103,802 research outputs found

    Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages

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    The purpose of this paper is to provide efficient algorithms that decide membership for classes of several Boolean hierarchies for which efficiency (or even decidability) were previously not known. We develop new forbidden-chain characterizations for the single levels of these hierarchies and obtain the following results: - The classes of the Boolean hierarchy over level Σ1\Sigma_1 of the dot-depth hierarchy are decidable in NLNL (previously only the decidability was known). The same remains true if predicates mod dd for fixed dd are allowed. - If modular predicates for arbitrary dd are allowed, then the classes of the Boolean hierarchy over level Σ1\Sigma_1 are decidable. - For the restricted case of a two-letter alphabet, the classes of the Boolean hierarchy over level Σ2\Sigma_2 of the Straubing-Th\'erien hierarchy are decidable in NLNL. This is the first decidability result for this hierarchy. - The membership problems for all mentioned Boolean-hierarchy classes are logspace many-one hard for NLNL. - The membership problems for quasi-aperiodic languages and for dd-quasi-aperiodic languages are logspace many-one complete for PSPACEPSPACE

    Width Hierarchy for k-OBDD of Small Width

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    In this paper was explored well known model k-OBDD. There are proven width based hierarchy of classes of boolean functions which computed by k-OBDD. The proof of hierarchy is based on sufficient condition of Boolean function's non representation as k-OBDD and complexity properties of Boolean function SAF. This function is modification of known Pointer Jumping (PJ) and Indirect Storage Access (ISA) functions.Comment: 8 page

    Symmetry in Critical Random Boolean Network Dynamics

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    Using Boolean networks as prototypical examples, the role of symmetry in the dynamics of heterogeneous complex systems is explored. We show that symmetry of the dynamics, especially in critical states, is a controlling feature that can be used both to greatly simplify analysis and to characterize different types of dynamics. Symmetry in Boolean networks is found by determining the frequency at which the various Boolean output functions occur. There are classes of functions that consist of Boolean functions that behave similarly. These classes are orbits of the controlling symmetry group. We find that the symmetry that controls the critical random Boolean networks is expressed through the frequency by which output functions are utilized by nodes that remain active on dynamical attractors. This symmetry preserves canalization, a form of network robustness. We compare it to a different symmetry known to control the dynamics of an evolutionary process that allows Boolean networks to organize into a critical state. Our results demonstrate the usefulness and power of using the symmetry of the behavior of the nodes to characterize complex network dynamics, and introduce a novel approach to the analysis of heterogeneous complex systems

    Spectra of Tukey types of ultrafilters on Boolean algebras

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    Extending recent investigations on the structure of Tukey types of ultrafilters on P(ω)\mathcal{P}(\omega) to Boolean algebras in general, we classify the spectra of Tukey types of ultrafilters for several classes of Boolean algebras, including interval algebras, tree algebras, and pseudo-tree algebras.Comment: 18 page
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