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Register a new userCharacterization Of any Non-linear Boolean function Using A Set of Linear Operators
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Global dynamics of a non-linear Cellular Automata is, in general irregular, asymmetric and unpredictable as opposed to that of a linear CA, which is highly systematic and tractable. In the past efforts have been made to systematize non-linear CA evolutions in the light of Boolean derivatives and Jacobian Matrices. In this paper two different efforts have been made: first we try to systematize non-linear CA evolution in the light of deviant states and non-deviant states. For all the non-deviant states the nearest linear rule matrix is applicable where as for the deviant states we have a set of other matrices. Second using algebraic manipulation, an efficient algorithm is proposed by which every Non-linear Boolean function can be characterized by a sequence of binary matrices.
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One successful model of interacting biological systems is the Boolean network. The dynamics of a Boolean network, controlled with Boolean functions, is usually considered to be a Markovian (memory-less) process. However, both self organizing features of biological phenomena and their intelligent nature should raise some doubt about ignoring the history of their time evolution. Here, we extend the Boolean network Markovian approach: we involve the effect of memory on the dynamics. This can be explored by modifying Boolean functions into non-Markovian functions, for example, by investigating the usual non-Markovian threshold function, - one of the most applied Boolean functions. By applying the non-Markovian threshold function on the dynamical process of a cell cycle network, we discover a power law memory with a more robust dynamics than the Markovian dynamics.
We propose a detailed proof of the fact that the inverse of Ackermann function is computable in linear time.
We show a partial Boolean function $f$ together with an input $xin f^{-1}left(*right)$ such that both $C_{bar{0}}left(f,xright)$ and $C_{bar{1}}left(f,xright)$ are at least $Cleft(fright)^{2-oleft(1right)}$. Due to recent results by Ben-David, G{o}{o}s, Jain, and Kothari, this result implies several other separations in query and communication complexity. For example, it gives a function $f$ with $C(f)=Omega(deg^{2-oleft(1right)}(f))$ where $C$ and $deg$ denote certificate complexity and polynomial degree of $f$. (This is the first improvement over a separation between $C(f)$ and $deg(f)$ by Kushilevitz and Nisan in 1995.) Other implications of this result are an improved separation between sensitivity and polynomial degree, a near-optimal lower bound on conondeterministic communication complexity for Clique vs. Independent Set problem and a near-optimal lower bound on complexity of Alon--Saks--Seymour problem in graph theory.
Open questions with respect to the computational complexity of linear CNF formulas in connection with regularity and uniformity are addressed. In particular it is proven that any l-regular monotone CNF formula is XSAT-unsatisfiable if its number of clauses m is not a multiple of l. For exact linear formulas one finds surprisingly that l-regularity implies k-uniformity, with m = 1 + k(l-1)) and allowed k-values obey k(k-1) = 0 (mod l). Then the computational complexity of the class of monotone exact linear and l-regular CNF formulas with respect to XSAT can be determined: XSAT-satisfiability is either trivial, if m is not a multiple of l, or it can be decided in sub-exponential time, namely O(exp(n^^1/2)). Sub-exponential time behaviour for the wider class of regular and uniform linear CNF formulas can be shown for certain subclasses.
In this paper we show that checking satisfiability of a set of non-linear Horn clauses (also called a non-linear Horn clause program) can be achieved using a solver for linear Horn clauses. We achieve this by interleaving a program transformation with a satisfiability checker for linear Horn clauses (also called a solver for linear Horn clauses). The program transformation is based on the notion of tree dimension, which we apply to a set of non-linear clauses, yielding a set whose derivation trees have bounded dimension. Such a set of clauses can be linearised. The main algorithm then proceeds by applying the linearisation transformation and solver for linear Horn clauses to a sequence of sets of clauses with successively increasing dimension bound. The approach is then further developed by using a solution of clauses of lower dimension to (partially) linearise clauses of higher dimension. We constructed a prototype implementation of this approach and performed some experiments on a set of verification problems, which shows some promise.
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