a class of ω-regular sets whose defining condition is more lenient than that of R_. In conclusion, we state that various hierarchies are reduced to four types of hierarchies

Exotic dynamic behavior of the forced FitzHugh-Nagumo equations

AbstractSpace-clamped FitzHugh-Nagumo nerve model subjected to a stimulating electrical current of form Io + I cos γt is investigated via Poincaré map and numerical continuation. If I = 0, it is known that Hopf bifurcation occurs when Io is neither too small nor too large. Given such an Io. If γ is chosen close to the natural frequency of the Hopf bifurcated oscillation, a series of exotic phenomena varying with I are observed numerically. Let 2πλγ denote the generic period we watched. Then the scenario consists of two categories of period-adding bifurcation. The first category consists of a sequence of hysteretic, λ → λ + 2 period-adding starting with λ = 1 at I = 0+, and ending at some finite I, say I∗, as λ → ∞. The second category contains multiple levels of period-adding bifurcation. The top level consists of a sequence of λ → λ + 1, period-adding starting with λ = 2 at I = I∗. From this sequence, a hierarchy of m → m + n → n, period-adding in between are derived. Such a regular pattern is sometimes interrupted by a series of chaos. This category of bifurcation also terminates at some finite I. Harmonic resonance sets in afterwards. Lyapunov exponents, power spectra, and fractal dimensions are used to assist these observations

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 ${\bf\Delta}^1\_2$

Borel Ranks and Wadge Degrees of Context Free Omega Languages

We show that, from a topological point of view, considering the Borel and the Wadge hierarchies, 1-counter B\"uchi automata have the same accepting power than Turing machines equipped with a B\"uchi acceptance condition. In particular, for every non null recursive ordinal alpha, there exist some Sigma^0_alpha-complete and some Pi^0_alpha-complete omega context free languages accepted by 1-counter B\"uchi automata, and the supremum of the set of Borel ranks of context free omega languages is the ordinal gamma^1_2 which is strictly greater than the first non recursive ordinal. This very surprising result gives answers to questions of H. Lescow and W. Thomas [Logical Specifications of Infinite Computations, In:"A Decade of Concurrency", LNCS 803, Springer, 1994, p. 583-621]

Polishness of some topologies related to word or tree automata

We prove that the B\"uchi topology and the automatic topology are Polish. We also show that this cannot be fully extended to the case of a space of infinite labelled binary trees; in particular the B\"uchi and the Muller topologies are not Polish in this case.Comment: This paper is an extended version of a paper which appeared in the proceedings of the 26th EACSL Annual Conference on Computer Science and Logic, CSL 2017. The main addition with regard to the conference paper consists in the study of the B\"uchi topology and of the Muller topology in the case of a space of trees, which now forms Section