Going higher in the First-order Quantifier Alternation Hierarchy on Words

We investigate the quantifier alternation hierarchy in first-order logic on finite words. Levels in this hierarchy are defined by counting the number of quantifier alternations in formulas. We prove that one can decide membership of a regular language to the levels $\mathcal{B}\Sigma_2$ (boolean combination of formulas having only 1 alternation) and $\Sigma_3$ (formulas having only 2 alternations beginning with an existential block). Our proof works by considering a deeper problem, called separation, which, once solved for lower levels, allows us to solve membership for higher levels

On Varieties of Ordered Automata

The Eilenberg correspondence relates varieties of regular languages to pseudovarieties of finite monoids. Various modifications of this correspondence have been found with more general classes of regular languages on one hand and classes of more complex algebraic structures on the other hand. It is also possible to consider classes of automata instead of algebraic structures as a natural counterpart of classes of languages. Here we deal with the correspondence relating positive $\mathcal C$-varieties of languages to positive $\mathcal C$-varieties of ordered automata and we present various specific instances of this correspondence. These bring certain well-known results from a new perspective and also some new observations. Moreover, complexity aspects of the membership problem are discussed both in the particular examples and in a general setting

Collaboration in Social Networks

The very notion of social network implies that linked individuals interact repeatedly with each other. This allows them not only to learn successful strategies and adapt to them, but also to condition their own behavior on the behavior of others, in a strategic forward looking manner. Game theory of repeated games shows that these circumstances are conducive to the emergence of collaboration in simple games of two players. We investigate the extension of this concept to the case where players are engaged in a local contribution game and show that rationality and credibility of threats identify a class of Nash equilibria -- that we call "collaborative equilibria" -- that have a precise interpretation in terms of sub-graphs of the social network. For large network games, the number of such equilibria is exponentially large in the number of players. When incentives to defect are small, equilibria are supported by local structures whereas when incentives exceed a threshold they acquire a non-local nature, which requires a "critical mass" of more than a given fraction of the players to collaborate. Therefore, when incentives are high, an individual deviation typically causes the collapse of collaboration across the whole system. At the same time, higher incentives to defect typically support equilibria with a higher density of collaborators. The resulting picture conforms with several results in sociology and in the experimental literature on game theory, such as the prevalence of collaboration in denser groups and in the structural hubs of sparse networks

On FO2 quantifier alternation over words

We show that each level of the quantifier alternation hierarchy within FO^2[<] -- the 2-variable fragment of the first order logic of order on words -- is a variety of languages. We then use the notion of condensed rankers, a refinement of the rankers defined by Weis and Immerman, to produce a decidable hierarchy of varieties which is interwoven with the quantifier alternation hierarchy -- and conjecturally equal to it. It follows that the latter hierarchy is decidable within one unit: given a formula alpha in FO^2[<], one can effectively compute an integer m such that alpha is equivalent to a formula with at most m+1 alternating blocks of quantifiers, but not to a formula with only m-1 blocks. This is a much more precise result than what is known about the quantifier alternation hierarchy within FO[<], where no decidability result is known beyond the very first levels

Submillimeter-Wave Measurements and Analysis of the Ground and ν2 = 1 States of Water

In order to facilitate further studies of water in the interstellar medium, the envelopes of late-type stars, jets, and shocked regions, the frequencies of 17 newly measured H_2 ^(16)O transitions between 0.841 and 1.575 THz are reported. A complete update of the available water line frequencies and a detailed calculation of unmeasured rotational transitions and transition intensities as a function of temperature are presented for the ground and ν_2 = 1 state levels below 3000 cm^(-1) of excitation energy. The new terahertz transitions were measured with a recently developed laser difference frequency spectrometer. Six of these transitions arise from the ν_2 = 1 state, and the other 11 are in the ground state; all have lower state energies from 700 to 1750 cm^(-1) and should be accessible to Stratospheric Observatory For Infrared Astronomy (SOFIA) through the atmosphere. The transitions near 0.850 THz are accessible from the ground with existing receivers. Observations of the newly measured ν_2 = 1 state transitions, which include the 1_(1, 1)-0_(0, 0) fundamental at 1.2057 THz and five other very low J transitions, should provide valuable insights into role played by the ν2 = 1 state in the cooling dynamics of jets, shocks, masers, and strongly infrared-pumped regions. The line list is presented to assist in the planning of observational campaigns with the Far-Infrared Space Telescope (FIRST) and other proposed space missions with which a full suite of water observations can be carried out