Family Of Hierarchies Research Articles

Separation for dot-depth two

The dot-depth hierarchy of Brzozowski and Cohen classifies the star-free languages of finite words. By a theorem of McNaughton and Papert, these are also the first-order definable languages. The dot-depth rose to prominence following the work of Thomas, who proved an exact correspondence with the quantifier alternation hierarchy of first-order logic: each level in the dot-depth hierarchy consists of all languages that can be defined with a prescribed number of quantifier blocks. One of the most famous open problems in automata theory is to settle whether the membership problem is decidable for each level: is it possible to decide whether an input regular language belongs to this level? Despite a significant research effort, membership by itself has only been solved for low levels. A recent breakthrough was achieved by replacing membership with a more general problem: separation. Given two input languages, one has to decide whether there exists a third language in the investigated level containing the first language and disjoint from the second. The motivation is that: (1) while more difficult, separation is more rewarding (2) it provides a more convenient framework (3) all recent membership algorithms are reductions to separation for lower levels. We present a separation algorithm for dot-depth two. While this is our most prominent application, our result is more general. We consider a family of hierarchies that includes the dot-depth: concatenation hierarchies. They are built via a generic construction process. One first chooses an initial class, the basis, which is the lowest level in the hierarchy. Further levels are built by applying generic operations. Our main theorem states that for any concatenation hierarchy whose basis is finite, separation is decidable for level one. In the special case of the dot-depth, this can be lifted to level two using previously known results.

ARCHITECTURE TRADITIONAL MAKASSAR WITH THE IDEAL FORM BASED TO RESIDENTIAL ENVIRONMENT HUMANISTIC (CASE: SETTLEMENT PROCESSING BRICKS GOWA IN SOUTH SULAWESI)

The traditional architecture is the cultural identity of a tribe because they have life of its people. Traditional architecture experiencing growth and renewal. Renewal of the traditional architecture of Makassar happens usually get influences from the outside architecture that fit into the traditional architecture thus experiencing blend elements of culture, but the culture of the traditional architecture of original tribes of Makassar maintain, this is called acculturation. Acculturation is a blend of one cultural and the other retaining local indigenous culture. The traditional architecture is experiencing a mix of cultures occurs in public housing environment is an element of the macro, mezo and micro. Incessant development in various sectors particularly the sectors of housing and settlement, this is what happen in the settlement processing brick Gowa district, where the development of the construction of the society processor brick has provoked aspects psychology, than it is that it raises the problem is how the role of ideal form of architecture Makassar who have undergone traditional blend of cultures can form needs humanist residential environment that is safe, convenient, and economic, hygienic. The purpose of this study was to analyze element the of original culture at traditional Makassar particularly in ideal form home to work place primary and support in shaping the residential environment humanist. This research is a descriptive exploratory and interpretative studied theoretically by means observe directly the ideal form of settlement processing bricks in Gowa of South Sulawesi. The results of this study indicate the ideal form of settlement processing in the hamlet brick- Giring-Giring Village Kalase'rena Bontonompo Gowa District of review the system orientation of the house, the tread pattern of at macro space occupancy humanist element appear in the system path, distance, direction of the wind , the working group, the increase in the economic, social and hierarchy family indicates comfort, security, economic to the location of key livelihood source and support to the community house brick processing.

Quasi-Linear Equations with a Small Diffusion Term and the Evolution of Hierarchies of Cycles

We study the long-time behavior (at times of order \(\exp (\lambda /\varepsilon ^2\))) of solutions to quasi-linear parabolic equations with a small parameter \(\varepsilon ^2\) at the diffusion term. The solution to a PDE can be expressed in terms of diffusion processes, whose coefficients, in turn, depend on the unknown solution. The notion of a hierarchy of cycles for diffusion processes was introduced by Freidlin and Wentzell and applied to the study of the corresponding linear equations. In the quasi-linear case, it is not a single hierarchy that corresponds to an equation, but rather a family of hierarchies that depend on the timescale \(\lambda \). We describe the evolution of the hierarchies with respect to \(\lambda \) in order to gain information on the limiting behavior of the solution of the PDE.