14,366 research outputs found
On reliable computation by noisy random Boolean formulas
We study noisy computation in randomly generated k-ary Boolean formulas. We
establish bounds on the noise level above which the results of computation by
random formulas are not reliable. This bound is saturated by formulas
constructed from a single majority-like gates. We show that these gates can be
used to compute any Boolean function reliably below the noise bound.Comment: A new version with improved presentation accepted for publication in
IEEE TRANSACTIONS ON INFORMATION THEOR
The relation between tree size complexity and probability for Boolean functions generated by uniform random trees
We consider a probability distribution on the set of Boolean functions in n
variables which is induced by random Boolean expressions. Such an expression is
a random rooted plane tree where the internal vertices are labelled with
connectives And and OR and the leaves are labelled with variables or negated
variables. We study limiting distribution when the tree size tends to infinity
and derive a relation between the tree size complexity and the probability of a
function. This is done by first expressing trees representing a particular
function as expansions of minimal trees representing this function and then
computing the probabilities by means of combinatorial counting arguments
relying on generating functions and singularity analysis
Noisy random Boolean formulae:a statistical physics perspective
Properties of computing Boolean circuits composed of noisy logical gates are studied using the statistical physics methodology. A formula-growth model that gives rise to random Boolean functions is mapped onto a spin system, which facilitates the study of their typical behavior in the presence of noise. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding macroscopic phase transitions. The framework is employed for deriving results on error-rates at various function-depths and function sensitivity, and their dependence on the gate-type and noise model used. These are difficult to obtain via the traditional methods used in this field
Finite size corrections to random Boolean networks
Since their introduction, Boolean networks have been traditionally studied in
view of their rich dynamical behavior under different update protocols and for
their qualitative analogy with cell regulatory networks. More recently, tools
borrowed from statistical physics of disordered systems and from computer
science have provided a more complete characterization of their equilibrium
behavior. However, the largest part of the results have been obtained in the
thermodynamic limit, which is often far from being reached when dealing with
realistic instances of the problem. The numerical analysis presented here aims
at comparing - for a specific family of models - the outcomes given by the
heuristic belief propagation algorithm with those given by exhaustive
enumeration. In the second part of the paper some analytical considerations on
the validity of the annealed approximation are discussed.Comment: Minor correction
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