Implementation complexity of Boolean functions with a small number of ones

We consider the class F n, k consisting of n-ary Boolean functions that take the value one on exactly k input tuples. For small values of k the class F n, k is splitted into subclasses, and for every subclass we find the asymptotics of the Shannon function of circuit implementation in the basis { x & y , x ‾ } $ \{x\&y,\overline x\} $ (or in the basis { x ∨ y , x ‾ } ) $ \{x\vee y,\overline x\}) $ ; the weights of the basic gates are arbitrary strictly positive numbers.

A Model of Secure Functioning of Computer Systems

We propose an automata model of computer system security. A system is represented by a finite automaton with states partitioned into two subsets: "secure" and "insecure". System functioning is secure if the number of consecutive insecure states is not greater than some nonnegative integer k. This definition allows one to formally reflect responsiveness to security breaches. The number of all input sequences that preserve security for the given value of k is referred to as a k-secure language. We prove that if a language is k-secure for some natural and automaton V, then it is also k-secure for any 0 < k < k and some automaton V = V (k). Reduction of the value of k is performed at the cost of amplification of the number of states. On the other hand, for any non-negative integer k there exists a k-secure language that is not k"-secure for any natural k" > k. The problem of reconstruction of a k-secure language using a conditional experiment is split into two subcases. If the cardinality of an input alphabet is bound by some constant, then the order of Shannon function of experiment complexity is the same for al k; otherwise there emerges a lower bound of the order nk.

Regional gravity field refinement for (quasi-) geoid determination based on spherical radial basis functions in Colorado

This study presents a solution of the ‘1 cm Geoid Experiment’ (Colorado Experiment) using spherical radial basis functions (SRBFs). As the only group using SRBFs among the fourteen participated institutions from all over the world, we highlight the methodology of SRBFs in this paper. Detailed explanations are given regarding the settings of the four most important factors that influence the performance of SRBFs in gravity field modeling, namely (1) the choosing bandwidth, (2) the locations of the SRBFs, (3) the type of the SRBFs as well as (4) the extensions of the data zone for reducing the edge effect. Two types of basis functions covering the same spectral range are used for the terrestrial and the airborne measurements, respectively. The non-smoothing Shannon function is applied to the terrestrial data to avoid the loss of spectral information. The cubic polynomial (CuP) function which has smoothing features is applied to the airborne data as a low-pass filter for filtering the high-frequency noise. Although the idea of combining different SRBFs for different observations was proven in theory to be possible, it is applied to real data for the first time, in this study. The RMS error of our height anomaly result along the GSVS17 benchmarks w.r.t the validation data (which is the mean results of the other contributions in the ‘Colorado Experiment’) drops by 5% when combining the Shannon function for the terrestrial data and the CuP function for the airborne data, compared to those obtained by using the Shannon function for both the two data sets. This improvement indicates the validity and benefits of using different SRBFs for different observation types. Global gravity model (GGM), topographic model, the terrestrial gravity data, as well as the airborne gravity data are combined, and the contribution of each data set to the final solution is discussed. By adding the terrestrial data to the GGM and the topographic model, the RMS error of the height anomaly result w.r.t the validation data drops from 4 to 1.8 cm, and it is further reduced to 1 cm by including the airborne data. Comparisons with the mean results of all the contributions show that our height anomaly and geoid height solutions at the GSVS17 benchmarks have an RMS error of 1.0 cm and 1.3 cm, respectively; and our height anomaly results give an RMS value of 1.6 cm in the whole study area, which are all the smallest among the participants.

Asymptotically best method for synthesis of Boolean recursive circuits

Models of multi-output and scalar recursive Boolean circuits of bounded depth in an arbitrary basis are considered. Methods for lower and upper estimates for the Shannon function for the complexity of circuits of these classes are provided. Based on these methods, an asymptotic formula for the Shannon function is put forward. Moreover, in the above classes of recursive circuits, upper estimates for the complexity of implementation of some functions and systems of functions used in applications are obtained.

The use of different spherical radial basis functions to combine terrestrial and airborne measurements for regional gravity field refinement

&lt;p&gt;The objective of this study is the combination of different types of basis functions applied separately to different kinds of gravity observations. We use two types of regional data sets: terrestrial gravity data and airborne gravity data, covering an area of about 500 km &amp;#215; 800 km in Colorado, USA. These data are available within the &amp;#8220;1 cm geoid experiment&amp;#8221; (also known as the &amp;#8220;Colorado Experiment&amp;#8221;). We apply an approach for regional gravity modeling based on series expansions in terms of spherical radial basis functions (SRBF). Two types of basis functions covering the same spectral domain are used, one for the terrestrial data and another one for the airborne measurements. To be more specific, the non-smoothing Shannon function is applied to the terrestrial data to avoid the loss of spectral information. The Cubic Polynomial (CuP) function is applied to the airborne data as a low-pass filter, and the smoothing features of this type of SRBF are used for filtering the high-frequency noise in the airborne data. In the parameter estimation procedure, these two modeling parts are combined to calculate the quasi-geoid.&lt;/p&gt;&lt;p&gt;The performance of our regional quasi-geoid model is validated by comparing the results with the mean solution of independent computations delivered by fourteen institutions from all over the world. The comparison shows that the low-pass filtering of the airborne gravity data by the CuP function improves the model accuracy by 5% compared to that using the Shannon function. This result also makes evident the advantage of combining different SRBFs covering the same spectral domain for different types of observations.&lt;/p&gt;

On the complexity of bounded-depth circuits and formulas over the basis of fan-in gates

We obtain estimates for the complexity of the implementation of n-place Boolean functions by circuits and formulas built of unbounded fan-in conjunction and disjunction gates and either negation gates or negations of variables as inputs. Restrictions on the depth of circuits and formulas are imposed. In a number of cases, the estimates obtained in the paper are shown to be asymptotically sharp. In particular, for the complexity of circuits with variables and their negations on inputs, the Shannon function is asymptotically estimated as $2\cdot {{2}^{n/2}};$this estimate is attained on depth-3 circuits.